32

M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

  • Upload
    others

  • View
    0

  • Download
    0

Embed Size (px)

Citation preview

Page 1: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

����������� ���������������������������� ���������

�� �!#"�$�!#%�& ')()*,+.-,*,/102"

3547698�:1;�;�<>=�?A@CBD6FE>G2HJILK 6MHONP6#QRSG#T 6MUWVX6MHZYAY�[]\M^A^2_

Page 2: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

`�a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~��Chf��jPa2eS�5b �]g q#nk��c�h��kjPv�qMdeF~

���M�o�A�k���M� ���2�o�M�M���M�]�7���A�#�����F�y� �,�¡ k�A� �#�M¢z���£� � ��¤#�k�z¥]�k�M�

¦�§7¨ª© ��«­¬9®��A�2�¯� �k�y°L�±��M�M�¡���£�  k�J²¦.³ �2� ���M¢m�J� �z´¡�M��µ,�J� � ²X¶]�M�·�·¢m�£�)�M�M�]�M� ¨ ���z�A¢s�¡�±��  k�k¸¦.³ �2� ���M¢m�J� �z´¡�M��µ,�J� � ²X¶]�M�·�·¢m�£�)�M�M�]�M�¹°L�±�£�F�M�¡�±��  k�

���M�}�#¢P�Oº»�k k��� �L�J¼2�A�#�A¢m�»�k�½¤�´¾��M�À¿Á�k�����M¢m M��Â7� ¼A�F M�¯���M� © �A¢m¥��ôy��A¢m� ¼MÄ �W�£�F�}�»�k M�2�M� � �1Ä#´�#¢P�#Ä ¢w�1��Å]ƹÇ1È�ÉkÊs˯ÌkÍFÎXÏÁÐ#ÈoÑ7ÇiÒwÍ1ÒwËÓÐ#Ô#ÕlËÓÔSÖ]Ñ2ÑÁÎÓ˯Ék×�Ø9Í2ÒPÙAÉkÈ�Í2ÒwËÓÌOÕ5ÚÛ#Û�Ü1Ý#Þ#ß#àMáAâ#Ü#ãk¸

¬M�A¢�¼A�#�i�±�»�k�W�¯���z�A¢s�¡�±�� ���)���L��M��¶�� äå�#�± MæL�#¢P�Oº»�k k�²X�¯�F M�¯¼F�Á�Ó�MÄ� M¼2¢»¢m�M�M�,�Ó�  k�M�F�J�¯�FÄ9 k� �M��� �� ���F�J²���k�¡�£�F�o¥]�M¤9�·�±Ä#�o�±�

çièAè �êé£ë2ëMìAì2ì�íïî�ðAñ·îlíð �Ãò èÁó ��íPò�î ë#ôÁ���A�M��� �M��ë ¸

õ¡ö7÷7ø�ùMúªû>ü�ýÿþ� ��������� ������� ö� ö���������� ù���� �! #"�"�$

% �M¢»��� ���O� � ��� ��Ä ¢z�#�M�s�k���s�y�¡�1æO�o�1�F���Á� ���¢»�Ó¤A¼F�»�Z�±�M¢s¤#���£�Ó�  k� �2� �k���M����A� �5¢P�M�#�2¢P���#¢m�1�#� �A�k�¡�£�F� k� ��´1¢s� Ä �F�l�M�#��  k�¡�#�M�L�£�2� �Z�#�M¢»��� ���J� � �L�M�#�£�  k��� �5�#¢P�k���M¢m���k��� � �1�¯�# k� �2� �k�k¸

Page 3: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

&'� � ý�ù�( þ ý

)+*-,�./,�0213./4�56,�./78,95;:<5=:8>?:<@A4B,�.B>�:<46CD7E,�CF>84G./13H�I-J94613./K?./,�0�13.L4�13>M4N*-,[email protected],�.B:<4BCA78,5B18@A78,�.L5 CT> �­�±�2�1��� �M� I-56CT>-RVU�>-CA4B,�H�CXWY,�./,�>-J9,95ZCF>-5B4B,�:[H\1<]^U�>-CA4B,_,9@D,�O�,�>84B5 J91<78,�.L,9H`0�.L,973CXa13I-5B@AK8bdc\,dJ913>-56CDH�,�.4N*-,d5B1E@TI-46CD1->P1<]e,9@A@ACF0-4BCAJ�0�:8.L4BCF:<@fH�CXWY,�./,�>E4BCF:<@^,9g�I�:<4BCA13>-513>hH�CXWi,�.L,�>84H�13O�:[CT>-5�bdc\,d5B18@A78,�4N*-,�./,95GI-@D46CT>-RP@ACF>-,�:8.,9g#I�:[4BCA13>h5BK3564B,�Od5 j+CA4N*\:'OdI-@A4BCAR3./CAHMCA4B,�.B:<4BCA13>13.+:_k ./K3@A1<7=CD46,�.B:<4BCA13>Vj+CA4N*�:�OdI-@A4BCAR3./CAHP0�./,9J913>-H�CA4BCA13>-,�.9b+)+*-,dOdI-@D46CDR-./CAH�./,95B4G./CAJ94BCA13>�:8>-H0�./18@A13>-R-:<4BCA13>\:8.L,;:<@A5B1�CFO�0-@A,�Od,�>E4B,9HlI-5BCF>-RVU�>-CD46,nmBH�CXWY,�./,�>-J9,9oP4/K#02,�5B4B,�>-J9CA@A5�b�)p*-,�U�./5645B4B,�0-5q:8./, R3I-CAH�,9H�rEK':dJ913I�0-@A,d1<]f,9s�:8O�0-@A,95t:E>-H�,9s-,�./J9CA5B,95�b

Page 4: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

uwv x+y�z{x+y�|

�<} ý�ù±öY~�� þ ý�ú ö }

� �<} ý9� ù���( þ � �

� � ü��\�5ö­ú ��� ö } ������( ý�úªö } �

� � � � ý�ú û­ùMú�~w�sö7ùd� } úªý9�\~�úA�{�iù�� } þ � � "

� � ü��\��� � ü�(�÷���~�~>öY�P(�ú } ���

$ õ¡ö } þ � � � ú ö } � $

�_�3���iù�� } þ � � ���

Page 5: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

����������� ���������������������������� ���������

� 6MHJ4Y�#H9�P�����Y�7UWV��XQN�Ã4��

RSG#T 6MUWVX6MHZYAY�[]\M^A^2_

� � x+y��YvZ _¡Z¢�y�£Bv¤x

)+*-,d5B18@FI-4BCA13>h1<]+0�:8./4BCF:<@�H�CXWY,�./,�>846CT:[@�,9g#I�:<46CD1->-5 1<]¥46,�>�@A,�:<H�54B1=4N*-,d5B18@FI-4BCA13>h1<]e,9g�I�:<4BCA13>5BK-5B4B,�Od5�b�¦�13. @F:8./RE,§0�./1-r-@D,�O¨5BCA©9,95�4N*-CA5�5B18@FI-4BCA13>M46,�>-H�5Q461�H�13OdCF>�:<46,'4N*-,;1<78,�.6:<@A@^J913Oa0-@A,9s3CA4/K?1<]�4G*-,'j�*-18@A,�5BCFOdI-@F:<4BCA13>�blªp,�>-J9,',G«;J9CA,�>E4;,9g�I�:<4BCA13>M5B1E@D7E,�.d@ACT¬[,�4N*-,�OdI-@A4BCAR3./CAHOd,94N*-1-Hl:8./,�>-,9,9H�,9HYbw)+*-,hCDH�,�:­CA5�4B1\J91->-5B4N.BI-J94;:E>MCD46,�.B:<4BCA78,'5B1E@D7E,�.�r�:<5B,9HS13>M5B,978,�.B:[@H�CA5BJ�./,94BCA©�:<46CD1->-5Z13>hH�CXWY,�./,�>84 56J�:<@A,95Z:<4Z:�46CTO�,8bQ)+*-,�OdI-@A4BCAR3./CAHnOd,94G*-13H\./,�:<J9*-,95 130-4BCFO�:[@@ACF>-,�:8.dJ913O�0-@A,9s-CA4®KMj�*-CAJ9*MCD5dJ913O_0�:8.B:8r-@A,=4B1PCT>�0�I-4G¯<13I-4N0�I-4�0�./1-J9,9H�I�./,95;:8>-HM184N*-,�.H�:<4G:*�:8>-H�@ACF>-R'CF>�:dJ913O_0�I-4N:<4BCA13>�b

DDSolver

SchwarzDDMultigrid

AddSchwarzDD

SymSchwarzDD

AddMultigrid NestedMultigrid

NonlinearMultigrid

FASMultigrid NestedFASMultigrid

CoarseAddCoarse SchwarzDD

¦�CAR3I�./,=°8±�ª+CA,�.B:E./J²*EK'1<]^OdI-@D46CDR-./CAH�:8>-H'H�13O_:<CF>'H�,9J913O�02185BCA4BCA13>hOd,94N*-1-H�5

³\I-@A4BCAR3./CAH�Od,94N*-1-H�5´:8>-HdH�13O_:<CF> H�,9J913O�02185BCA4BCA13>�O�,94N*-13H�5�:8./,!CFO�0-@A,�Od,�>846,9HQCF> �7���A�#�����F�CF>�:qJ913O�O�13>]�.B:8Od,9j{13.B¬ :E0�0-@DCAJ�:8r-@A, 4B1 CA4B,�.6:<4BCA78,{5B18@A78,�./5�µ30�./,9J913>-H�CA4BCA13>-,�.L5�:E>-H;>-13>-@ACF>-,�:8.5B18@A78,�.L5�bf)+*-,+I-5B,�.�*�:<5�4B1�:<H�H�:80�0�.L1¶s-CFO�:<4BCA78,!5B18@A78,�./5�13>¤4N*-,^H�CXWY,�./,�>E4fH�CA5BJ�./,946CD©�:[4BCA13>-5e:8>-HR3./CAHM4N.B:8>-5L]¥,�.1302,�.B:<4B13.L5Q0�./1E·®,9J94BCF>-R\:8>-HMCF>84B,�.60�18@F:<4BCF>-Rn.L,95BCAH�I�:<@A5§:8>-HMJ913.B.L,9J94BCA13>-5]¸./1-O13>-,�H�CA5BJ�.L,94BCA©�:<4BCA13>­4B1�:8>-184N*-,�.¹b;)+*-,95B,_J913O�0�1->-,�>84B5d:8./,�5G0�,9J9CXU�,9H?CF>\4N*-, �2��º2î�ñ<» ó8¼<½ ��¾CF>846,�.�]¸:[J9, CT> �­�±�A�#��� �M� b)+*-,;OdI-@A4BCAR3./CAH?:<@AR813./CA4N*�O¿CA4B5B,9@X]t:80�0-@ACA,95�4N*-,;:E0�0�./1<s3CFO�:<46,Q5618@A78,�./5Z1->P4N*-,�H�CXWY,�./,�>E4ZH�CA5/aJ�./,94BCA©�:<46CD1->-5�:E>-H;I-5B,95{J913:8./56,QÀÂÁwJ9*-,�:80�Ã!H�CA5BJ�./,94BCA©�:<4BCA13>-5{4B1J913.B./,9J94!5B18@FI-4BCA13>-5{13>U�>-,�. ÀÂÁ,9s�0�,�>-5BCA78,�ÃpH�CA5BJ�./,94BCA©�:<4BCA13>-5�be)+*-,5B4N:8>-H�:E./H'j+:�K_4B1dH�1;4N*-CA5+CA5qJ�:<@A@A,9H�Äpa�J9K-J9@D,8b

ÅÆ�ÇÉÈfÊYË�Ì�ÍÏÎ�Î�ÐÒÑÔÓ/ÕÖ ×NØ�Ù<Ó®Ú{×GØ�ÑÔÛ/Ü/Ý�Þ9߶à�áNâ�ãåä�æGçGè¶àGç9é�êÂëLì9ß9íNì¶î9ïLè¶ð/߶àGñ6è�ê�î�á/ò²ñ�æGó�êÉò¹ô�Ý

°

Page 6: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

õ

ö

÷

ø6ù ú

ú6ù û

ú6ù ø

û6ù ú

¦�CAR3I�./,Zü�±�³`[email protected]Ä+a®ý�K3J9@A,

þ CA78,�>_:q5B,9J913>-H�13.LH�,�.�H�CÔWY,�./,�>E4BCF:<@�130�,�.B:[4B13.�ÿ\:8>-H=:qH�13O�:<CF>�� µEj{,e@A1-13¬¤]¥1-.�4N*-,{5B18@FI-4BCA13>1<]

ÿ�� Á � 13>��� Á ��� 13>�� ������� � Á ��� 13>�������13>':dr�1-I�>-H�,9H�H�1-O�:<CF>�� bl)+*-,ZOdI-@A4BCAR3./CAH\:<@AR813.LCD4G*�Oå]¥1-.e:E>=CT>-CA4BCF:<@�R3I-,95B5���:8>-HP:Q.LCDR-*84 ��� ����� � !#"�$*�:8>-H'5BCAH�,�%p4B1�5B18@A78, 4G*-, ,9g#I�:<46CD1->'5BK-5B4B,�O�O_:�K=r�,Zjq./CA4B4B,�>'./,9J�I�./56CD7E,9@DKP:<5

� � Á & � À'�)(*%9Ã� � Á � �,+.-0/21 �43 /65�/*1 �¶À'78( -0/ 3 /*1 �¹À9%;:Mÿ / � � ÃBÃ5�/ À'�)(*%9à Á & � À'� � (2%9Ãj�*-,�./,�&\H�,�>-184B,95�4N*-,+:80�0�./1<s-CFO�:<4BCA78,!5B18@A78,�./5�:8>-H -0/*1 ��3 / :8>-H -0/*1 ��3 / :8./,!4N*-,^R3.LCDHd4N.6:8>-5/]Â,�.1302,�.B:<4B13.L5�b�)+*-,�,97<:<@FI�:<4BCA13>­1<]e4N*-,=./,95BCAH�I�:<@{CA5ZH�,�>-184B,9H?rEK<%�:?ÿ;�Yb=)+*-,;:<@AR81-./CA4N*�O 13>@A,978,9@Y13>-,J�:8>'r�,H�,GU�>-,9H�:<5 5 �¶À'�)(*%9ÃMÁ=&tÀ'�)(*%9Ãc\,P:<5B5NI�Od,V]¸:EOdCA@DCF:8.LCD4/K?j+CA4N*S5B1-Od,'1<]Z4N*-,Pr�:<5BCAJPJ913>-J9,�0-4B5=1<] �­�±�A�#��� �M� > ?A@CBEDGF bw¦�1-.:\Od1-./,�H�,94N:[CD@A,9H�0�./,95B,�>E4N:<46CD1->?1<]Z4N*-,POdI-@D46CDR-./CAH O�,94N*-13HSj{,�./,G]Â,�.d4B1M4B,9s-4;r�1-13¬E5;@ACF¬<,> ªq:<J�HJI F µ > ª�:<J BLK µ!J²*�:80-46,�.'°�7 F bNMt>?:<@A4B,�.6>�:<4BCA78,�CF>E4N./1-H�I-J94BCA13>?461nOdI-@D46CDR-./CAH Od,94N*-1-H�5�CF>�7���A�#�����F� µ�r�:<5B,9HP13>§U�>-CA4B,d,9@D,�O�,�>84H�CD56J�./,94BCA©�:<4BCA13>-5 J�:8>Pr2,Z]Â13I�>-HPCF> > O I�O BJDPF 4B18R8,94G*-,�.j+CA4N*P:<H�H�CA4BCA13>�:<@{./,G]Â,�./,�>-J9,95�b�QÂ4ZO_:�KPr�,;*-,9@F0E]�I-@^4B1�*�:�78,�:<J9J9,9565 4B1'4N*-, �­�±�A�#��� �M� O�:8>EaI�:<@�0�:[R8,95 ô·��R��Mò j�*-CA@D,\./,�:[H�CT>-RM4N*-CA5=4NI-4B13./CF:<@Âb S-1-Od,'1<] 4N*-,�H�1-J�I�Od,�>E4N:<4BCA13>M1<]qU�>-CA4B,H�CXWY,�./,�>-J9,95�CF> �­�±�A�#��� �M� J�:8>S13>-@AK r2,h]¥13I�>-HwCF>w4N*-,P13>-@ACF>-,nO�:E>3I�:<@µp5B,9,P,8bÒR�b ôM�TR��MòU � èTV2è�W � :E>-H ô·��R��·ò VAè�W � �#º b3)p*-, 5B13I�./J9,qJ913H�,95+:8>-H�:[@D@�4N*-,qCF>�0�I-4{U�@A,95�:8./, :�7¶:<CA@F:8r-@A,Q:[4XM� VTY ëAð ¼ �Aë·�·�A�­ë±��ô ó ë·ôAô#� ó R­ëAð ¼ �2ë b)+*-,�./,�0213./4qCD5 1-./R3:8>-CA©9,9H\:<5p]¥18@A@A1<j+5�±pc\,dCT>E4N./1-H�I-J9,d4N*-,�OdI-@D46CDR-./CAHn:8>-HPH�13O�:<CF>hH�,9J913Oa0�1E5BCA4BCA13>?5B1E@D7E,�.ZCF>E4B,�.�]�:<J9,'CF> �7���A�1�����F� b�Zp,9s34�j{,'H�,978,9@A130?:�U�>-CD46,'H�CÔWY,�./,�>-J9,\[�18CA5B5B13>,9g�I�:<4BCA13>P5BCFOdI-@F:<4B13.r�:<5B,9H\1->P4N*-, �7���A�1�����F� J9@F:<5B5 U � è�VAè]W � jpCD4G*\4N*-,_5B4N:8>-H�:8.LH�J9*-18CAJ9,1<]{H�CT.L,9J94Q:8>-HPCA4B,�.B:<46CD7E,�5B18@A78,�./5 1[Wi,�.L,9H`rEK�^ �Ãò�_]`EaiôLR b c\,�,9s-4B,�>-H\4N*-CA5 5BCFOdI-@F:<4B13.t]¥1-. :OdI-@A4BCAR3./CAH;CA4B,�.6:<4BCA78,+5B1E@D7E,�.e:E>-H;:ZOdI-@A4BCAR3./CAH'0�./,9J913>-H�CA4BCA13>-,�.!]Â13.^k .LK3@A1<7dCD46,�.B:<4BCA13>-5�bbQ�>�4N*-,@F:<5B4!J²*�:80-46,�.�j{,+5N*-1<j *-1<jS4B1Q.BI�>_4N*-,+56CTOdI-@F:<4B1-.^1->�:8> @ a�5G*�:80�,9H'H�13O_:<CF>�CF>-5B4B,�:[H;1<]Y4N*-,I�>-CA4d5Bg#I�:E./,8b�Mq@A@e4G*-,�J913H�,95�:<@A13>-Rhj+CA4N*PH�,94N:[CD@A,9H?,9s�0-@F:8>�:<4BCA13>-5�:8>-H\5G:8O�0-@A,;CF>�0�I-4U�@A,95:8./, RECD7E,�>�bcS-,978,�.6:<@�,9s#02,�./CFOd,�>8465 j+CA4N*=4N*-,ZJ91-H�,Q:E./,Z0�./1302185B,9H'4B1�,9s�0-@A13./,4N*-,Q0�./1-0�,�./4BCA,951<]f4G*-,QOd,94N*-1-H�5�I�>-H�,�.pJ913>-5BCAH�,�.B:<46CD1->�b

ü

Page 7: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

d � x+y�z��Ce�fp¢�z)+*-, �7�±�2�1�����M� CFO�0-@A,�Od,�>E4N:<4BCA13>=1<]�4N*-, [email protected]'Od,94N*-1-H�CA5+r�:<5B,9H�1->�4N*-, �2��º2î�ñ<» ó8¼<½ ��¾CF>846,�.�]¸:[J9,8b�c\, j+CA@A@f,9s�0-@F:<CF>P*-1¶j 4B1dCTO_0-@D,�O�,�>84 4N*-,Z>-,9J9,9565N:8./K_]¸I�>-J946CD1->-5�b g�g�h �*� ikjl�nm g�oprqrs2trt�u ô9â2v9æGçxw try ã qrpï9â9à9î9î trt�u ô9â2v�æGç�w try ã{z9ì9í¶â�áNï}|�à6ò�é�â9æG~Bé�z9ì9í¶â�áNï�ãtrt�u ô9â2v�æGç�w try��l���r�v<áBç9ñGì¶à9â�� trt�u ô9â2v�æGçlw try��l�J�

v<áBç9ñGì¶à9â u z¶à9ï9æG~Bé���æGñr��ôr�Gó u z�à9ï9æNî ��� ï9ô6ò¶îGñ���� �v<áBç9ñGì¶à9â�v�ô�áBé=î9æGñ u ñ�àGç9ñ �'� á/ò9Þ��r��æ9ïGñ�ô6çr����� u z¶à9ï9æG~Bé�î4z¶à9ï9æT� u ñ�àGç9ñr��æ9ïGñ�ô6ç2�¹ôGé�æîGñ�àGç9ñ �J�v<áBç9ñGì¶à9â���ô9ô � æ9à6ò;î9ô9â2v�æ u ì9í u4� îGñ²æ6ß �� á/ò9Þ��r��æ9ïGñ�ô6çr� í�� � á/ò9Þ��r��æ9ïGñ�ôBçr�}��� u z¶à9ï9æG~Bé�î4z¶à9ï9æT�u ñ�àGç9ñr��æ9ïGñ�ô6ç2�¶ôBé�æîGñ�àGç9ñ�� trt�u ô9â2v�æGç2�¶ôBé�æt߶ôGé�æ2� u w�� u4�6�ry Þ � ��� �prp ç�æGñGì�çGò�v�à9â6ì¶æ=á/ò�é<áNï9àGñ�æ9îZï6è¶à6ò���æ9îdôGó=ñGè¶æ�î9ô9â6ì�ñ<áNô6ò�v�æ9ïGñ�ôGçv<áBç9ñGì¶à9â�v�ô�áBé_ç�æ9î�áBéGì¶à9â �� á/ò9Þ��r��æ9ïGñ�ô6çr� í�� � á/ò9Þ��r��æ9ïGñ�ôBçr�}��� � á/ò9Þ��r��æ9ïGñ9ôGçr�Zç�� u z¶à9ï9æG~Bé�î4z¶à9ï9æ �J�v<áBç9ñGì¶à9â�v�ô�áBé�߶àGñr��æ9ï �ï9ô6ò¶îGñ � á/ò9Þ��r��æ9ïGñ9ôGçr� í�� � á/ò9Þ��r��æ9ïGñ9ôGç*����� u z¶à9ï9æG~BéQî4z¶à9ï9æ �J�

v<áBç9ñGì¶à9â���ô9ô � æ9à6ò�ñ9ç�à6ò¶îGó�æGç �ï9ô6ò¶îGñ � á/ò9Þ��r��æ9ïGñ9ôGçr�Zórv�� u z¶à9ï9æG~BéQó<áJ�� á/ò9Þ��r��æ9ïGñ9ôGçr�Zñrv�� u z¶à9ï9æG~BéQñ<áJ���ô9ô � æ9à6ò;àGé9é��Gñ�ôr�Gñr�é2zr� �r��u Þ�� trtr� ç�à6ò¶îGó�æGç2��ôGé²æ2� �2 r� � u �GÞ  �� ��� �prp á/ò�é<áNï9àGñ�æ9îZï6è¶à6ò���æ9îdôGó=ñGè¶æ�î9ô9â6ì�ñ<áNô6ò�v�æ9ïGñ�ôGç

v<áBç9ñGì¶à9â�á/ò�ñ¡��æGñ2¢¶ôGç2£ � ç²à6ò¶î6ó�æGç �ku z¶à9ï9æG~Béó<áJ� u z¶à9ï9æG~Bédñ<áJ��ï9ô6ò¶îGñ � ç�æ9ï9ô6ò�é2¢¶ô6ç2£¥¤�ôGç2£¦�Gñ2z �ï9ô6ò¶îGñ���� �v<áBç9ñGì¶à9âZç�æ9à9â���æGñ u ñ�ôGç�à2�²æ � ç9à6ò¶îBó�æ6ç �ku z¶à9ï9æG~BéZó<áJ� u z¶à9ï9æG~BéZñ<á � ï9ô6ò¶îGñ���� �v<áBç9ñGì¶à9â�á/ò�ñ¡��æGñ2¢¶ôGç2£ u ôNâ2v�æ �ku z¶à9ï9æG~BéQî4z¶à9ï9æT��ï9ô6ò¶îGñ � ç�æ9ï9ô6ò�é2¢¶ôGç2£¥¤�ôGç2£¦�Gñ2z �ï9ô6ò¶îGñ���� �v<áBç9ñGì¶à9âZç�æ9à9â���æGñ u ñ�ôGç�à2�²æ u ôGâ2v�æ �ku z¶à9ï9æG~Bédî4z¶à9ï9æ � ï9ô6ò¶îGñ��§� �

v<áBç9ñGì¶à9â u ñ9ç<á/ò���ï9ô6ß9߶æ6ò�ñ �l�J��T�prqr¨2trt�u ô9â2v9æGçxw try ã qrp)+*-,�]�I�>-J94BCA13>ª© óMè�« î W ��ºM��� � ó ð .L,94NI�.B>-5Y4N*-,{>3I�Odr2,�.�1<]�R3./CAH�5�¬�bf)+*-,�]�I�>-J94BCA13> ð·î�ñ¶» ó º�­T®2ºE¯Áð èÁó RCFO�0-@A,�Od,�>E4B5d4N*-,�5NOd1-184N*-,�.°&tÀ'�)(*%9Ãp13>P4N*-,_R3./CAHn>-I�OQr2,�. ð���� � ó b�)+*-, ð è � ¼1è :8.LR3I�Od,�>E4O�:�K'CF>-H�CAJ�:<4B,':=©9,�./1'5B4N:8.L4+78,9J94B13.ª�ÏbªM ©9,�.L1'5B4N:8./4q78,9J94B1-.q5BCFO�0-@ACÔU�,95�0�:8.L4B5�1[]e4N*-,dJ913Oa0�I-4N:<46CD1->':8>-H'CA5�4N*-,�.L,G]¥13.L, 4N./,�:<4B,9H'H�CXWY,�./,�>E4B@AK8b�¦�1-.{,9s#:EO�0-@A,QOdI-@A4BCF0-@DK-CF>-R'j+CA4N*�:_O�:<4N./CAsrEKn©9,�./1­CD5=,�:<5BK8b R�îAô ó CD5'I-5B,9HS4B1nO�:8¬[,§:PH�CA5B46CT>-J946CD1-> r2,94/je,9,�> 4N*-,�0�./,Ga�& � :8>-H?4N*-,0�1E5B4/a�5NOd1-184N*-CF>-R�& � CX]^>-,9J9,95B5N:8.LK8b;± >-, 5N*-13I-@AH�r2,Z:8r-@A, 4B1dJ913O�0�I-4B, 4G*-,Z./,95BCAH�I�:<@,%b:\ÿ / �rEK�4N*-,]�I�>-J94BCA13> ¼ ó ði�MôL­���ñ b)+*-,dR3./CAHP4N.B:8>-5/]Â,�.pCD5H�13>-,d73CF:=4N*-, è�¼ �·ò7ðF� ó[¼ ]�I�>-J94BCA13>�b�QÂ4CFO�0-@A,�Od,�>E4B5t:�4N.B:8>-5/]Â,�.

4G.B:8>-5/]Â,�.9À'��(4²®Ãp±J³µ´ª¶·¸³�¹º

Page 8: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

Q�>�4N*-,�OdI-@A4BCAR3./CAH;J�:[5B,+CA4^CA5{13>-@AK�I-5B,9Hd]Â13.�4N*-,q0�./18@A13>-R3:<46CD1->ª¬;:'°}·»¬d:8>-H�4N*-,q./,95B4N./CAJ94BCA13>¬¼·»¬�:S°8bc\,?>-,9,9H 5B13Od,­,�>3I�Od,�.6:<4BCA13>¾½�:<R85h4B1wCF>-H�CDJ�:[4B,?H�CXWi,�.L,�>84\Od1-H�,95�µ¤]¥13.h4N*-,M5NOd1-184N*-,�.L5�A�#ºAî ñ<» ó[¼EU îAô ó :8>-H?4N*-,hJ913O�Od13> º è � ¼1è�¿Áó � è î ¼EU î2ô ó :8>-H �A��À ¼ �Mò7ðF� ó8¼JU î2ô ó ]Â13.d5B1-Od,5N02,9J9CÔU�J_4N.B:8>-5/]Â,�.{1302,�.B:<4B1-./5�b{¦�1-.{4N*-,�OdI-@D46CDR-./CAH`O�,94N*-13Hhj{,Z13>-@AK�>-,9,9HP5B1-Od, 1<]^4N*-,7¶:<@XaI-,95�b�c­,Z:<@A5B1;I-5B,d:d5N0�,9J9CF:<@�4/K#02, ,�>3I�O�,�.B:<4BCF>-R=4N*-, R3./CAH�5�b

jxÁ�Â�$prqrsru z¶à9ï9æG~/é�ã qrpñ � z¶æGé�æGó=á/ò�ñ u z¶à9ï9æG~Bé �prqr¨ru z¶à9ï9æG~/é�ã qrpprqrs2trt�u ô9â2v9æGç��¶ôGé9æ�ã q*pæ6ò9ì9ß trt�u ô9â2v�æGç2��ôGé�æ� u w�� u4���ry Þ¥��Ã�� prp ô6ò¶â � ô6ò¶æ=î9ô9â2v�æGç�ã{ï9ô9àGç�î9æQî9ô9â2v�æGç��+î � ß9߶æGñ9ç<áNïî9ô9â2v�æGç`ê9ê9êu w�� u4���ry Þ¦�l�r¢ t �§ÄT� prp ó<áBç�îGñ;î9ô9â2v�æGç���z�ç�æ9î6߶ô9ôGñGè<á/ò6���dê9ê9êu w�� u4���ry Þ¦��� �ry2Å �§Æ prp î9æ9ï9ô6ò�é�î9ô9â2v�æGç���z¶ô9îGñ�î6߶ô9ôGñGè¶á/ò��T�Qê9ê9ê�T�æ6ò9ì9ß trtr� ç�à6ò¶îGó�æ6ç2�¶ôBé�æ� �r r� � u �9Þ   ��Ã�� prp îGñ�à6ò�é�àGç9édñ9ç�à6ò¶îGó�æGç�r r� � u �9Þ   ���GÞ u4� Þ t ��Ä prp è8á��Gè¶æGç�ôGç9é�æGç�ñ9ç�à6ò¶îGó�æGçdó�ôGçdò¶æ9îGñ�æGé=áBñ�æGç�àGñ<áNô6ò�T�prqr¨2trt�u ô9â2v9æGç��¶ôGé9æ�ã q*p

Ç È¾É zËÊMv�£6|�|�v¤x¨z�Ì¡�f{y�£Bv¤xc\, j+:8>E4{4B1�J913O�0�I-46,Z:8>':80�0�./1<s-CTO_:<4BCA13>=1<]

:�ͼ� Á � CF>��?Á > 7�(�° F �� Á 7 13>����¦�CF./5B4!je,{CF>E4N./1-H�I-J9,t:¤5BCFOQI-@F:<4613.�]Â13.�4G*-,+@ACT>-,�:E.^56J�:<@F:8.;[�18CA5B5613>d,9g�I�:<4BCA13>=r�:<5B,9H�13>_:qU�>-CA4B,H�CXWY,�./,�>-J9,Z5BJ9*-,�Od,8b�c­,QI-5B,Z:U�7E,Z0�18CF>E4�5B46,�>-J9CD@�4B1�./,�0�./,956,�>84q4N*-, @ :80-@F:<J9CF:8>

ÎÏÐ

: °: ° K : °: °Ñ ÒÓ

13>=:dI�>-CÔ]Â13.BOå@F:<4B46CDJ9, R-./CAH'13>�4G*-,tI�>-CA4+5Bg�I�:8./, > 7�(�° F � b!)+*-,�R-./CAH'5N0�:<J9CF>-RdCD5�Ô;Á °GÕJÖ'j+CA4N*À'Ö�:w°�Ã0×lÀ'Ö�:w°�ÃqCF>E4B,�./CA13.Z0218CF>8465�b�)+*-,U�>-CA4B,;H�CXWY,�./,�>-J9,�1-0�,�.B:<4613.qCA5Z./,�0�./,95B,�>E4B,9HnrEK\:U � èTV2è�W � 0�18CF>E4�1-0�,�.B:<4613.{O�:<4N./CAs=CF> �7���A�#�����F� b)+*-, �7�±�2�1�����M� J913H�,\CA5hCT> 4N*-,hU�@A,95 U]Ø �iôLR,ÙXí ç :8>-H U�Ø � ôLR)Ù>í®¾ � b�QÂ4'CA5�5BCFOdCA@F:8.h4B1?4N*-,5B4N:E>-H�:8./H �7�±�2�1�����M� 5BCFOdI-@F:<4B13.L5q]¥13.eU�>-CD46,�,9@A,�Od,�>8465 > @ :8> B�KLF :8>-HhCA4 CD5Zr�:[5B,9HPCT>P0�:8.L4�13>4N*-, 5N:EO�0-@A,ZJ91-H�,95�R8CA78,�>=CF>V4N*-,Z13>-@ACF>-,QO_:8>3I�:[@ ôM�TR��Mò U � èTV2è�W � b ÚÜÛLÝßÞ

$áà4â "ã�ä*å4æèç ÑÒÐÒÐ�é�ê¶Õ Ø�Ù<Ó�Û å Õ�Ó�Ñëêª��ä9ó9éGß�à pK

Page 9: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

4

−1

−1

−1−1 ì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkìì9ì9ìkì9ì9ì9ìkì9ì9ì9ìkì

stencil iterator interior

iterator boundary

¦�CAR3I�./, º ±�)p*-,�CF>-H�,9sw5B,94B5;:8>-HS4N*-,hU�>-CA4B,PH�CÔWY,�./,�>-J9,P5B46,�>-J9CD@qJ9K-J9@DCF>-RS1¶7E,�.�:<@A@�CF>E4B,�./CA13.0�1ECT>E4B5�b

í áBóGò�é�æGó}��ä9ó9éGß�Ã��6è¦�r~ u �G~�� yr� w t Þ tí é�æGó<á/ò¶æ°��ä9ó9éGß�Ã��6è¦�r~ u �G~�� yr� w t Þ tí á/ò¶ï9â6ì�é�æ s ��éGß � á/ò � éGß�êÉè ¨í á/ò¶ï9â6ì�é�æ s �¶æ6ò9ì*w try êÉè ¨í á/ò¶ï9â6ì�é�æ sru ñ�ôGç�æ2î � â9ô6ñ9ñ<á�ò��#êÉè ¨í á/ò¶ï9â6ì�é�æ s2t ç�à2¤2� t êÉè ¨í á/ò¶ï9â6ì�é�æ s w�ò8á���w<ñ<áNâ�êTè ¨ï9â9à9î9î°��ä9ó9éGß�Ã8ãïz9ì9í¶â�áNï}�¶æ6ò9ì*w try �bz9ì9í¶â�áNï u ñ�ôGç�æ2î � â9ôGñ9ñ�á/ò6��z�ç�ôGñ�æ9ïGñ�æGé�ãprp ��æ6ò¶æGç�à9âé�àGñ�à�ãç�æ9à9â è � prp ߶æ9î6è§î�á�ð�æ}z¶àGç�à6߶æGñ�æGçá/ò�ñ ò � prp z¶àGç9ñ<áBñ<áNô/ò;ï9ô6ò¶îGñ�à6ò�ñ|�à6ò�é�â9æ � ~/ò²é�æ2� u æGñ � á/ò�é � prpòñ á/ò�ñ�æGç<áNôBç ñ á/ò�é�æ2�;î9æGñ|�à6ò�é�â9æ � ~/ò²é�æ2� u æGñ � í¶ô6ì � prpòñ í¶ô6ì9ò�é�àGç ��ñ á/ò�é�æ2�;î9æGñ|�à6ò�é�â9æ � �¶àBñ � ñr�4z � ç�æGà9â �2� ��� prp z¶ô�á/ò�ñ�ô4z¶æGç�àGñ�ôGç|�à6ò�é�â9æ � ä9ç�áBé � àGñ9ñ�áNïGæ � �9ç<áBé � prp ñGè¶æ��9ç<áBé|�à6ò�é�â9æ � �8áLæ9âGé4� t�� ì � prp ñGè¶ædé<áBó9ó�æGç�æ6ò¶ïNædà4zrz�ç�ô2�<á/߶à6ñ<áLô6ò|�à6ò�é�â9æ � �8áLæ9âGé4� t�� í � prp ñGè¶ædç<á��Gè�ñdè¶à6ò�é;î�áBé�æ|�à6ò�é�â9æ � ��éBß � á�ò � éBß � â�á/ò¶æ2� �Ëprp á/ò�ñ�æGç9ó�à9ï9æñ�ô�æ2�Gì¶àGñ<áNô6ò;î9ô9â2v�æGç�îv<áBç9ñGì¶à9âZç�æ9à9â¨ó �ó� ñrv � ç�æNà9â � � � ï9ô6ò¶îGñ �v<áBç9ñGì¶à9âZç�æ9à9âQì¦� �ó� ñrv � ç�æNà9â � � � ï9ô6ò¶îGñ �v�ô�áBé;î9ï9à6ò�ä9ç<áBé � �¶æ6ò9ì u2� îBñ�æ6ßr�߶æ6ò9ì �J�v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æG~/ò�é²æ2� u æGñ� |�à6ò�é�â9æ � ~/ò�é9æ2� u æGñ � ��á/ò�ñ�æGç<áNôGç���|�à6ò�é�â9æ � ~/ò�é�æ�� u æGñ � � í¶ô6ì9ò�é�àGç ���J�

z9ì9í¶â�áNï�ã��ä9ó9éGß�à �l�¦��l��ä9ó9éGß�à �l�è�r�v<áBç9ñGì¶à9â�v�ô�áBé=àGéGß � �¶æ6ò9ì u2� îGñ�æLß�� ߶æ6ò9ì �J�v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æ � �¶æ6ò9ì u2� îGñ�æLß�� ߶æ6ò9ì��¤á/ò�ñ;â9æ2v�æ9â��¼� � ~�� �J�v<áBç9ñGì¶à9â�v�ô�áBé=î9ï9à6ò � �¶æ6ò9ì u2� îGñ�æLß�� ߶æ6ò9ì �J�v<áBç9ñGì¶à9â�v�ô�áBé=î9ô9â2v�æ � ç�ô6í¹â9æ6ß �l�J�

I

Page 10: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

v<áBç9ñGì¶à9â�v�ô�áBé_ç�æ9î6ì¶âGñ   æ4z¹ôGç9ñ �l�J��T�í æ6ò�é<áBó)+*-,+CF>-H�,9sd5B,94 �kò�ô J913>E4N:<CF>-5^:<@A@�CT>E4B,�./CA13.!0�18CF>E4B5^1[]�4N*-,+@F:<464BCAJ9,8µ8j�*-,�./,p4N*-,{U�>-CA4B,+H�CXWY,�./,�>-J9,5B4B,�>-J9CA@YCA5{4B1�r�,Z:E0�0-@DCA,9HYb+)+*-, r�1-I�>-H�:8./K'>-13H�,95+jpCD4G*�ô CF./CAJ9*-@D,94 r�1-I�>-H�:8./K=J913>-H�CA4BCA13>-5t:E./,J913>E4N:<CF>-,9H�CF>V4N*-,ZCF>-H�,9s'5B,94 ®�îL­ b ? 184G*'CF>-H�,9s'5B,94B5 :8./, CF>-CA4BCF:<@ACA©9,9H\CF> ô ó �­�Ãò óöõ ò�ô ó�÷ º óFè b)+*-,�J91-O�0-@A,94B,;@F:<464BCAJ9,;R3.LCDHMCA5�./,�0�./,95B,�>E4B,9H?r8K\© ¼ �Mô µÏj�*-CAJ²*­CA5dCT>-CA4BCF:<@ACA©9,9HwCF> ðA�·�·ò Ø3¼ �Fô b)+*-,Z./CAR3*E4�*�:8>-Hh5BCAH�,Z1<]�4N*-, @ACF>-,�:8.p,9g#I�:<46CD1->'5BK-5B4B,�O CA5�:[5B5B,�Odr-@A,9H�CF>�% :8>-H'4G*-, 5B18@FI-4BCA13>CA5 CF> ­ bd)+*-,�./CAR3*E4Z*�:8>-H\5BCAH�,�1[]e4N*-,dH�CXWY,�./,�>846CT:[@�,9g#I�:<46CD1->PCA5ZCFO�0-@A,�Od,�>E4B,9H :[5+]�I�>-J94BCA13>� :8>-Hw4N*-,PCF>-CA4BCF:<@Z5B1E@TI-46CD1-> I-56,9H r8Kl:E>lCA4B,�.6:<4BCA78,P5B18@A78,�.=CA5'R8CA78,�>wrEK ­�ø b )+*-,P@ACF>-,�:8.,9g�I�:<4BCA13>?56K35B46,�O :8>-HS4N*-,�,9g�I�:<4BCA13> 5B18@A78,�.=:8./,�O�:E>�:<R8,9H rEK ñ �kò ó ` j�*-CAJ9*lCA5=H�,�./CA78,9H]�./13Où^ �Ãò�_]`EaiôLR b ÚÜÛLÝßÞ

$áà4â oí á/ò¶ï9â6ì�é�æ s ��ä9ó9éGß�Ã8êÉè ¨��ä9ó9éGß�Ã8ã9ãá��ä9ó9éGß�à �l��r�v�ô�áBé���ä9ó9éGß�Ã8ã9ã!àGéGß � �¶æ6ò9ì u2� îGñ9æ6ß�� ߶æ6ò9ì �úprp àGéGß8á/ò8áNîGñ�æBçdñGè¶æ߶æ6ò9ì� �¶æ6ò9ì*w try ã9ã àGñ9ñ9à9ï6è � ߶æ6ò9ì �J��prp æ6ò¶à6í¶â9æ9îZâ9àGñ�æGç;à9ï9ï9æ9î9îñ�ô�߶æ6ò9ì§àGçr��ê+à9î�߶æ6ò9ì¦�9î � îGñ�æ/ߦû ¨é�æGó<á/ò¶æ � ߶æ6ò9ì �J� prp é�æGó<á/ò¶æ p íNì8áNâ6édñGè¶æ߶æ6ò9ì߶æ6ò9ì�ênz�ç�ô6ß*z�ñ ���J� prp z�ç�ô6ßrz�ñZì¶î9æGç��pç�æ9àGéd߶æ6ò9ì§à6ò¶î2¤�æGç�î_á/ò�ñ�ô߶æ6߶ôGç �î9ï9à6ò � ߶æ6ò9ì �J� prp ç�æ9àGéd߶æ6ò9ì§à6ò¶î2¤�æGç�î_á/ò�ñ�ô�ï9â9à9î9î}v�àGç<áNà6í¶â9æ9îZà6ò�éhá/ò8áBñ�

v�ô�áBé���ä9ó9éGß�Ã8ã9ã�é�æGó<á/ò¶æ � �¶æ6ò9ì u2� îGñ²æ6ßr�߶æ6ò9ì��qá/ò�ñ§â9æ2v�æ9â ��߶æ6ò9ì�êÂàGé9é�~®ñ�æ6ß � â9æ�v�æGâT��ü/òJüJ�°ü/òJüJ�ü���ô�ôGóhá/ò�ñ�æGçrv�à9â9îdá/ò ý'�T��Ãlþ�üJ�ü�Ã�ÿGüJ�lür~rü �J���éGß � á/ò � éGß2ã9ã�é9æGó<á�ò¶æ u ñ�à6ñ<áLï � ß�æ6ò9ì]��âGæ2v�æGâ��¦Ã �J�u ñ�ôGç�æ2î � â9ôBñ9ñ<á�ò���ãNã�é9æGó<á�ò¶æ u ñ�à6ñ<áNï � ß�æ6ò9ì���âNæ2v�æGâ��Jà �J��

v�ô�áBé���ä9ó9éGß�Ã8ã9ã!î9ï9à6ò � �¶æ6ò9ì u�� îGñ9æ6ß�� ߶æ6ò9ì �� � ê�ç�æ6í8á/ò�é � ò¶æ2¤¼�¶àGñ � ñ��4z � ç�æ9àGâ �r�¦��9ç<áBé�ê�ç�æ6í8á�ò�é � ò¶æ2¤�ä9ç<áBé � àGñ9ñ<áNï9æ �J�ì�ê�ç�æ6í8á/ò�é � ò¶æ2¤¼�8áNæ9âGé2� t��¦�í�ê�ç�æ6í8á/ò�é � ò¶æ2¤¼�8áNæ9âGé2� t��¦�â�á/ò¶æ2��ê�ç�æ6í¶á/ò�é � ò¶æ2¤���éGß � á/ò � éGß �¦�ò§�_߶æ6ò9ì�êó��æGñ � ü/ò¦ü � êó��æ6ñ�~�ò�ñ ���J�è§�\Ã8ê'� p ç�æ9à9â � ò �J�é�æGó<á/ò¶æG~/ò�é²æ2� u æGñ � á/ò9é���í�ô6ì �J�î9ï9à6ò�ä9ç<áBé � ߶æ6òGì �J��

v�ô�áBé���ä9ó9éGß�Ã8ã9ã!î9ï9à6ò�ä9ç<áBé � �¹æ6ò9ì u2� îBñ�æLß�� ߶æ6ò9ì �D

Page 11: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

�â�á/ò¶æ2��û ¨ î9ïNà6ò � ߶æ6òNì ���u ñ�ôGç�æ2î � â9ôBñ9ñ<á�ò���ãNãÂîGï9à6ò � ß¹æ6ò9ì���Ä �J� prp é<á/ß

prp z¶ô�á/ò�ñ;ô4z¶æGç�àGñ�ôGç��!ñr¤�ôdé<á/߶æ6ò¶î�áNô6ò¶àNâ���l� ê�ç�æGé<á/ß � á/ò9é �l� ��à �J� prp ¤<áBñGè'à6òVôGó9ó�î9æGñ�ôGó'ô6ò¶ædó9ç�ô6ßprp ñGè¶æ;ï9æ6ò�ñ9ç�à9âdæ9â9æ6߶æ6ò�ñ ��� �T�k� � ê���l�J� �T�kû�à � ��û�à ����l�J� û�Ã��{� � ��û�à �Ë���l�J� �T�ª� � � î �Ë���l�J� Ã��ª� � ��û�à ����l�J� �T��à � ��û�à ��9ç<áBé �l� êÂî9ïNà6ò � àGó�ôBçGß � üBé2��Ä�ý'�T��Ãlþr��ý'�T��Ã�þ�á/ò�é�æ2��ãrý'��ã��9é2þr�]ý'��ã��9é*þ�üJ��ò���ò �r�J�ì �l� ê�ç�æGé<á/ß � �9ç�áBé �k� ��ü/ìJü �J�í �l� ê�ç�æGé<á/ß � �9ç�áBé �k� ��ü/íJü �J�ì �l� êóv�à9â6ì¶æNî �l� �§� �í �l� êóv�à9â6ì¶æNî �l� �§� �prp á/ò8áBñ<áNà9â�á�ð9àGñ<áLô6ò�ôGó;ñGè¶æ�ç<á��Gè�ñdè¶à6ò�é;î�áBé�æ� ñrv � á/ò�ñ � áNî � Ä �J�� ñrv � ç�æ9à9â � z¶î � Ä �J�á/ò�é �l� êÂîGñ�àBç9ñ�~/ñ�æGç²àGñ9ôGç � áNî �J�¤Gè8áNâ9æ � á/ò�é �l� ê�áBñ²æGç9àGñ�æ �l�*�� z¶î � à � �_è q áNî � à �J�z¶î � Ä � �_è q áNî � Ä �J�

í �l� êóv�à9â6ì¶æG~/ò²é�æ4� � áBî � à � ��áNî � Ä �2� ��è q è q ó � z¶î ����

prp á/ò8áBñ<áNà9â�á�ð9àGñ<áLô6ò�ôGó;ñGè¶æ í¶ô6ì9ò�é�àGç � ï9ô6ò�é<áBñ<áNô6ò¶îí¶ô6ì �l� êÂîGñ�àBç9ñ�~/ñ�æGç²àGñ9ôGç � áNî �J�¤Gè8áNâ9æ � í¶ô6ì �l� ê�áBñ²æGç9àGñ�æ �l�*�� z¶î � à � �_è q áNî � à �J�z¶î � Ä � �_è q áNî � Ä �J�

ì �l� êóv�à9â6ì¶æG~/ò²é�æ4� � áBî � à � ��áNî � Ä �2� ��ì¦� � z¶î �J��â�á/ò¶æ2��û ¨ àGñNñ�à9ïLè �'���l� ��ì ��� �Fí �l�2�J��

v�ô�áBé���ä9ó9éGß�Ã8ã9ã�é�æGó<á/ò¶æG~/ò�é�æ�� u æ6ñ � |�à6ò�é�â9æ � ~/ò�é9æ2� u æGñ � ��á/ò�é<áJ�0|�à6ò�é�â9æ � ~/ò²é�æ2� u æBñ � �_á/ò�éGí �� ��ô2��~/ò�é<áNï9æNî2�_á/ò�ñ�æGç<áNôGç¥� q ò¶æ2¤���ô2��~/ò�é<áNï9æ9î �á/ò�é<á3ê�ç�æ6í8á�ò�é � á/ò�ñ�æGç<áNôGç �¦�á/ò�ñ�æGç<áNôGç�ê î9ï9àLò � àBó�ô6çGß � ü*Ä � �9éT���9é � � �9é����9é � üJ��Ã���ÃG��ò¦ûGÃ��Fò¦û�à �r�¦�~/ò�é�æ2� u æGñ�~�ò�é<áBï9æ9î��¤í¶ô6ì9ò�é�àGç � � q ò¶æ2¤�~/ò�é�æ2� u æGñ�~�ò�é<áBï9æNî � ÿ �J�á/ò�éGí�ê�ç�æ6í8á�ò�é � í¶ô6ì9ò�é�àGç ���¦���ô2��~/ò�é<áNï9æNî¤í¶ô2�r�Jà �í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T� � � � �9é��k� � üJ��ò �r�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T���9é � � �9é����9é � ü¦��ò��Éò���ò �r���í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�

Page 12: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T�xà � � �T���9é � üJ��òPû�à �2�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é���à � � �9é����9é � ü¦��ò��Éò���òPû�à �r�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J��

v�ô�áBé���ä9ó9éGß�Ã8ã9ãÂîGô9â2v²æ � ç�ô6í�â9æ/ß �l��â�á/ò¶æ2��û ¨ ߶àl£¶æ u4� îGñ²æ6ß �l�J� prp ߶à4£¶æ�ñGè¶æ�æ2�Gì¶àGñ<áNô6ò;î � îGñ�æ6ßâ�á/ò¶æ2��û ¨ î9ôNâ2v�æ �l�J� prp î9ô9â2v�æ_æ2�Gì�ñ<áNô6ò�î � îGñ�æ6ßá/ò�ñdò8áBñ�æGç�àGñ<áNô6ò¶î � ��ô9ô � æ9à6ò;ï � prp ó�ôGçVáBñ�æGç�àGñ<á�v�æZî9ô9â2v�æGç;îGñ�àGñ<áNîGñ<áNïGîáBó � â�á/ò¶æ2��û ¨ â¹á/ò � éGß �l� ên��æBñ u ñ9àGñ<áLîGñ�áNï9î � ò¶áBñ�æ6ç�àGñ�áNôLò¶îT�Âï �*� prp áBñ�æGç�àGñ<á�v�æî9ô9â2v�æGç�îr�9ô srs ôGó�ôGçGß � ü�6ò�6ò qrqrq î9ô9â2v�æGç ��î9ï9ô/ò�v�æ6çr��æBé_á/ò���ÆGéháBñ�æGç�àGñ<áNô6ò�î qrqrq 6ò�6òJüJ�

ï���ü ü\ã°üdò¶ôGñ�üJ��ò8áBñ�æGç�àGñ<áLô6ò�î �J�u ñ�ôGç�æ2î � â9ôBñ9ñ<á�ò���ãNã�é6ì9ßrz � ì �l�r��� prp éGì9ßrz;ó�ôGç;â9àGñ�æGç�v<áNî6ì¶à9â�á�ð�à6ñ<áNôLòâ�á/ò¶æ y ì�çrv�æNî � ì �l�r�¦��

v�ô�áBé���ä9ó9éGß�Ã8ã9ã�ç�æ9î6ì¶âGñ   æ4z¶ôBç9ñ ����r�ç�æ9à9â}��ä9ó9éGß�Ã8ã9ã�ó �ó� ñrv � ç�æ9à9â � � � ï9ô6ò¶îGñ�ç�æGñGì�çGò Ã8ê ��

ç�æ9à9â}��ä9ó9éGß�Ã8ã9ã´ì¦� �ó� ñrv � ç�æ9àNâ � � � ï9ô6ò¶îGñ�ç�æGñGì�çGò¼��ê ��

)+*-,^0�./1-J9,9H�I�./,95 ô ó �7�kò ó :8>-H ð2�#�Mò *�:8>-H�@A,e4G*-,�CT>�0�I-4�Od,�>-I�b�)+*-,^0�./1-J9,9H�I�./, ¼�ó ðG­7ñ èTY�ó ��î ¼1èCA5+: 4B,�O�0-@F:<46,+]Â13.�13I-4N0�I-4{1<]Y4N*-,p5B18@FI-4BCA13>':8>-H�564N:<4BCA5B4BCAJ95�b!)+*-,{U�78, 0�18CF>E4Aô CÔWY,�./,�>-J9,q5B4B,�>EaJ9CA@^CA5Z56,94QI�0­CT>`4N*-,;0�./1-J9,9H�I�./, ð2�#�Mò Ø-¼ �Mô b�)p*-,;./CAR3*E4Q*�:8>-H\56CDH�,=CA5Z5BJ�:[@D,9H?rEK Ô � b;)+*-,:<@A4B,�.B>�:[4BCA78,Zj{13I-@AH\r�,d4B1=5BJ�:<@A,d4N*-,ZH�CXWY,�./,�>-J9,d5B4B,�>-J9CA@�CF>-5B4B,�:[HYµ�j�*-CAJ²*hCA5Z:<@A5B1=g#I-CA4B,dJ913OaOd13>�b

� ��Á %pj+CA4N*�%���Á Ô � �´À9���ÂÃj+CA4N*h@F:<4B4BCAJ9,;0218CF>8465����e:E>-H�7E,9J94B13./5ª�\:8>-H�%<bZ)+*-, ô ó �7�Ãò óTõ ò�ô ó¦÷ º óMè 0�./13J9,9H�I�.L,�H�,GU�>-,954N*-,dCA4B,�.B:<4613./5q13>'4N*-,�>-1-H�,95�µ �Ãò�ô�� J9K-J9@ACT>-RP1<78,�.q:<@A@�CF>84B,�.LCD1-. >-13H�,95 > °E(*Ö :w° F � :E>-H �Ãò�ô�®J9K-J9@DCF>-R\1<78,�. :[@D@�ô CF./CAJ²*-@A,94;r213I�>-H�:8./Kn>-1-H�,95 > 7�(2Ö F � � > °J(*Ö :�° F � b')+*-CA5�:[@D561�H�,GU�>-,95d4N*-,13./H�,�.p1<]^4N*-,dCA4B,�.B:[4BCA13>�µ�j�*-CAJ²*hCA5 @A,9s3CAJ918R-.B:80�*-CAJ8b¥S-1-Od,�:<H�OdCF>-CA5B4N.B:<46CD1->�CA5Z*-CAH�H�,�>­CT>h4N*-,J9@F:<5B5�� ôLR ^ �Ãò]a ôLR j�*-CAJ²*V,9s346,�>-H�5�5B1-Od, 1<]�4N*-, J�:80�:8r-CA@ACA4BCA,95Z1<];^ �Ãò�_]`EaiôLR bܱ >-,]Â,�:<4NI�./, CA54N*-,+J91->878,�.L5BCA13>�1[]i4G*-,+U�>-CA4B, H�CXWY,�./,�>-J9, 4/K#02,+1302,�.B:<4B13. U � èTV2è�W � CF>E4B1Q:dr�:8>-H�,9H'O�:<4N.LCDsdCF>4N*-,dJ�:<5B,K813Ih,�O�0-@A1<K�: þ :8I-565BCF:8>h,9@DCFOdCF>�:<46CD1->\,9g�I�:<4BCA13>h5B18@A78,�.p13.qK81-I�./,9g�I-,95B4Z:8>-184G*-,�.O�:<4G./CAs;5B4613.B:<R8,q4/K#02,8b)+*-,�O_:<CF>;0�./1ER3.B:8O CA5{13>-@AK�R8CA78,�>�]¥1-.�4N*-CA5{,9s#:EO�0-@A,8µ3CF>-CA4BCF:<@ACA©9CF>�R �­�±�2�1��� �M� µ<J91->-5B4N.BI-J94BCF>-R4N*-,�56CTOdI-@F:<4B1-. J9@F:<5B5�:8>-Hn.6I�>�>-CT>-RPCA4�b')+*-,'OdI-@D46CT0-@A,'@A1-130P,�>E73CF./13>�O�,�>84d,�>�:8r-@A,95d5B,978,�.B:[@H�CXWY,�./,�>84q.BI�>-5tI-56CT>-R=:<@A@fJ913Odr-CF>�:<4BCA13>-5p1<]�:[@D46,�.B>�:<4BCA78,95+1[]´CF>�0�I-4qH�:<4N:�µ�56,9, > @ [ BLK�F b

$��l� Á2â oí á/ò¶ï9â6ì�é�æ s ��ä9ó9éGß�Ã8êÉè ¨

H

Page 13: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

á/ò�ñd߶à�á/ò � á/ò�ñò¶àGçr��îT�+ï9ô6ò¶îGñ;ï6è¶àGç qrq àGçr��î ��á/ò8áBñ t ~l�r� �6�ryrÅT� ò¶àBçr�9îT��à6çr�²î �J���â9ô6í¶à9âr�6߶æ/ò9ì�ê¸á/ò8á®ñ � üBó<á�ò8á®ñ�æ¤é<áBó9ó�æGç�æ6ò¶ï9æ � ô�áNî9î9ô6ò;æ2�Gì¶àGñ<áNô6ò�î�á/ß9ì¶â9àGñ�ôGç�üJ��ül�²ä9ó9é6ß�Ã2ü �J���ä9ó9éGß�Ã�z�ç�ô6í¶â9æ6ß � prp ߶à4£¶æ�à'î�á/ß9ì¶â9àGñ�ôBç�ô6í��9æ9ïGñ��+ï9à9â9â9æGé}z�ç�ô6í¶â9æ6ß��â9ô6í¶à9âr�6߶æ/ò9ì�êTß9ì¶âBñ<á�z¶â9æ � ôNô4z � z�ç�ô6í¶â9æ6ß �J� prp î9ô9â2v�æ�ô6ò¶æ�ôGç'î9æ2v�æGç�à9âªz�ç�ô6í¶â9æ6߶ît � ��� üNâ9æ9à2v�á/ò��¤ß¶à�á/òJü �J�ç�æGñGì�çGò¼� ��

)+*-,q]Â18@A@A1¶jpCT>-R=CF>�0�I-4t0�:E.B:8Od,94B,�.L5�O�:�K;r�,5B13Od, R-I-CDH�,9@ACF>-,d]Â13.!K813I�.{,9s�0�,�.LCTO�,�>84B5 � bc\, R8CA78, 4G*-,ZCF>�0�I-4t0�:E.B:8Od,94B,�.L5^]Â13.+:_.BI�>=1<]�4N*-, 5BCFOdI-@T:[4B13.!]Â13.+:d5618@FI-4BCA13>=j+CA4N*�:�J913>�·LIEaR3:<46,9H;R3.6:<H�CA,�>84+CA4B,�.6:<4BCA13>':8>-H�:8>=:8r-5B1E@TI-46,Z4B,�.BO�CT>�:[4BCA13>=J�./CA4B,�./CA13>�µ�56,9, 4N:8r-@A,�°8µ-CF>�0�I-4+U�@A,èÁó ð è Ù>íJ� b

��� } ��úªý9��� ( }i��� �iù> ° DO�:<4N.LCDs=4/K#02, ³\:<4*[�4*± 0r�:<5BCAJQO�,94N*-13H ý{1->�· þ .B:<H0�./,9J913>-H�CA4BCA13>-CF>-R�4/K�0�, [^./,9J�Zp13>-,� °8±�J91->878,�.LR8,�>-J9,ZOd13>-CA4B13.p>�:8Od, ý{³\M r-5N)f.6I-, ��,95BCAH�I�:<@� °8±^O_:<s�,�.B./1-. °8b 7<,Ga �

)�:8r-@A,;°8±�ý{13>�·BI-R3:<46,9H'R3.B:<H�CA,�>E4B5+13>':U�>-CA4B,H�CXWi,�.L,�>-J9,QH�CA5BJ�.L,94BCA©�:<4BCA13>�µ è�ó ð è Ù>íO�[�185B56CTr-@A,,9s#02,�./CFOd,�>8465�CF>-J9@FI-H�,8±

¦ 4N*-,d>3I�Odr2,�.{1<]�CD46,�.B:<4BCA13>-5+H�,�02,�>-H�CF>-R�13>=4N*-, R-./CAH�5BCA©9,¥Ö¦ 4N*-,+>-I�Odr�,�.�1<]�CA4B,�.B:[4BCA13>-5�H�,�0�,�>-H�CF>-Rd13> 4G*-,eCA4B,�.6:<4BCA78,{5B18@FI-4BCA13>d0�./1-J9,9H�I�./,+@ACF¬<,�S�±��:8>-H��3:<J913r-C

¦ :dJ91-O�0�:8./CA5B13>V1<]fH�CF./,9J94 :8>-H�CA4B,�.6:<4BCA78, 5B18@A78,�.L5

¦ :d564NI-H�K;1[]´H�CXWY,�./,�>E4�4B,�.6OdCF>�:<4BCA13>=J�./CA4B,�./CF:=:8>-H'4B18@A,�.B:E>-J9,95

¦ :d564NI-H�K;1[]�0�./,9J91->-H�CD46CD1->-,�./5q]¥1-.{4N*-, J913>�·BI-R3:<4B,9H'R-.B:<H�CA,�>84qOd,94N*-1-H

� � ¡ �/y�£"!+�Y£B =e�vp�$#;xZ£/y�z  d£&%§z��Yz{xZ¢�z^|

c\,qj+CA@A@i>-1<jwCFO�0-@A,�Od,�>E4�:[email protected]�0�.L,9J913>-H�CA4BCA13>-,�.!]Â13.�4N*-,q5BCFOdI-@F:<4B13. U�Ø � ô�R,Ù bfc­,tI-5B,U�>-CA4B, H�CXWY,�./,�>-J9,95�461�.L,�0�./,95B,�>E4+4N*-, H�CXWY,�./,�>846CT:[@Y130�,�.6:<4B13.!13>_,�:<J9*=R3./CAH�:8>-H'j{, J913>-5B4N.6I-J94./,95B4G./CAJ94BCA13>Z:8>-H�0�./1E@D1->-R3:<4BCA13>¤130�,�.B:[4B13./5YCFO�0-@A,�Od,�>E4B,9H�CF>d:p5BCFOdCA@T:E.�j+:�K�j+CA4N*hmBH�CXWY,�./,�>-J9,9o5B4B,�>-J9CA@A5�b�c­,�J�./,�:<4B,{:+>-,9jn56CTOdI-@F:<4B1-. U�Ø � ôLR(' Od1-H�CÔ]ÂK-CT>-R :8>-H,9s34B,�>-H�CF>-R 4N*-,!13./CAR8CF>�:<@-J913H�,U�Ø � ô�R,Ù b�¦�CF./564 je,�>-,9,9H\461;CF>-J9@FI-H�,'4N*-, �2��ºAî ñ<» ó8¼ ./,9@F:<4B,9HhU�@A,95�b�)+*-,dJ9@F:<5B5CA5 CT>�*-,�.LCD46,9H) é�ÐÔÓ/Üf×"*�Ó�Ñëêª��ä9ó9éGß�à p ��æ6ç<áBó ��p

B

Page 14: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

]�./13O �A��º2î�ñ¶» ó8¼<½ �#¾ 4B1dCFO�0-@A,�Od,�>84p4N*-,�H�1-O�:<CF>�H�,9J91-O�0�1856CD46CD1->=CT>E4B,�.�]�:<J9,8b^)p*-,tOdI-@A4BCAR3./CAH:<@AR813.LCD4G*�Od5�:�7¶:<CA@F:8r-@A,dCF>'4N*-, @ACFr�.B:8./K�I-56,Z4N*-CA5+CF>E4B,�.�]�:<J9,8b)+*-, J91-H�,Z1<]�4N*-, 5BCFOdI-@F:<4B13.e]¥18@A@A1<j+5,+8± ÚÜÛLÝßÞ

$.-Pâ "í áBóGò�é�æGó}��ä9ó9éGߦÄr�6è¦�r~ u �G~�� yr� w t Þ tí é�æGó<á/ò¶æ°��ä9ó9éGߦÄr�6è¦�r~ u �G~�� yr� w t Þ tí á/ò¶ï9â6ì�é�æ s ��éGß � á/ò � éGß�êÉè ¨í á/ò¶ï9â6ì�é�æ s �¶æ6ò9ì*w try êÉè ¨í á/ò¶ï9â6ì�é�æ sru ñ�ôGç�æ2î � â9ô6ñ9ñ<á�ò��#êÉè ¨í á/ò¶ï9â6ì�é�æ s2t ç�à2¤2� t êÉè ¨í á/ò¶ï9â6ì�é�æ s w�ò8á���w<ñ<áNâ�êTè ¨í á/ò¶ï9â6ì�é�æ s ��æ9ï u á/ßrz¶â9æGîGñ��4|�à/ò�é�âGæ�êÉè ¨}prp ��æ9ï u á/ßrz¶â9æ9î6ñ ñ îqò¶æ9æGé�æGéí á/ò¶ï9â6ì�é�æ s2trt�u ô9â2v�æGç�êTè ¨ prp�trt�u ô9â2v�æGçí á/ò¶ï9â6ì�é�æ s2trt�u ô9â2v�æGç�w try êTè ¨ prp á/ò�ñ�æGç9ó�à9ï�á/òr�dñ�ô trt�u ô9â2v�æGçí á/ò¶ï9â6ì�é�æ s2trt�u ô9â2v�æGç���z�çGß�êÉè ¨ prp�trt�u ô9â2v�æGç}z¶àGç�à6߶æGñ�æGç9îï9â9à9î9î°��ä9ó9éGߦÄ�ãïz9ì9í¶â�áNï}�¶æ6ò9ì*w try �bz9ì9í¶â�áNï u ñ�ôGç�æ2î � â9ôGñ9ñ�á/ò6���ïz9ì9í¶â�áNï trt�u ô9â2v�æGçxw try�z�ç�ôGñ�æ9ïGñ�æGé�ãprp ��æ6ò¶æGç�à9âé�àGñ�à�ã��æ9ï u á/ßrz¶â9æNîGñ � |�à6ò²é�âGæ � äGç<á®é � à6ñ9ñ<áLï9æ �r� �9ç<áBé �»prp ñGè¶æ��9ç<áBé�î|�à6ò�é�â9æ � �8áLæ9âGé4� t�� ì � prp ñGè¶ædé<áBó9ó�æGç�æ6ò¶ï9æZà4zrz�ç�ô2�<á/߶à6ñ<áNôLò

z�çGß �ó� ç�æ9ï9ô/ò�é � z�ç�æ9ï9ô6ò�é � ç6ß � prp z�çGß;ó�ôGç trt z�ç�æ9ï9ô6ò�é<áBñ<áLô6ò¶æ6ç|�à6ò�é�â9æ � ��éBß � á�ò � éBß � â�á/ò¶æ2� �Ëprp á/ò�ñ�æGç9ó�à9ïNæñ�ô=æ2�Gì¶àGñ<áNô6ò�î9ô9â2v�æGç�î��æ9ï u á/ßrz¶â9æNîGñ � |�à6ò²é�âGæ � �9éGß � á/ò � éGß �r� î6߶ô9ôGñGè �Ëprp æ2�Gì¶àGñ<áNô6ò�î9ô9â2v�æGç�îv<áBç9ñGì¶à9âZç�æ9à9â¨ó �ó� ñrv � ç�æNà9â � � � ï9ô6ò¶îGñ �v<áBç9ñGì¶à9âZç�æ9à9âQì¦� �ó� ñrv � ç�æNà9â � � � ï9ô6ò¶îGñ �v<áBç9ñGì¶à9â�v�ô�áBé=î9ï9à6ò�ä9ç<áBé � �¶æ6ò9ì u2� îGñ�æ6ßr� ߶æ6ò9ì �J�v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æG~/ò�é²æ2� u æGñr� � |�à6ò�é�â9æ � ~�ò�é�æ�� u æGñ � � í¶ô6ì9ò�é�àGç � �qá/ò�ñdò �J�v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æG~/ò�é²æ2� u æGñ�~ � |�à6ò�é�â9æ � ~�ò�é�æ�� u æGñ � ��á/ò�ñ�æGç<áNôGç��qá/ò�ñdò��á/ò�ñ;î2�Jà �J�prp ß9ì¶âGñ<á��9ç<áBéç�æ9â9àGñ�æGéQé�àGñ�à�ãá/ò�ñ ò¶ôr�9ôGó��2�9ç�áBé�î � prp ß9ì¶âGñ<á��9ç<áBé�â9æ2v�æ9â9îá/ò�ñ ï9ô9àGç�î9ær�2�Nç<áBé � prp î�á�ð�æ=ï9ô9àGç�î9æ}�9ç<áBéá/ò�ñ z�ç�æ u ߶ô9ôGñBè � prp ò¶ô=ôGó�áBñ�æGç�àGñ<áNô6ò¶îá/ò�ñ z¶ô9îGñ u ߶ô9ôBñGè � prp ò¶ô=ôGó�áBñ�æGç�àGñ<áNô6ò¶îz�çGß �'trt�u ô9â�v�æGç � é9é�î9ô9â2v�æGç6�4z�ç6ß � prp z¶àGç�à6߶æGñ�æGç�îtß9ì¶âGñ<á��9ç<áBéQî9ô9â2v�æGç��æ9ï u á/ßrz¶â9æNîGñ � |�à6ò²é�âGæ �ó� ç�ô�� �r� z�ç�ô �r�4z � prp z�ç�ô �9æ9ïGñ<áNô6ò�ô4z¶æGç�àGñ�ôGç�î��æ9ï u á/ßrz¶â9æNîGñ � |�à6ò²é�âGæ �ó� ç�ô�� �r� z�ç�ô �r�Gç � prp z�ç�ô �9æ9ïGñ<áNô6ò�ô4z¶æGç�àGñ�ôGç�î��æ9ï u á/ßrz¶â9æNîGñ � |�à6ò²é�âGæ �ó� ç�ô�� �r� z�ç�ô �r�6ò¶æ9îGñ ��prp ò¶æ9îGñ�æGé�ñ9ç�à6ò¶îGó�æGç|�à6ò�é�â9æ �'trt6u ô9â4v�æGç � é9é�î9ô9â2v�æGç � prp ß9ì¶âGñ<á��9ç<áBé�î9ô9â2v�æGçv<áBç9ñGì¶à9â�v�ô�áBéVá/ò8áBñ � ç�ô � �k�J� prp î9æGñGìrz�z�ç�ô �v<áBç9ñGì¶à9â�v�ô�áBéVá/ò8áBñ2�¶àGñ9ç<áLï9æ9î �l�J� prp á/ò8áBñd߶àGñ9ç<áNï9æ9î ��� á �v<áBç9ñGì¶à9â�v�ô�áBé�߶à4£¶æ4�¶àGñ9ç<áx� �l��� prp î9æGñdìrz�â�á/ò¶æ2���¶àGñ9ç<á��v<áBç9ñGì¶à9â�v�ô�áBé�߶à4£¶æ4�¶àGñ9ç<áx� � á�ò�ñ�á �J� prp î9æGñdìrz�î6߶ô9ôGñGèd߶àGñ9ç<á��v<áBç9ñGì¶à9â�á/ò�ñ¼�9ç<áBé u á�ð�æ �ku z¶à9ïGæG~Bé�á �J� prp �9ç<áBé;î�á�ð�æQò��tâ9àGñ9ñ<áNï9æ¤ò q ò

z9ì9í¶â�áNï�ã��ä9ó9éGß¦Ä �l�¦�/lä*å4æèç ÑÒÐÒÐ�é�ê¶Õ Ø�Ù<Ó�Û å Õ�Ó�Ñëêª��ä9ó9éGß¦Ä p

°�7

Page 15: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

�l��ä9ó9éGß¦Ä �l�è�r�v<áBç9ñGì¶à9â�v�ô�áBé=àGéGß � �¶æ6ò9ì u2� îGñ�æLß�� ߶æ6ò9ì �J�v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æ � �¶æ6ò9ì u2� îGñ�æLß�� ߶æ6ò9ì��¤á/ò�ñ;â9æ2v�æ9â��¼� � ~�� �J�v<áBç9ñGì¶à9â�v�ô�áBé=î9ï9à6ò � �¶æ6ò9ì u2� îGñ�æLß�� ߶æ6ò9ì �J�v<áBç9ñGì¶à9â�v�ô�áBé=î9ô9â2v�æ � ç�ô6í¹â9æ6ß �l�J�v<áBç9ñGì¶à9â�v�ô�áBé_ç�æ9î6ì¶âGñ   æ4z¹ôGç9ñ �l�J�prp�trt�u ô9â2v�æGç�w tryu z¶à9ï9æG~Bé¥��æGñr��ôr�Gó u z�à9ï9æGî �k� ï9ô6ò¶îGñ � prp ò¶ôr�9ôGó��2�9ç<á/é�î��ô9ô � æ9à6ò_î9ô9â2v�æ u ì9í u4� îGñ9æ6ß �'� á/ò9Þ��r��æ9ïGñ9ôGç*�í�� � á/ò9Þ��r��æ9ïBñ�ôGç2�����u z¶à9ï9æG~Bé�î4z¶à9ï9æT� u ñ�àGç9ñr��æ9ïGñ²ôGç2��ôGé²ædîGñ�àGç9ñ��trt�u ô9â2v�æGç2��ôGé²æ¤ß¶ôGé�æ2� u w�� u����*y Þ ���v�ô�áBé�ç�æ9î�áBéGì¶à9â �'� á/ò9Þ��r��æNïGñ�ô6çr� í�� � á/ò9Þ��r��æ9ïGñ�ôBçr�}��� � á/ò9Þ��r��æ9ïGñ9ôGçr�Zç�� u z¶à9ï9æG~Bé�î4z¶à9ï9æ �J���ô9ô � æ9à6ò�ñ9ç�à6ò¶îGó�æGç � ï9ô6ò¶îGñ � á/ò9Þ��r��æ9ïGñ9ôGç*�dórv�� u z¶à9ï9æG~Bédó<áJ�� á/ò9Þ��r��æ9ïGñ9ôGç*�dñrv�� u z¶à9ï9æG~Bédñ<áJ���ô9ô � æ9à6ò;àGé9é��Gñ�ôr�Gñr�Zé2zr� �r��u Þ�� trtr� ç�à6ò¶îGó²æGç2��ôGé�æ4� �* r� � u �NÞ  ���� prp à4zrz¶â � z�ç�ô �v<áBç9ñGì¶à9â�á/ò�ñ¡��æGñ2¢¶ôGç2£ � ç²à6ò¶î6ó�æGç �ku z¶à9ï9æG~Béó<áJ� u z¶à9ï9æG~Bédñ<áJ��ï9ô6ò¶îGñ � ç�æ9ï9ô6ò�é2¢¶ô6ç2£¥¤�ôGç2£¦�Gñ2z � ï9ô6ò¶îGñ �v<áBç9ñGì¶à9âZç�æ9à9â���æGñ u ñ�ôGç�à2�²æ � ç9à6ò¶îBó�æ6ç �ku z¶à9ï9æG~BéZó<áJ� u z¶à9ï9æG~BéZñ<á � ï9ô6ò¶îGñ �v<áBç9ñGì¶à9â�á/ò�ñ¡��æGñ2¢¶ôGç2£ u ôNâ2v�æ �ku z¶à9ï9æG~BéQî4z¶à9ï9æT��ï9ô6ò¶îGñ � ç�æ9ï9ô6ò�é2¢¶ôGç2£¥¤�ôGç2£¦�Gñ2z � ï9ô6ò¶îGñ �v<áBç9ñGì¶à9âZç�æ9à9â���æGñ u ñ�ôGç�à2�²æ u ôGâ2v�æ �ku z¶à9ï9æG~Bédî4z¶à9ï9æ � ï9ô6ò¶îGñ �u ñ9ç<á/ò��¨ï9ô6ß9߶æ6ò�ñ �l�J��T�í æ6ò�é<áBó

c\,;>-,9,9H?:�*-CA,�.B:E./J²*EKP1<]+@F:<4B4BCAJ9,�R3.LCDH�5�© ¼ �Mô CF>­J913>E4N.B:8./Kh4B1'4N*-,=13>-,�R3.LCDH­CT>­4N*-,;0�.L,973CXa13I-5J�:<5B,8b�M�@A4N*-13I-R-*\j{,':8./,�13>-@AK\CF>E4B,�./,95B46,9H?CT>­4N*-,=5B18@FI-4BCA13> ­ 13>­4N*-,_U�>-,95B4dR3./CAHYµ�j{,j+CA@A@^0�,�.�]Â13.BO J91-O�0�I-4N:<4BCA13>-5 1->�:[@D@�R3./CAH�5�bd)+*-,dR3./CAH�5Z:8.L,�>-I�Odr�,�./,9Hh]�./13O °�À�J913:8.L5B,�Ã+461ò�î î1� © ¼ �Mô ð À U�>-,�Ã9b�c\,+*�:�7E,e1->-,�*�:E>-H�@D,p4B1 � ôLR ^ �Ãò]a ôLR�ñ �kò ó ` 4B1 O�:8>�:<RE,e4G*-,e,9g�I�:<4BCA13>5BK-5B4B,�O j{, j+:8>E4{4B1�5B18@A78,8bïM�H�H�CA4BCA13>�:<@A@AKPj{,Z*�:�7E, 13>-, � ôLR ^ �Ãò]a ôLRWð�R�îAî è1ç ]Â13.!,�:<J9*'R3.LCDH4B1;O_:8>�:<R8, 4N*-, 5GOd131E4N*-CF>-R;13>V4N*-, R3./CAHYbAM0� ôLR ^ �Ãò]a ôLR r�:<56CDJ�:[@D@AKh5B4B13./,95p4N*-,Z./,�0�./,95B,�>E4N:�a4BCA13>d1[]i4G*-,�O�:<4N.LCDs�./,�0�.L,95B,�>846CT>-Rd4N*-,pH�CÔWY,�./,�>E4BCF:<@�1302,�.B:<4B13.¹µ<5B13Od,+7E,9J94B13./5{:8>-H;: ,9g�I�:<4BCA13>5B18@A78,�.¹b)+*-, ,9g�I�:<4BCA13>V5B18@A78,�.{1<] ñ �kò ó ` I-5B,95�:d,9g�I�:<4BCA13>V5B18@A78,�. �A� õMèÁó[¼ µ-j�*-CAJ²*VCD5 :d4B,�O_0-@T:[4B,q]¥1-.4N*-,ZOdI-@A4BCAR3./CAH\:<@AR813./CA4N*�O�5�µ�13.+:_0�./,9J913>-H�CA4BCA13>-,9H\k ./K-@D1<7'CA4B,�.B:[4BCA13>\À�,8bÒR�b ¾2î·ò(1 Ø-¼ �Aô Ã!j+CA4N*:dOdI-@D46CDR-./CAH�0�./,9J913>-H�CA4BCA13>-,�. V�¼�ó �M�2� b3)p*-,Z5NOd1-184N*-,�.L5�:E./,�CFO�0-@A,�Od,�>E4B,9H\rEK;4N*-, ,9g�I�:<4BCA13>5B18@A78,�.L5;CF> ð�R�îAî è1ç b )+*-CA5�I-5NI�:<@A@AKwCD5�:`U�s-,9H >-I�Odr�,�._1<]ZCA4B,�.B:<46CD1->-5'1<]Z5B13Od,hCA4B,�.B:<4BCA78,Od,94N*-1-H @ACF¬<, º WEY 1-.32 ���#î�®X� :E>-H ]Â13.�@T:E./R8,�.ÏJ913:8./5B,�R-./CAH�5f5B1-Od,�H�CF./,9J94f5618@A78,�.YCF> ð�R�î2î è1ç54 Ù76 b)+*-,!U�>-CA4B,+H�CXWY,�./,�>-J9,+5B46,�>-J9CD@�CA5!5B4B13./,9H¤]¥1-.�,�:<J²*8� ô�R ^ �kò�aiôLR 13r-·/,9J94^:<@A13>-R j+CA4N*d:8>CA4B,�.B:<4B1-.1<78,�.^:[@D@�CF>84B,�.LCD1-.e>-1-H�,95�b^)p*-CD5qOd,�:8>-5^4G*�:<4eI-5NI�:[@D@AK'4N*-,p5N:8Od,qH�CXWi,�.L,�>-J9, 5B4B,�>-J9CA@�CD5+0�./,956,�>845B,978,�.6:<@+4BCFOd,95�µ{jq*-CDJ9*SH�13,95'>-184;.L,9g#I-CF./,\OdI-J²*SOd,�Od13.LK8b�)+*-,�./,�:E./,'J�:<5B,95�µ{jq*-,�./,�4N*-,5B4B,�>-J9CA@A5^H�CXWY,�.9b:9�7E,�> 4N*-,!U�>-CA4B,{H�CXWi,�.L,�>-J9,+130�,�.6:<4B13.ÏCF> ðGR�î2î è#ç;4 ò�î î1� © ¼ �Mô�ð<6 :8>-H ñi�Ãò ó `H�1;>-184q>-,9,9H�4B1dJ918CF>-J9CAH�,�CX]´j{,ZI-5B,dOdI-@D46CDR-./CAH�:<5�:_0�./,9J913>-H�CA4BCA13>-,�.9b)+*-,'./CAR3*E4;*�:8>-H?5BCAH�, � :8>-H?4G*-,'CT>-CA4BCF:<@pR3I-,95B5 ­�ø ./,�O�:[CT>?I�>-J9*�:8>-R8,9HYb?)+*-, ðA�·�·ò Ø-¼ �Fô0�./1-J9,9H�I�./,dCA5Z>-1¶j .L,95N0�13>-56CTr-@A,�461;J�./,�:[4B,d4N*-,dj�*-18@A,�*-CD,�.6:8./J9*8Kh1<]^R3./CAH�5�bqc\,dH�13I�r-@A,�4N*-,

°8°

Page 16: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

R3./CAHd5BCA©9, Ö�]¸.L13O 13>-,{R3./CAHd4B1 4N*-,+>-,9s-4�R3./CAHd5B4N:8./46CT>-R jpCD4G*Q:qR-./CAHQ56CD©9,p1<] Ö=Á �·îA� ¼ ð ó © ¼ �Mô µj�*-CAJ9*�Od,�:8>-5dÀ'Ö�:S°�à � CT>E4B,�./CA13.p>-13H�,95�b{)+*-CA5+CA5qJ913H�,9HhCT>V4N*-,q]�I�>-J94BCA13>�© ¼ �Mô-º��>= ó µ8jq*-CDJ9*./,94NI�.6>-5+4N*-,Z>-I�OQr2,�. ÖV]Â13.{,�:<J9*=@A,978,9@Âbe)p*-,ZR3./CAH�5�:E./,Z>-,95B4B,9HYbc\,{5N0-@ACA4^4N*-,{J�./,�:[4BCA13>1<]�CA4B,�.B:[4B13./5�CF>84B1 1->-,�0�.L13J9,9H�I�./,�]¥13.ÏCF>846,�./CA13.^>-1-H�,95^I-5B,9Hd]Â13. ñi�Ãò ó ` µðGR�îAî è1ç :8>-HS4N*-,�.L,95B4N./CAJ94BCA13>S:8>-Hl0�./18@A13>-R3:[4BCA13>?1302,�.B:<4B1-./5;:8>-Hl:8>-1E4N*-,�.�0�./1-J9,9H�I�./,h]¥1-.r�1-I�>-H�:8./KQ>-1-H�,95�I-5B,9HdCT> ñ �kò ó ` b�)+*-,+0�:8.B:8Od,946,�.@?{H�,�>-184B,95�4N*-,{5B4B,�0¤5BCA©9,!]¥13.Ï4N*-,!CD46,�.B:<4B13.¹b)+*-, H�,G]�:8I-@A4�CA5d°8µ�:8>=CA4B,�.6:<4B13.{1<78,�.{:<@A@f>-1-H�,95�b{)+*-,Z564B,�0=5BCA©9,Qü_CF>',�:<J9*=H�CF./,9J94BCA13>hCA5tI-5B,9HCF>;.L,95B4N./CAJ94BCA13>=:8>-H;0�./1E@D1->-R3:<4BCA13>d1-0�,�.B:<4613./5�µ-:80�0-@AK3CF>-R;:¤5B4B,�>-J9CA@�]Â13.�,�:[J²*�>-1-H�,8µ-j�*-CAJ²*=*�:<5:dJ913I�>E4B,�.B0�:E./4{13>=4N*-,Z>-,9s-4�J913:E./5B,�.!R3./CAHYb)+*-,n0�:E.B:8Od,94B,�.L5 � ¼ ó ºLR�îAî è1ç :E>-H ��î�ð è ºLR�îAî è#ç :8.L,`I-56,9H�4B1?1<78,�.B./CAH�,\4G*-,n>3I�Odr2,�.'1<]CA4B,�.B:<46CD1->-5�]Â13.=:<@A@�4N*-,P,9g�I�:<4BCA13> 5B18@A78,�./5=CF> ð�R�îAî è1ç ,9s-J9,�0-4=]Â13.d4N*-,PJ913:8.L5B,�R3.LCDHS5B18@A78,�.ðGR�îAî è1ç;4 ÙA6 bq)+*-,�0�:8.B:8O�,94B,�. ôAô�ð·î�ñ¶» ó8¼ 5B4B13./,4N*-,d4/K#02,Q1[][email protected]:<@AR813./CA4N*�O :8>-HPCA4B50�:8.B:EOd,94B,�./5�bbQ¸4+CA5t./,�:[H=]¸./1-O 4G*-,QOd,�>-I�bc\,{CF>84G./13H�I-J9,p5B13Od,+>-,9jl0�./1-J9,9H�I�./,95^461 5B4N.BI-J94NI�.L,e4G*-,eCF>-CA4BCF:<@ACA©�:<4BCA13>'0�./1-J9,95B5�b�)+*-,q0�./13J9,GaH�I�./, �kòX� èTV�¼ îB1 CF>-CA4BCF:<@ACD©9,954N*-,Q0�.L18@A13>-R3:<4BCA13> � ¼ îC1 � µ�.L,95B4N./CAJ94BCA13> � ¼ îC1 ¼ :E>-H'CT>E4B,�.B0218@F:�a4BCA13> � ¼ îB1 ò ó ð è À ]¥13.{>-,95B46,9H�OdI-@A4BCAR3./CAH�Ã+1-0�,�.B:<4613./5�b�)+*-, �Ãò � èTU � è#¼ �1� ó ð 0�./13J9,9H�I�.L,ZJ�:<@A@A5R��#� ó¦U � è#¼ � ÷:4 6 4B1 5B,94eI�0�4N*-,+,9g�I�:<4BCA13>-5^461 5B18@A78,+CF> ñi�Ãò ó ` :E>-H R��1� ó¦U � è�¼ � ÷:4 �Ãò è 6 4B1 5B,94I�0=4N*-, ,9g�I�:<4BCA13>-5+7-CA5BCFr-@A, ]Â13.!4N*-, 5NOd1-184N*-,�./5{13>=,�:[J²*_@A,978,9@Âbe)+*-,qU�>-CA4B,H�CXWi,�.L,�>-J9,Z5B4B,�>-J9CA@0�./,95B,�>E4B,9H',�:8.L@DCA,�.qCA5+CFO�0-@A,�Od,�>E4B,9H\*-,�./,8b)+*-,d0�./13J9,9H�I�.L,95 �2ôLR µ ô ó �7�kò ó µ ð2�#�·ò µ ð·î�ñ<» ó�V#¼ îL®­ñ ó R :8>-H ¼ ó ð�­7ñ è�YÁó ��î ¼#è :E./,Z4N*-,d5B4G:8>EaH�:8./H�r�I-CA@AH�CF>-R�r-@A1-J²¬E5�1[] �­�±�A�#��� �M� 5BCFOdI-@F:<4B13.L5�:8>-H�:E./,Z:<H�:80-4B,9H�:[J9J913./H�CF>-R8@AK8b)+*-, �A�#ºAî�ñ¶» ó[¼[½ �#¾ CF>E4B,�.�]�:<J9,�CA5fCFO�0-@A,�Od,�>E4B,9Hd4N*�./1-I-R3*+4G*-,�]Â18@A@D1<j+CF>-R 0�./1-J9,9H�I�./,95�±C© óMè�« î W ��ºM��� � ó ð./,94NI�.6>-5Y4N*-,^>-I�OQr2,�.�1<]�R3./CAHZ@A,978,9@A5�5B4B13.L,9H CT> ò�î î#� © ¼ �Mô ð bY)+*-,^0�./1-J9,9H�I�./, ð#î�ñ¶» ó ºL­T®2ºE¯�ð èÁó RCFO�0-@A,�Od,�>E4B5Z4N*-,d5NO�13184G*-,�.9µ�jq*-CDJ9*hCA5Z4N*-,�:80�0�./1<s-CFO�:<4B,5B18@FI-4BCA13>h13>P:=R8CA78,�>hR3./CAH ð������ ój+CA4N*\.LCDR-*84Z*�:8>-H­5BCAH�, ® :8>-H\CF>-CA4BCF:<@{R3I-,95B5 ÷ b�)+*-,�5618@FI-4BCA13>PCA5Q./,94GI�.B>-,9H\CF> ÷ b�)+*-,�½�:[Rð è � ¼#è CF>-H�CAJ�:<4B,95dj�*-,94G*-,�.q4N*-,dCT>-CA4BCF:<@!R3I-,95B5CD5 7<:<@ACAHn13.q4G*-,QCA4B,�.6:<4BCA13>h5N*-13I-@AHnr2,�5B4N:8.L4B,9Hj+CA4N*h©9,�./1�bq)+*-,}½�:<R R�îAô ó ,�>�:Er-@D,95dI-5 4B1=5B,�0�:8.B:<46,Q0�./,Ga�5NO�13184G*-CT>-R�:E>-H�0�1E5B4/a�5NOd1-184N*-CF>-R:8>-H=4B1�I-5B,q4N*-,Z:80�0�./130�.LCT:[4B,t>-I�Odr�,�.{1[]iCA4B,�.6:<4BCA13>-5�b^)p*-,Z./,94NI�.B>_7¶:<@FI-, CF>-H�CAJ�:<4B,95qj�*-,94N*-,�.4N*-,�7E,9J94B13. ÷ *�:<5Qr2,9,�>M4B13I-J9*-,9HYb�QÂ4ZH�1-,95Q>-1E4Q*�:�78,�4B1Pr�,�4613I-J²*-,9H­]Â13.q©9,�./1�CA4B,�.6:<4BCA13>-5�µ,9CA4N*-,�.p©9,�./1;0�./,Ga^13.�©9,�./1;0�1E5B4/a�5NOd1-184N*-CF>-R�564B,�0-5�b)+*-,�0�./1-J9,9H�I�./, è�¼ �·ò7ðF� ó[¼ CFO�0-@A,�Od,�>E4B5d4N*-,�0�./1E·®,9J94BCA13>P:8>-H\./,9564N./CAJ94BCA13>P1302,�.B:<4B1-./5�bZ)+*-,78,9J94B1-. �-» 13>R3./CAH �­� *�:[5�4B1tr2,+4N.B:E>-5/]¥,�.6./,9HZ4B1qR3./CAH è � b�)+*-,+./,95NI-@A4�CA5�5B4B1-./,9HZCF>d78,9J94613. è » b)+*-CA5^J�:E>Qr2184N*dr�,q0�./18@A13>-R3:<46CD1->EDá·FD + °�:<5^: .L,95B4N./CAJ94BCA13>8D + °ª·FD6bG9�CA4N*-,�.�4N*-,{1302,�.B:<4B1-.� ¼ îB1 � 13. � ¼ îC1 ¼ CD5pJ�:<@A@D,9HYb{)+*-,0½�:<R �AôAô è î è CF>-H�CDJ�:[4B,95�jq*-,94N*-,�.!4N*-,�0�./1E·®,9J94B,9H=78,9J94B1-.5N*-13I-@AH�r�,­1¶7E,�./j�./CA4B4B,�> 13>E4B1�²IH 13._CD4'5G*-13I-@AH r�,\:<H�H�,9HYb )p*-CD5 ½�:[RnCA5�0�:<5B56,9H 461n4N*-,0�./1E·®,9J94BCA13>=1-0�,�.B:<4613./5�µ�jq*-CDJ9*h4N*-,�Od5B,9@A78,95 0�:<5B5p4N*-CA5�½�:<R�4B1�4G*-,QO�:<4G./CAs§OdI-@A4BCF0-@AK`I-5GI�:<@A@DK,�O�0-@A1<K8,9HYb�)p*-,ª½�:<R R�îAô ó J�:8>hr�,�I-5B,9HP4B1�56,�0�:8.B:<4B,4N*-,ZJ�:[5B,Z1<]^564N:8>-H�:8./H�0�.L18@A13>-R3:<4BCA13>1<]�J913.B./,9J946CD1->-5t:8>-Hh5B13Od,184N*-,�.pCF>84B,�.60�18@F:<4BCA13>h1<]^:_5B18@FI-4BCA13>'CF>P:=>-,95B4B,9H�CA4B,�.6:<4BCA13>'J9K-J9@A,8bQ�>`4N*-CA5ZJ�:<5B,_je,dJ�:8>­I-5B,�4N*-,�5G:8Od,d130�,�.6:<4B13.9b�)p*-CD5j{13I-@AHn>-184Zr�,=4N.BI-,dCX]+j{,;*�:<HnI-5B,9H:8>-184G*-,�.{5BJ�:<@ACF>-R'1<]�4N*-, U�>-CA4B,H�CÔWY,�./,�>-J9,d5B46,�>-J9CD@Âb�Q�>'5GI-J²*h:�J�:<56,Zr�184N*V130�,�.6:<4BCA13>-5qje1-I-@DH:<@Aj+:�K35{H�CXWY,�.9b)+*-,P@F:<5B4�0�./1-J9,9H�I�./,95�© óMèCJ î ¼ �TÀ ¼ �·ò­ðM� ó8¼ µA© óMè º è î ¼ � © ó À ¼ �Mò7ðM� ó8¼ µ�© óFèBJ î ¼ �#ºAî ñ<» ó :8>-H© óFè º è î ¼ � © ó ºAî ñ<» ó 0�./1<7-CDH�,M5B13O�,�564N:<4BCA5B4BCAJ95hj�*-CAJ²*wCA5P0�:<5B5B,9H�461?4N*-,\@ACF>-,�:8.h5B18@A78,�.'1-.0�./,9J913>-H�CA4BCA13>-,�.¤5B4N:<46CD564BCAJ95t:8>-HPH�,95BJ�.LCTr2,95Z,95B46CTO_:<4B,95q]Â13.+4G*-,Q5GOd131E4N*-,�./5 :8>-H�4N.6:8>-5/]Â,�.+130Ea

°�ü

Page 17: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

,�.B:<4613./5�µ-r�184N*=CF>'Od,�O�13./K;./,9g�I-CF./,�Od,�>E4�:8>-H�>-I�Odr�,�.{1<]�130�,�.B:[4BCA13>-5�bbM >=CAH�,�>84BCXU�,�.p5B4N./CF>-R�#î�RTR ó ò è 5N*-13I-@AH�:[@D561;r�,CTO_0-@D,�O�,�>84B,9HYb ÚÜÛLÝßÞ

$.-Pâ oí á/ò¶ï9â6ì�é�æ s ��ä9ó9éGߦÄ�êÉè ¨í á/ò¶ï9â6ì�é�æ s ïGç�æ9àGñ�æ trt�u ô9â2v9æGç#êÉè ¨}prp ïGç�æ9àGñ<á/ò��¤ß9ì¶âGñ<á��9ç<áBéQô6í��9æ9ïGñí á/ò¶ï9â6ì�é�æ s2trt ~Bñ�æGç�êÉè ¨í á/ò¶ï9â6ì�é�æ s4� ç�æ9ï trt êÉè ¨í á/ò¶ï9â6ì�é�æ s �¶àGñ � ñ   �4z¦�6ç�æ9àGâ�êAè ¨��ä9ó9éGߦÄ�ã9ãá��ä9ó9éGß¦Ä �l��r�v�ô�áBé���ä9ó9éGߦÄ�ã9ã!àGéGß � �¶æ6ò9ì u2� îGñ9æ6ß�� ߶æ6ò9ì �úprp àGéGß8á/ò8áNîGñ�æBçdñGè¶æ߶æ6ò9ì� �¶æ6ò9ì*w try ã9ã àGñ9ñ9à9ï6è � ߶æ6ò9ì �J��prp æ6ò¶à6í¶â9æ9îZâ9àGñ�æGç;à9ï9ï9æ9î9îñ�ô�߶æ6ò9ì§àGçr��ê+à9î�߶æ6ò9ì¦�9î � îGñ�æ/ߦû ¨é�æGó<á/ò¶æ � ߶æ6ò9ì �J� prp é�æGó<á/ò¶æ p íNì8áNâ6édñGè¶æ߶æ6ò9ì߶æ6ò9ì�ênz�ç�ô6ß*z�ñ ���J� prp z�ç�ô6ßrz�ñZì¶î9æGç��pç�æ9àGéd߶æ6ò9ì§à6ò¶î2¤�æGç�î_á/ò�ñ�ô߶æ6߶ôGç �î9ï9à6ò � ߶æ6ò9ì �J� prp ç�æ9àGéd߶æ6ò9ì§à6ò¶î2¤�æGç�î_á/ò�ñ�ô�ï9â9à9î9î}v�àGç<áNà6í¶â9æ9îZà6ò�éhá/ò8áBñ�

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�é�æGó<á/ò¶æ � �¶æ6ò9ì u2� îGñ²æ6ßr�߶æ6ò9ì��qá/ò�ñ§â9æ2v�æ9â ��߶æ6ò9ì�êÂàGé9é�~®ñ�æ6ß � â9æ2v�æ9âT�ü/ò¶ô�ôGó �9ç<áBé=â9æ2v�æ9â9îGüJ� prp ߶æ6ò9ì§ï9ô6ß9߶à6ò�é p ò¶à/߶æüNâ9æ2v�æ9âGüJ� prp ï9ô6ß9߶à6ò�é�â�á/ò¶æ�ô4z�ñ<áNô6ò�ãK��â9æ2v�æ9âü/ò¶ô�ôGó�ì9ò8áBó�ôGçGß�ç�æGó<á/ò¶æ6߶æ6ò9ñ�îGü¦�ü�î�üJ� prp é�æGó�à6ì¶âGñ�à6ò¶î2¤�æGçür~�Ã2ü �J� prp v�à9â�áBé=à6ò¶î2¤�æGç�ã Ã_á/ò�ñ�æ2��æGç߶æ6ò9ì�êÂàGé9é�~®ñ�æ6ß � â9æ2v�æ9âT�üNï9ô9àGç�î9æZâ9àGñ9ñ<áNï9æGüJ� prp ߶æ6ò9ì�ï9ô6ß9߶à6ò�é p ò¶à6ß�æüNâ9àGñ9ñ<áNï9ærüJ� prp ï9ô6ß9߶à6ò�é�â�á/ò¶æ�ô4z�ñ<áNô6ò�ãK��â9æ2v�æ9âüNâ9àGñ9ñ<áNï9ædáNî�ò q òJüJ�ü*ÄGüJ� prp é�æGó�à6ì¶âGñ�à6ò¶î2¤�æGçür~�Ã2ü �J� prp v�à9â�áBé=à6ò¶î2¤�æGç�ã Ã_á/ò�ñ�æ2��æGç߶æ6ò9ì�êÂàGé9é�~®ñ�æ6ß � â9æ2v�æ9âT�üNî2¤�æ9æ4z¶îGü¦� prp ߶æ6ò9ì'ï9ô6ß9߶à6ò�é p ò¶à/߶æüNî2¤�æ9æ4z¶îGü¦� prp ï9ô6ß9߶à6ò�é�â�á/ò¶æ�ô4z�ñ<áNô6ò¶î�ãK��î2¤�æ9æ4z¶îüNîGñ9ç<á/ò��Qâ�ál£¶æòý'ÄT�kÄ2þ¥�¼z�ç�æ¼��z¶ô9îGñ;î6߶ô9ôGñGè8á/ò��dî2¤�æ9æ4z¶îGüJ�ü�ýlÃ���Ãlþ�üJ� prp é�æGó�à6ì¶âGñ�à6ò¶î2¤�æGç�ã��JÃ���ÃZï � ï9â9æü u ü �J� prp v�à9â�áBé=à6ò¶î2¤�æGç�ãeîGñ9ç<á/ò����éGß � á/ò � éGß2ã9ã�é�æGó<á/ò¶æ u ñ�àGñ�áNï � ߶æ6òNì��Ââ9æ2v9æ9â,�Jà ���z�çGß �'trt�u ô9â�v�æGç � ã9ã�é�æGó<á/ò¶æ u ñ�à6ñ<áNï � ߶æ6ò9ì��+â9æ2v�æ9â��Jà �J�kp2p ß9ì¶âGñ<á��9ç<áBé}z¶àGç�à6߶æGñ�æBç�î߶æ6ò9ì�êÂî9æGñ y ô6ß9ß�à6ò�é � ç9æGó<ál� � üNî6ß�ô9ôGñBè¶æ6ç�ü �����éGß � á/ò � éGß2ã9ã�é�æGó<á/ò¶æ u ñ�àGñ�áNï � ߶æ6òNì��Ââ9æ2v9æ9â,�Jà ���߶æ6ò9ì�êÂî9æGñ y ô6ß9ß�à6ò�é � ç9æGó<ál� � üNï9ôGàGç�îNæ°�9ç<áBé�ü �J���éGß � á/ò � éGß2ã9ã�é�æGó<á/ò¶æ u ñ�àGñ�áNï � ߶æ6òNì��Ââ9æ2v9æ9â,�Jà ���߶æ6ò9ì�êÉì9ò¶î9æBñ y ôLß9߶à/ò�é � ç�æ6ó<áx� �l���u ñ�ôGç�æ2î � â9ôBñ9ñ<á�ò���ãNã�é�æGó<á/ò¶æ u ñ�àGñ<áLï � ߶æ6òGì��¥â9æ2v9æ9â��¦Ã ����

v�ô�áBé���ä9ó9éGߦÄ�ã9ã!î9ï9à6ò � �¶æ6ò9ì u�� îGñ9æ6ß�� ߶æ6ò9ì ��° º

Page 18: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

prp â9ô9àGé;à6ò¶î2¤�æGç�îó9ç�ô6ß;ñGè¶æQ߶æ6ò9ì�ãò¶ôr�9ôGó��2�9ç<á®é�î°�=߶æ6ò9ì�êó��æGñ � ü/ò¶ô�ôGó �9ç<áBé;â9æ2v�æ9â9îGü � êó�9æGñ�~/ò�ñ �l�¦�ï9ô9àGç�î9ær�2�9ç�áBé}�=߶æ6ò9ì�êó��æGñ � üNï9ô9àGç�î9ædâ9àGñ9ñ<áNï9æGü � êó�9æGñ�~�ò�ñ �l�J��9ç<áBé�ê�ç�æGé<á�ß � ò¶ôr�9ôGó��2�9ç�áBé²î �J�î6߶ô9ôGñGè�ê�ç�æBé<á/ß � ò¶ôr�9ôGó��2�9ç�áBé�î �J�z�ç�ô �r�4z�ê�ç�æBé<á/ß � ò¶ôr�9ôGó��2�9ç�áBé�î2û�à �¦�z�ç�ô �r�Gç�ê�ç�æBé<á/ß � ò¶ôr�9ôGó��2�9ç�áBé�î2û�à �¦�z�ç�ô �r�6ò¶æ9îGñ#ê�ç�æ6é<á/ß � ò¶ôr�9ôGó��2�9ç�áBé�î*û�à �J�é9é�î9ô9â2v�æGç��lz�çGß�êÂî9ïNà6ò � ߶æLò9ì �J�é9é�î9ô9â2v�æGç���ïGç�æ9àGñ�æ trt�u ô9â�v�æGç � é9é²î9ôGâ2v�æ6ç��lz�çGß �J�é9é�î9ô9â2v�æGç�û ¨ àGñGñ�à9ï/è*w<î9æGç y ôBé�æ �'q ñGè¶áNî �J�â�á/ò¶æ2��ê�ç�æ6í¶á/ò�é � ò¶æ2¤���éGß � á/ò � éGß �¦�â�á/ò¶æ2��û ¨ î9ïNà6ò � ߶æ6òNì ���u ñ�ôGç�æ2î � â9ôBñ9ñ<á�ò���ãNãÂîGï9à6ò � ß¹æ6ò9ì���Ä �J� prp é<á/ßî9ï9à6ò�ä9ç<áBé � ߶æ6òGì �J�z�ç�æ9ï9ô6ò�é � çBß�êÂîGï9à6ò � ß�æ6ò9ì �J�â�á/ò¶æ2��û ¨ â�á�ò � é6ß �l� êÂà6ñ9ñ�àGï6è � z�ç�æ9ï9ô6ò�é � ç6ß �¦�~Nî'áNî � ߶æ6ò9ì�êó�9æGñ � üNî2¤�æ9æ4z¶îGü �2�J�áNîrû ¨ á��Gò¶ôGç²æ ��ñ ý ñ��J�áNîrû ¨ ��æGñ � z�ç�æ u ߶ô9ôGñ6è �J�áNîrû ¨ á��Gò¶ôGç²æ ��ñ � ñ��J�áNîrû ¨ ��æGñ � z¶ô9îGñ u ߶ô9ô6ñGè ����

á/ò�ñ���ä9ó9éGߦÄ�ã9ã��9ç<áBé u á�ð�æ �'u z¹à9ï9æP~Bé�á ��ç�æGñGì�çGò � ï9ô9àGç�î9ær�2�9ç�áBé p Ä ��srs á ��

v�ô�áBé���ä9ó9éGߦÄ�ã9ã!î9ï9à6ò�ä9ç<áBé � �¹æ6ò9ì u2� îBñ�æLß�� ߶æ6ò9ì ��ó�ôGç � á/ò�ñ'á��Jà � á s �Gò¶ôr�9ôGó6�2�9ç�áBé�î � á,� � �¥�á/ò�ñ_ò§� �9ç<áBé u á�ð�æ � á �J��9ç<áBé � á � ê�ç�æLí8á/ò²é � ò¶æ2¤�ä9ç<áBé � àGñ9ñ<áNï9æ �J��9ç<áBé � á � û ¨ îGï9à6ò � à6ó�ôGç6ß � üBér�rÄ\ý'�T��Ãlþr��ý'�J��Ãlþ_á/ò�é�æ2��ãrý'��ãL�9érþ2��ý#��ã��Gérþ�ü���ò��pò �r�J�áBó � á ¨ à �߶æ6ò9ì�êÂî9æGñ y ô6ßN߶àLò�é � ç�æBó<á�� � üNî/߶ôGôGñGè�æGç�ü �J�

æ9â9î9æ߶æ6ò9ì�êÂî9æGñ y ô6ßN߶àLò�é � ç�æBó<á�� � üNïNô9à6ç�î9æª�9ç<áBé�ü �J�

î6߶ô9ôGñGè � á � êÉç�æ6í¶á/ò9é � ò¶æ2¤���éGß � á/ò � éGß �J�î6߶ô9ôGñGè � á � û ¨ î9ïNà6ò � ߶æLò9ì �J�߶æ6ò9ì�êÉì9ò¶î9æGñ y ô6ßN߶àLò�é � ç�æBó<á�� �l�J���

v�ô�áBé���ä9ó9éGߦÄ�ã9ã´ß¶à4£¶æ4�¶àGñ9ç<á�� �l��á/ò�ñdò�� �9ç<áBé u á�ð�æ � ò¶ôr�GôGó6�2�9ç�áBé�î �J�|�à6ò�é�â9æ � �¶àBñ � ñr�4z � ç�æGà9â �2� � �_ò¶æ2¤��¶àGñ � ñ��4z � ç�æ9àNâ ���|�à6ò�é�â9æ � ~/ò²é�æ2� u æGñ � á/ò�é � prpòñ á/ò�ñ�æGç<áNôGç ñ á/ò�é�æ2�;î9æGñ

° K

Page 19: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

|�à6ò�é�â9æ � ~/ò²é�æ2� u æGñ � í¶ô6ì � prpòñ í¶ô6ì9ò�é�àGç ��ñ á/ò�é�æ2�;î9æGñé�æGó<á/ò¶æG~/ò�é²æ2� u æGñ�~ � á�ò�é���ò �J�é�æGó<á/ò¶æG~/ò�é²æ2� u æGñr� � í�ô6ì���ò �J�

prp z¶ô�á/ò�ñ�ô4z¶æGç�àGñ�ôGç��!ñr¤�ô�é<á/߶æ6ò¶î�áNô6ò�à9â� û ¨ ç�æGé<á/ß � á/ò�é �l� �xà ��� prp ¤<áBñGè§à6ò�ôGó9ó�î9æGñ;ôGó;ô6ò¶æ_ó9ç�ô6ßprp ñGè¶æ�ï9æ6ò�ñ9ç�à9âdæ9â9æ6߶æ6ò�ñ ��� �T�k� � ê�kq2���J� �T�kû�à � �§û�à ��kq2���J� û�Ã��c� � ��û�à � �kq2���J� �T�è� � � î �¡�kq2���J� Ã��è� � ��û�à ��kq2���J� �T�¥Ã � �§û�à ��8áNæ9âGé2� t�q í��_ò¶æ2¤��8áNæ9âGé2� t�� �9ç<á®é � ò¶ôr�GôGó6�2�9ç�áBé�î �J��� �lü�íJü �J�AprpNM Mí¦û ¨ v�à9â6ì¶æ9î �l� ��� �ì�ê�ç�æ6í8á/ò�é � ò¶æ2¤¼�8áNæ9âGé2� t�� �9ç<á/é � ò¹ôr�GôGó��4�9ç�áBé�î �J�l� �lü�ìJü �2�J�ì¦û ¨ v�à9â6ì¶æ9î �l� ��� �prp á/ò8áBñ<áNà9â�á�ð9àGñ<áLô6ò�ôGó;ñGè¶æ�ç<á��Gè�ñdè¶à6ò�é;î�áBé�æ� ñrv � á/ò�ñ � áNî � Ä �J�� ñrv � ç�æ9à9â � z¶î � Ä �J�ç�æ9à9âè§�\Ã8ê'� p ç�æ9à9â � ò �J�á/ò�é �l� êÂîGñ�àBç9ñ�~/ñ�æGç²àGñ9ôGç � áNî �J�¤Gè8áNâ9æ � á/ò�é �l� ê�áBñ²æGç9àGñ�æ �l�*���z¶î � à � �_è q áNî � à �J�z¶î � Ä � �_è q áNî � Ä �J�í¦û ¨ v�à9â6ì¶æG~/ò9é�æ2� � áBî � à � ��áNî � Ä �r� �_è q è q ó � z¶î �J��

prp á/ò8áBñ<áNà9â�á�ð9àGñ<áLô6ò�ôGó;ñGè¶æQí¶ô6ì9ò�é�àGç � ï9ô6ò�é<áBñ<áNô6ò�îí¶ô6ì �l� êÂîGñ�àBç9ñ�~/ñ�æGç²àGñ9ôGç � áNî �J�¤Gè8áNâ9æ � í¶ô6ì �l� ê�áBñ²æGç9àGñ�æ �l�*���z¶î � à � �_è q áNî � à �J�z¶î � Ä � �_è q áNî � Ä �J�ì¦û ¨ v�à9â6ì¶æG~/ò9é�æ2� � áBî � à � ��áNî � Ä �r� �_ì¦� � z¶î �J��

â�á/ò¶æ2��û ¨ àGñNñ�à9ïLè �kq�� � q ì�� q í �J��v�ô�áBé���ä9ó9éGߦÄ�ã9ã´ß¶à4£¶æ4�¶àGñ9ç<á�� � á/ò9ñ=á ��á/ò�ñdò�� �9ç<áBé u á�ð�æ � á �J�|�à6ò�é�â9æ � �¶àBñ � ñr�4z � ç�æGà9â �2� ���� ê�ç�æ6í8á/ò�é � ò¶æ2¤¼�¶àGñ � ñ��4z � ç�æ9àGâ ���J�|�à6ò�é�â9æ � ~/ò²é�æ2� u æGñ � á/ò�é � prpòñ á/ò�ñ�æGç<áNôGç ñ á/ò�é�æ2�;î9æGñ|�à6ò�é�â9æ � ~/ò²é�æ2� u æGñ � í¶ô6ì � prpòñ í¶ô6ì9ò�é�àGç ��ñ á/ò�é�æ2�;î9æGñé�æGó<á/ò¶æG~/ò�é²æ2� u æGñ�~ � á�ò�é���ò �J�é�æGó<á/ò¶æG~/ò�é²æ2� u æGñr� � í�ô6ì���ò �J�

prp z¶ô�á/ò�ñ�ô4z¶æGç�àGñ�ôGç��!ñr¤�ô�é<á/߶æ6ò¶î�áNô6ò�à9â� û ¨ ç�æGé<á/ß � á/ò�é �l� �xà ��� prp ¤<áBñGè§à6ò�ôGó9ó�î9æGñ;ôGó;ô6ò¶æ_ó9ç�ô6ßprp ñGè¶æ�ï9æ6ò�ñ9ç�à9âdæ9â9æ6߶æ6ò�ñ ��� �T�k� � ê�kq2���J� �T�kû�à � �§û�à ��kq2���J� û�Ã��c� � ��û�à � �kq2���J� �T�è� � � î �¡�kq2���J� Ã��è� � ��û�à �°�I

Page 20: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

�kq2���J� �T�¥Ã � �§û�à ��8áNæ9âGé2� t�q çGè¶î¼��ò¶æ2¤¼�8áNæ9âGé2� t�� �Nç<á/é � á �J�k� �lü/çGè¶îrü ����prpOM MçGè¶îrû ¨ v�à9â6ì¹æ9î ��� �§� ��8áNæ9âGé2� t�q î9ô9â¼��ò¶æ2¤¼�8áNæ9âGé2� t�� �Nç<á/é � á �J�k� �lüBî9ô9ârü ����prpOM Mî9ô9ârû ¨ v�à9â6ì¹æ9î ��� �§� �prp á/ò8áBñ<áNà9â�á�ð9àGñ<áLô6ò�ôGó;ñGè¶æ�ç<á��Gè�ñdè¶à6ò�é;î�áBé�æ� ñrv � á/ò�ñ � áNî � Ä �J�� ñrv � ç�æ9à9â � z¶î � Ä �J�ç�æ9à9âè§�\Ã8ê'� p ç�æ9à9â � ò �J�á/ò�é �l� êÂîGñ�àBç9ñ�~/ñ�æGç²àGñ9ôGç � áNî �J�¤Gè8áNâ9æ � á/ò�é �l� ê�áBñ²æGç9àGñ�æ �l�*���z¶î � à � �_è q áNî � à �J�z¶î � Ä � �_è q áNî � Ä �J�çGè¶îrû ¨ v�à9â6ì¶æP~/ò�é²æ2� � áNî � à � �/áBî � Ä �r� ��è q è q ó � z¶î �J��

prp á/ò8áBñ<áNà9â�á�ð9àGñ<áLô6ò�ôGó;ñGè¶æQí¶ô6ì9ò�é�àGç � ï9ô6ò�é<áBñ<áNô6ò�îí¶ô6ì �l� êÂîGñ�àBç9ñ�~/ñ�æGç²àGñ9ôGç � áNî �J�¤Gè8áNâ9æ � í¶ô6ì �l� ê�áBñ²æGç9àGñ�æ �l�*���z¶î � à � �_è q áNî � à �J�z¶î � Ä � �_è q áNî � Ä �J�î9ô9ârû ¨ v�à9â6ì¶æP~/ò�é²æ2� � áNî � à � �/áBî � Ä �r� ��ì¦� � z¶î �J��

î6߶ô9ôGñGè � á � û ¨ à6ñ9ñ�àNï6è �kq2� � q î9ô9âT� q çGè¶î �J�� á/ò9Þ�� � éGߥ��î¼��î6߶ô9ôGñGè � á � û ¨ â�á/ò � é6ß �l���é9é�î9ô9â2v�æGç�û ¨ àGñGñ�à9ï/è � á/ò   è¶î � î�êTí¶â �k� �páJ��é2z �r  w�Þ �J�é9é�î9ô9â2v�æGç�û ¨ àGñGñ�à9ï/è � á/ò u ô9â � î�ên��â �k� �pá �J��

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�é�æGó<á/ò¶æG~/ò�é�æ�� u æ6ñ�~� |�à6ò�é�â9æ � ~�ò�é²æ2� u æGñ � �_á/ò�é<áJ� á/ò�ñ_ò�� á/ò�ñ§îGñ�æ4z �� ��ô2��~/ò�é<áNï9æNî q á/ò�ñ�æGç<áNôGç¥�_ò¶æ2¤ ��ô2��~/ò�é<áNïNæ9î �á/ò�é<á3ê�ç�æ6í8á�ò�é � á/ò�ñ�æGç<áNôGç �¦�� ñrv � á/ò�ñ � îGñ�æ4z¶î � Ä �J�îGñ�æ4z¶î��§îGñ�æ4z �á/ò�ñ�æGç<áNôGç�û ¨ î9ïGà6ò � àGó9ôGçGß � ü�Ä � �Gé����Né � � �9é����9é � üJ�¥îGñ9æ4z��ÂîGñ²æ4z��Éò¦ûNîGñ�æ�z���ò�û9îBñ�æ4z �r�¦�á/ò�ñ�æGç<áNôGç�û ¨ î9æ6ñ u ñ²æ4z�î � î6ñ�ælz¶î ����

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�é�æGó<á/ò¶æG~/ò�é�æ�� u æ6ñr�� |�à6ò�é�â9æ � ~�ò�é²æ2� u æGñ � �_á/ò�éGí�� á/ò�ñ_ò �� ~/ò�é�æ2� u æGñ�~�ò�é<áBï9æ9î��¤í¶ô6ì9ò�é�àGç � � q ò¶æ2¤�~/ò�é�æ2� u æGñ�~�ò�é<áBï9æNî � î �J�á/ò�éGí�ê�ç�æ6í8á�ò�é � í¶ô6ì9ò�é�àGç ���¦���ô2��~/ò�é<áNï9æNî¤í¶ô2�r�Jà �í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T� � � � �9é��k� � üJ��ò �r�J�

° D

Page 21: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T���9é � � �9é����9é � ü¦��ò��Éò���ò �r���í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T�xà � � �T���9é � üJ��òPû�à �2�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é���à � � �9é����9é � ü¦��ò��Éò���òPû�à �r�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J��

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�á/ò8áBñ � ç�ô � �l�èprp î9æGñGìrz�z�ç�ô ��ô4z¶æGç�àGñ�ôGç�î�ó�ôGç � á/ò�ñ'á��Jà � á s ò¶ôr�9ôGó����9ç<á/é�î � á,� � �¥�á/ò�ñ_ò§� �9ç<áBé u á�ð�æ � á �J�|�à6ò�é�â9æ � ~/ò�é9æ2� u æGñ � á/ò�éJà � prp ñ á/ò�ñ�æGç<áNôGç ñ á/ò�é�æ2�;î9æGñ�ó<á/ò¶æé�æGó<á/ò¶æG~/ò�é�æ4� u æBñ�~ � á/ò�éJÃ��A�9ç<áBé u á�ð�æ � áI�Jà � ��Ä �J�|�à6ò�é�â9æ � ~/ò�é9æ2� u æGñ � á/ò�é�Ä � prp ñ á/ò�ñ�æGç<áNôGç ñ á/ò�é�æ2�;î9æGñ;ï9ô9àGç�î9æé�æGó<á/ò¶æG~/ò�é�æ4� u æBñ�~ � á/ò�é�ÄT�!ò��¥Ã �J�|�à6ò�é�â9æ � �¶àGñ � ñ   �4z � ç�æGà9â �r� � ���ê�ç�æ6í8á/ò�é � ò�æ2¤ª�¶àGñ � ñ   �4z � ç9æ9à9â �r����¦û ¨ ç�æGé<á/ß � á�ò�éJà �l� �qá/ò�é�Ä �l� �ªÃ �J��¦û ¨ î9æGñr��ô   ô4¤�î � î6ß�ô9ôGñ6è � á,�Jà � û ¨ â�á�ò � é6ß �k� êó�9æGñ � á/òGÞ�� u4� îBñ�æ6ß �l� ê���l� êÉ߶àGñ �l� ê ��æGñ2��ô   ô2¤9î �l�2�J��kq � �J� û�Ã��kû�à � � ê'Ä P �¸�kq � �J� û�Ã���� � �?ê�P � �kq � �J� û�Ã���à � �?ê'Ä P ��kq � �J� �T�kû�à � � ê�P � �kq � �J� �T�ª� � �\Ã8ê � �kq � �J� �T��à � �?ê�P ��kq � �J� Ã��kû�à � � ê'Ä P �¸�kq � �J� Ã��ª� � �?ê�P � �kq � �J� Ã���à � �?ê'Ä P ��¦û ¨ ô4z�ñ<á/ß8á�ð9æ �l�¦�|�à6ò�é�â9æ �'� á/òGÞ��2�¹àGñGç<á�� � �r� ��r��ê�ç�æ6í8á/ò�é � ò¶æ2¤ � á/ò9Þ��2�¶àGñ9ç�á�� � q � �r�J�prp ç�æ9îGñ9ç<áNïGñ<áNô/òz�ç�ô �r�Gç � á � êÉç�æ6í¶á/ò9é � ò�æ2¤ � ç�ô �4�¶àGñ9ç<á�� �l�*�J�z�ç�ô �r�Gç � á � û ¨ ç�æ/í8á�ò�é2��àGñNç<á�� �kq �*� ���z�ç�ô �r�Gç � á � û ¨ á/ò¶áBñ �l�J�prp z�ç�ô9â9ô6ò���àGñ<áLô6òz�ç�ô �r�4z � á � êÉç�æ6í¶á/ò9é � ò�æ2¤ � ç�ô �4�¶àGñ9ç<á�� �l�*�J�z�ç�ô �r�4z � á � û ¨ ç�æ/í8á�ò�é2��àGñNç<á�� �kq �*� ���z�ç�ô �r�4z � á � û ¨ á/ò¶áBñ �l�J�prp ò¶æ9îGñ�æGéz�ç�ô �r�6ò¶æ9îGñ � á � êFç�æLí8á/ò9é � ò¶æ2¤ � ç�ô �4�¶àGñ9ç<áx� �l�2�J�z�ç�ô �r�6ò¶æ9îGñ � á � û ¨ ç9æ6í8á�ò�é��¶àGñGç<á�� �kq �r� ���z�ç�ô �r�6ò¶æ9îGñ � á � û ¨ á�ò8áBñ �l�¦���

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�á/ò8áBñ2�¶àGñ9ç<áNïNæ9î ����ó�ôGç � á/ò�ñ'á��Jà � á s �Gò¶ôr�9ôGó6�2�9ç�áBé�î � á,� � �¥�߶à4£¶æ4�¶àGñ9ç<á�� � á �¦�î6߶ô9ôGñGè � á � û ¨ ߶àl£¶æ u2� î6ñ�æ/ß �l����

߶à4£¶æ4�¶àGñ9ç<áx� �l���° �

Page 22: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

â�á/ò¶æ2��û ¨ ߶àl£¶æ u4� îGñ²æ6ß �l�J��v�ô�áBé���ä9ó9éGߦÄ�ã9ãÂîGô9â2v²æ � ç�ô6í�â9æ/ß �l��á/ò8áBñ2�¶àGñ9ç<áLï9æ9î �l�J�á/ò8áBñ � ç�ô � �k�J�áBó � â�á/ò¶æ2��û ¨ â¹á/ò � éGß �l� ên��æBñ u ôGâ2v�æBç ��� ê�é9æ9îNïGç<á�z�ñ<áLô6ò �l� ê�ï9ô/ò�ñ�à¹á/ò¹î � ü t ô6ß�à�á�ò t æ9ï9ô6ßrz¶ô9î�á/ñ<áLô6òJü �r�è���à9î�áNïG~Bñ u ô9â4v�æGç*�dî9ô9â�� yr��u2� �   Þr� � â¹á/ò¹æ2��û ¨ â�á�ò � éGß ��� ê ��æGñ u ôNâ2v�æ6ç �l� �;��à9î�áNïG~Bñ u ô9â2v9æGç �J�trt ~Bñ�æGçr�dé9é�î9ô9â � yr��u2� �   Þr� � îGô9âJ� trt ~Bñ�æGç �J�é9é�î9ô9â�êÂàGñ9ñ�àGï6è � q éGé�î9ôGâ2v²æGç ����

� ç�æ9ï9ô6ò�é¥�2z�ç�æ9ï���â�á/ò¶æ2��û ¨ â�á/ò � éGß �l� êó��æ6ñ � ç�æ9ï �l�J�áBó � z�ç�æ9ï�ê�é�æ9îGïGç<ákz�ñ�áNô6ò �l� êÂï9ôLò�ñ�à9á/ò�î � ü t ô/߶à�á�ò t æ9ï9ô6ßrz¶ô9î9áBñ<áBô6ò¦ü �r�ª�� ç�æ9ï trt ��î9ô9â¼� yr��u2� �   Þr� � z�ç�æNïT� � ç�æ9ï trt��J�î9ô9â�ê�á/ò8áBñ �kq é9é�îNô9â4v�æGç �J��

â�á/ò¶æ2��û ¨ î9ôNâ2v�æ �l�J� prp î9ô9â2v�æ_æ2�Gì�ñ<áNô6ò�î � îGñ�æ6ßá/ò�ñdò8áBñ�æGç�àGñ<áNô6ò¶î � ��ô9ô � æ9à6ò;ï � prp ó�ôGçVáBñ�æGç�àGñ<á�v�æZî9ô9â2v�æGç;îGñ�àGñ<áNîGñ<áNïGîáBó � â�á/ò¶æ2��û ¨ â¹á/ò � éGß �l� ên��æBñ u ñ9àGñ<áLîGñ�áNï9î � ò¶áBñ�æ6ç�àGñ�áNôLò¶îT�Âï �*� prp áBñ�æGç�àGñ<á�v�æî9ô9â2v�æGç�îr�9ô srs ôGó�ôGçGß � ü�6ò�6ò qrqrq î9ô9â2v�æGç ��î9ï9ô/ò�v�æ6çr��æBé_á/ò���ÆGéháBñ�æGç�àGñ<áNô6ò�î qrqrq 6ò�6òJüJ�

ï���ü ü\ã°üdò¶ôGñ�üJ��ò8áBñ�æGç�àGñ<áLô6ò�î �J�u ñ�ôGç�æ2î � â9ôBñ9ñ<á�ò���ãNã�é6ì9ßrz � ì �l�r��� prp éGì9ßrz;ó�ôGç;â9àGñ�æGç�v<áNî6ì¶à9â�á�ð�à6ñ<áNôLòâ�á/ò¶æ y ì�çrv�æNî � ì �l�r�¦��

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�ç�æ9î6ì¶âGñ   æ4z¶ôBç9ñ ����r�ç�æ9à9â}��ä9ó9éGߦÄ�ã9ã�ó �ó� ñrv � ç�æ9à9â � � � ï9ô6ò¶îGñ�ç�æGñGì�çGò Ã8ê ��

ç�æ9à9â}��ä9ó9éGߦÄ�ã9ã´ì¦� �ó� ñrv � ç�æ9àNâ � � � ï9ô6ò¶îGñ�ç�æGñGì�çGò¼��ê ��

u z¶à9ï9æG~Bé}��ä9ó9éGߦÄ�ã9ãb��æGñr��ôr�Gó u z¶àGï9æ9î �l� ï9ô6ò¶îGñ�ç�æGñGì�çGòQò¶ôr�9ôGó��2�9ç<áBé9î ��

��ô9ô � æ9à6ò���ä9ó9éGߦÄ�ã9ã^î9ô9â2v�æ u ì9í u2� îGñ�æ/ß �� á/ò9Þ��r��æ9ïGñ�ô6çr� í�� � á/ò9Þ��r��æ9ïGñ�ôBçr�}���u z¶à9ï9æG~Bé�î4z¶à9ï9æT� u ñ�àGç9ñr��æ9ïGñ²ôGç2��ôGé�æZîGñ�àGç9ñ�� trt�u ô9â2v�æGç2�¶ô6é�æq߶ôGé�æ �� z�çGß �'� á/ò9Þ�� u ô9â4v�æGç � ��î6߶ô9ôGñGè¦�4z�çGߥ� � z�çGß �'� á/ò9Þ�� u ôGâ2v�æ6ç � � �î6߶ô9ôGñGè � î4z¶àGï9æ � û ¨ â�á/ò � éBß �l� êó��æBñ u ô9â2v9æGç � çGß �l�J�

áBó � î4z¶à9ï9æ ¨ à �ª�áBó �r� ߶ôGé�æ2�r� u w�� u����ry Þ¦�l�r¢ tG��Q Qx� ß�ôGé�æ4�r� u w�� u4���*y Þ �r�î6߶ô9ôGñGè¦�4z�çGß�êA߶à4����á/ñ�æBç�àGñ�áNô6ò¹î°� z�ç�æ u ߶ô9ôGñGè �

æ9â9î9æ=áBó � ߶ôGé�æ2�r� u w�� u4�6�ry Þ��2� �*yrÅG�î6߶ô9ôGñGè¦�4z�çGß�êA߶à4����á/ñ�æBç�àGñ�áNô6ò¹î°� z¶ô9îGñ u ߶ô9ôGñBè �

°�H

Page 23: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

æ9â9î9ædó�àGñ�à9â9æGç9ç�ôGç4� ��� ül�²ä9ó9é6ߦÄ�ãNã!î9ô9â2v�æ u ì9í u2� îBñ�æLßJüJ��ü/ß¹ôGé�æ�üJ�+߶ôGé�æ �J�áBó � î6߶ô9ôGñGè¦�4z�çBß�êT߶à2�r��á®ñ�æGç9àGñ<áLô6ò�î}�r� � �ç�æGñGì�çGò;é2zr� �r��u Þ �cprp î9ô9â6ì�ñ<áNô6òdè¶à9îò¶ôGñ'ï6è¶à6ò���æGé�

î6߶ô9ôGñGè¦�4z�çBß�êÂî6ñ�àGçNñGß�ôGé�æª�'îGñ�àGç9ñ �� á/ò9Þ�� u2� îGñ²æ6ß � ç�æ9ïª��â�á/ò �'î6߶ô9ôGñGè � î4z¶à9ï9æ � û ¨ â�á�ò � éGß ��� ê ��æGñ � á�ò9Þ�� u2� î6ñ�æ/ß �l�J�� á/ò9Þ�� u ô9â2v²æGç ��î9ô9â��'î6߶ô9ôGñGè � î4z¶à9ï9æ � û ¨ â�á�ò � éGß ��� ê ��æGñ u ôNâ2v�æ6ç �l���â�á/ò�êÂàGñ9ñ�à9ï/è � ���{í �J�î9ô9â�êÂî9ô9â2v�æ � â�á/ò �J�ç�æGñGì�çGò�é2z �r  w�Þ ��prp î9ô9â6ì�ñ<áNô6òdè¶à9î�ï6è¶à6ò���æGé�

v�ô�áBé���ä9ó9éGߦÄ�ã9ã�ç�æ9î�áBéGì¶à9â �� á/ò9Þ��r��æ9ïGñ�ô6çr� í�� � á/ò9Þ��r��æ9ïGñ�ôBçr�}��� � á/ò9Þ��r��æ9ïGñ9ôGçr�Zç�� u z¶à9ï9æG~Bé�î4z¶à9ï9æ �� � á/ò9Þ�� u2� îGñ²æ6ß � ç�æ9ïª��â�á/ò �'î6߶ô9ôGñGè � î4z¶à9ï9æ � û ¨ â�á�ò � éGß ��� ê ��æGñ � á�ò9Þ�� u2� î6ñ�æ/ß �l�J�â�á/ò�êÂàGñ9ñ�à9ï/è � ���{í �J�â�á/ò�ê�ç�æ9î�áBéBì¶à9â � ç �J��

��ô9ô � æ9à6ò���ä9ó9éGߦÄ�ã9ã�ñ9ç�à6ò¶îGó�æGç �ï9ô6ò¶îGñ � á/ò9Þ��r��æ9ïGñ�ô6çr�Zórv�� u z¶à9ï9æG~Bédó<áJ� � á/ò9Þ��r��æ9ïGñ�ô6çr�Zñrv�� u z¶à9ï9æG~Bédñ<áJ���ô9ô � æ9à6ò�àGé9é��Gñ�ôr�Gñ�� trtr� ç�à6ò¶îGó�æBç2��ôGé�æq߶ôGé�æ ��áBó � ߶ôGé�æ}�r� �r r� � u �9Þ  ��è�áBó � ó<á��r�=ñ<á*û�à � prp z�ç�ô9â9ô6ò���àGñ�áNô6òz�ç�ô �r�4z � ó<á � û ¨ à4zrz�â ��� órv��+ñrv������ � � �r r� � ul� � u Þ t �eàGé9é��Gñ�ôr�Gñ �J�æ9â9î9æ=áBó � ó<á��r��ñ<á,�Jà � prp ç�æ9îGñ9ç<áNïGñ<áBô6òz�ç�ô �r�Gç � ñ<á � û ¨ à4zrz�â ��� órv��+ñrv�� �r r� � u4� � u Þ t �{àGé9é��Gñ�ôr�Gñ �¦�æ9â9î9ædó�àGñ�à9â9æGç9ç�ôGç4� ��� ül�²ä9ó9é6ߦÄ�ãNã�ñ9ç�à6ò¶îGó�æGç�üJ�lü®ó9ç9ô6ßR�9é=ñ�ôS�9é�üJ�+ó<áJ�+ñ<á �J�� æ9â9î9æ=áBó � ߶ôGé�æ��r� �r r� � u �9Þ   ���GÞ u2� Þ t��è�áBó � ó<á��r�=ñ<á*û�à � prp ò¶æ9îGñ�æGé=á/ò�ñ�æGç2z¶ô9â9àGñ�áNô/òz�ç�ô �r�6ò¶æ9îGñ � ó<á � û ¨ à4z*z¶â ��� órv��+ñrv������ � � �r r� � u4� � u Þ t �{àGé9é��Gñ�ôr�Gñ �J�æ9â9î9ædó�àGñ�à9â9æGç9ç�ôGç4� ��� ül�²ä9ó9é6ߦÄ�ãNã�ñ9ç�à6ò¶îGó�æGçò¶æ9îGñ�æGé�üJ�lü®ó9ç�ôLßR�9é=ñ�ôE�9é�üJ�pó<áJ��ñ<á �J��

æ9â9î9ædó�àGñ�à9â9æGç9ç�ôGç�� ��� ül�9ä9óNéGߦÄ-ã9ã�ñ9ç�à6ò¶îGó�æGç߶ôGé�æ �GüJ�lü/ß�ôGé²æT�9é�üJ�{߶ôGé�æ �J�ç�æGñGì�çGò�é2z �r  w�Þ ��

á/ò�ñ���ä9ó9éGߦÄ�ã9ã���æGñ2¢¶ôGç2£ � ç9à6ò¹îGó�æ6ç �ku z¶à9ï9æG~BéZó<áJ� u z¶à9ï9æG~BéZñ<áJ� ï9ô6ò¶îGñ � ç�æ9ï9ô6ò�é2¢¹ôGç2£ � ï9ô6ò¶îGñ�áBó � ó<á��r�=ñ<á*û�à �ç�æGñGì�çGò�z�ç�ô �r�4z � ó<á � û ¨ ��æGñ4¢¶ôGç�£ ���J�

áBó � ó<á��r�=ñ<á,�Jà �ç�æGñGì�çGò�z�ç�ô �r�4z � ñ<á � û ¨ ��æGñ4¢¶ôGç�£ ���J�

ç�æGñGì�çGò¼� ��ç�æ9à9â}��ä9ó9éGߦÄ�ã9ãï��æGñ u ñ�ôGç�à2��æ � ç�àLò¶îGó²æGç �ku z¶à9ï9æG~Bédó<áJ� u z¶à9ï9æG~Bédñ<á � ï9ô6ò¶îGñ�áBó � ó<á��r�=ñ<á*û�à �ç�æGñGì�çGò�z�ç�ô �r�4z � ó<á � û ¨ ��æGñ u ñ�ôBç�à4��æ ���J�

ç�æGñGì�çGò¼� ��° B

Page 24: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

á/ò�ñ���ä9ó9éGߦÄ�ã9ã���æGñ2¢¶ôGç2£ u ôGâ2v²æ �ku z¶à9ï9æG~BéQî4z¶à9ï9æT�qï9ô6ò¶îGñ � ç�æ9ï9ô6ò�é2¢¶ôGç4£ � ï9ô6ò¶îGñ� ç�æGñGì�çGò yr��u2� � y �2� u2� � � ¢ � U�� ��äGó9éGßPÄ � û ¨ îL߶ôNôGñGè � î4z¶à9ï9æ � û ¨â�á/ò � éGß �l� êó�9æGñ u ô9â4v�æGç �l� êó��æ6ñ2¢¶ôBç2£ �l�J�c�

ç�æ9à9â}��ä9ó9éGߦÄ�ã9ãï��æGñ u ñ�ôGç�à2��æ u ô9â4v�æ �ku z¶à9ï9æG~Bé�î4z¶à9ï9æ � ï9ô6ò¶îGñ� ç�æGñGì�çGò yr��u2� � y �2� u2� � � ¢ � U�� ��äGó9éGßPÄ � û ¨ îL߶ôNôGñGè � î4z¶à9ï9æ � û ¨â�á/ò � éGß �l� êó�9æGñ u ô9â4v�æGç �l� êó��æ6ñ u ñ²ôGç9à2��æ �l�¦���

u ñ9ç<á/ò��}��ä9ó9éGߦÄ�ã9ã{ï9ô6ß9߶æ6ò�ñ �l�� ç�æGñGì�çGò§ül��ä9ó9éGߦĤß9ì¶âGñ<á��9ç<áBéñ�æ9îGñ�ü � �)+*-,{U�>-CA4B,qH�CXWY,�./,�>-J9,�564B,�>-J9CA@�CF> R��1� ó¦U � è�¼ � ÷:4 6 :E>-H R��1� ó¦U � è�¼ � ÷:4 �Ãò è 6 ./,�0�.L,95B,�>846CT>-Rd4N*-,@ :80-@F:<J9CF:8>VCA5t:<R-:<CF>=J²*-1856,�>'4B1�r�,d:U�78,Z0218CF>84q5B4N:8.¹b

ÎÏÐ

: °: ° K : °: °Ñ ÒÓ

)+*-,^0�./1E@D1->-R3:<4BCA13>:8>-HZ:<H<·/18CF>84f.L,95B4N./CAJ94BCA13>¤130�,�.B:[4B13./5f:E./,�H�,GU�>-,9HZCF>d0�./1-J9,9H�I�./, �Ãò � èTV#¼ îC1 bc\,ZI-5B,4N*-, CF>846,�.B0�18@F:<46CD1->'5BJ9*-,�Od,Q:E>�:<@A18R3I-, 4B1;r-CA@ACF>-,�:8.pU�>-CA4B,Z,9@A,�Od,�>E4B5�b

ÎÏÏÐ

�V

�� �V�� ° ���

V�� �V

Ñ ÒÒÓ

Q�> ]�:<J94'4N*-,P5B,94B46CT>-R?1[]�>-,95B4B,9HSU�>-CA4B,\,9@A,�Od,�>E4�5G0�:<J9,95�:8>-H :M>�:<4NI�.B:[@t0�./1E@D1->-R3:<4BCA13>�µqCX]�XW˳)�. ³���4G*-,�>�:<@A5B1 �XW˳��hH�1-,95�>-184�:80�0-@AKwCF>�4G*-,�J�:[5B,\1<]U�>-CD46,\H�CXWi,�.L,�>-J9,95�µ5BCF>-J9,h]¸I�>-J946CD1->-5§:8.L,'13>-@AKl¬3>-1<j�>S:<4�J9,�./4N:[CT>S>-13H�,95=13>?:­R3./CAHYbwª+1<j{,978,�.9µ!j{,'J�:8>lI-5B,4N*-,�1-0�,�.B:<4613./5 H�,�./CA78,9HM]�./13O OdI-@A4BCAR3./CAH­]¥13.tU�>-CA4B,;,9@A,�Od,�>E4B5�bVc\,�J�:8>\:[@D561�I-5B,=H�CXWi,�.L,�>841302,�.B:<4B13.L5�µ-j�*-CAJ²*VH�1;>-184�*�:�78, :dJ913>E4BCF>3I-1-I-5�J913I�>E4B,�.B0�:E./4�bc\, R8,�>-,�.6:<4B,Z:�./,9J94G:8>-R3I-@F:8.{5N*�:E0�,9H�O_:<4N./CAs U 1<]�5BCA©9,

U WY�[Z \^]3_ )` Z \^]G_ ãj�*-CAJ9*?H�,GU�>-,95=4N*-,�0�./18@A13>-R3:[4BCA13>�¯hCT>E4B,�.B0218@F:<4BCA13>�bw)+*-,'4N.B:E>-5N0�1856,9H?CD5=I-5B,9H?]Â13.d./,95B4N.LCDJGa4BCA13>�b!c\,QI-56,Q:�0�./1E·®,9J946CD1->'1<]�4/K#02, V�¼ îB1 U � è#¼ � ÷ j�*-CAJ²*VCD5qCF>-CA4BCF:<@ACD©9,�HnrEK'4N*-,QO�:[4N./CAs�bAQ�>4N*-,¤U�>-CD46,Z,9@A,�Od,�>E4 J913>846,9s34+j{,dI-5B,Z5B1-Od,Q0�./1E·®,9J946CD1->=J9@T:[5B5B,95 j+CA4N*hr�I-CD@A4/a�CF>PCF>E4B,�.B0218@F:<4BCA13>:<@AR813.LCD4G*�Od5+CF>-5B4B,�:<HYbc\,�J91->-5B4N.BI-J94Z:'O�:[4N./CAsP1<]e4/K�0�, U � èTVAè�Y�W � jq*-CDJ9*­CD5d:'./,9J94N:8>-R-I-@T:E. J913I�>E4B,�.B0�:8.L4 1<]e4N*-,O�:<4G./CAs�4/K�0�, U � èTV2è�W � ]¥1-.¥Ö × ÖSO�:<4N./CAJ9,95�I-5B,9H­]¥1-. 4N*-,_U�>-CA4B,;H�CXWY,�./,�>-J9,'5B46,�>-J9CD@Âb\)+*-,O�:<4G./CAs U � èTV2èTY�W � CA5^CF>-CA4BCF:<@ACA©9,9H�j+CA4N*�4®j{1qCF>-H�,9sQ56,94B5^CF>-5B4B,�:[H�1<]�1->-,8±Yc\,�I-56,e1->-,+CA4B,�.B:<4B1-.]Â13.!4N*-,t.L1¶jp5c³���:E>-H'13>-,�CA4B,�.6:<4B13.�]Â13.!4N*-,�J91E@TI�O_>-5 ³)�¤À�56,9,+U�R3I�.L, K Ã9b ? 184N*_CD46,�.B:<4B13.L5�:E./,I-5B,9Hn:[4 4N*-,d5N:8Od,�46CTO�,�:E>-Hn*�:�78,d4B1'CA4B,�.B:<4B,_1¶7E,�.+,9s�:<J94B@AKP4N*-,�5N:EOd,�>3I�Odr�,�.¤1<]�>-1-H�,95�b)Ï1§:<J9*-CA,978,�4N*-CA5�µ�j{,CD46,�.B:<4B,d1<78,�.q:<@A@´>-1-H�,95Z13>V4N*-,dJ913:8./56,ZR3./CAH ³)�9b )+*-,dJ913.B./,95N0213>-H�CF>-RCA4B,�.B:<4613.q13>P4N*-,_U�>-,�R3.LCDH.³��CD46,�.B:<4B,951¶7E,�. ,978,�./KP5B,9J91->-H >-1-H�,�CF>\,�:[J²*­H�CF./,9J94BCA13>�b�)+*-,O�:<4G./CAs;OdI-@D46CT0-@AK'CA5+H�13>-,CF>':Zj+:�K�4N*�:[4e4G*-, 5B4B,�>-J9CA@YCA5�:80�0-@ACA,9H�4B1d,978,�.LK§>-1-H�,Z0�18CF>E4B,9H�461rEK'4N*-,dCD46,�.B:<4B13.¹b�)+*-, º × º 5B4B,�>-J9CA@�j+CA4N*hR8CA78,�>hJ9,�>846,�. 0�18CF>E4 CD5 4G*-,�>�:E0�0-@DCA,9H\1->'4N*-,�³��

üJ7

Page 25: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

V1 V2

1

.5

.5

.5.5

.25

.25.25

.25

a�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�aa�a�a�aba�a�a�a�a�a�a�a

stencil

iterator 1 iterator 2

¦�CDR-I�./, K ±b[!./18·/,9J94BCA13>':8>-H�./,9564N./CAJ94BCA13>'jpCD4G*'5B4B,�>-J9CA@A5 I-5BCF>-R U � èTVAè�Y�W � b5BCAH�,qj�*-CAJ²*_J913.B./,95G0�13>-H�5{4B1 4N*-, >-1-H�,t0218CF>84B,9H=4B113>d4N*-, ³)�^56CDH�,8b�c *-CAJ9*�56CDH�,qCA5+./,�:<H;:8>-Hj�*-CAJ9*'CA5+j�./CA4B46,�>'H�,�0�,�>-H�5q13>Vj�*-,94N*-,�.{j{,Z:80�0-@AK'4N*-, 1302,�.B:<4B1-.^1-.{CA4B5�4G.B:8>-5N02185B,9HYb)+*-,d0�./13J9,9H�I�.L, ð·î�ñ<» ó�V#¼ îL®­ñ ó R 0�,�.®]¥13.6Od5�4/j{1 �A�#ºAî�ñ¶» ó[¼ 5G0�,9J9CXU�J�4N:<5G¬85�b�S-CF>-J9,;:;OdI-@A4BCXaR3./CAH'5B1E@D7E,�.!]¥1-.BO 4N*-,ZOd,�>-I=CA5�1->-@DK�:�4B,�O�0-@F:<4B, �A� õMèÁó[¼ :E>-H�:dOdI-@D46CDR-./CAH`0�.L,9J913>-H�CA4BCA13>-,�.CA5t:�4B,�O�0-@F:<4B, V#¼�ó �M�A� µ-j{,Z*�:�78, 4B1;0�:<565t:�.L,G]¥,�./,�>-J9,4B1�4N*-,Z:[J94NI�:<@f:<@AR81-./CA4N*�O�J91->84N:[CT>-,9HCF>�: �A��º2î�ñ¶» ó8¼ 13r-·/,9J94�b�)+*-CA5�CA5+H�1->-,Zj+CA4N*h:8> � èAè � � ç ./,95N02,9J94BCA78,Q:E> �kòX� è J�:<@A@Âb)+*-,q]Â18@A@A1¶jpCT>-R=CF>�0�I-4t0�:E.B:8Od,94B,�.L5�O�:�K;r�,5B13Od, R-I-CDH�,9@ACF>-,d]Â13.!K813I�.{,9s�0�,�.LCTO�,�>84B5 V bc\, R8CA78,4N*-, CF>�0�I-4t0�:E.B:8Od,94B,�.L5{]¥1-.{4N*-,Z>-,9j 5BCFOQI-@F:<4613.{4B1�.BI�>V4N*-, 5N:8Od, ,9s�:8O�0-@A,95 :<5+CF>4N*-,Z0�./,97-CA13I-5qJ²*�:80-46,�.9µ�5B,9, 4N:Er-@D,dü�µ-CF>�0�I-4+U�@A, è�ó ð è Ù>íO� bf)p*-CD5q5B4N:E>-H�:8./H'CA4B,�.B:<46CD7E, 5B18@A78,�./5J�:8>'r2,QI-5B,9HV]¥13.!J913O_0�:8./CA5B13>-5+jpCD4G*'4N*-,q]Â18@A@A1¶jpCT>-R'OdI-@A4BCAR3./CAHP5B18@A78,�./5

��� } ��úªý9��� ( }i��� �iù>-1�1<]�R3./CAH�@A,978,9@A5 KJ913:8./56, @T:[4B4BCAJ9, üO�:<4N.LCDs=4/K#02, ³\:<4*[�4*± 0r�:<5BCAJQO�,94N*-13H ý{1->�· þ .B:<H0�./,9J913>-H�CA4BCA13>-CF>-R�4/K�0�, [^./,9J�Zp13>-,� °8±�J91->878,�.LR8,�>-J9,ZOd13>-CA4B13.p>�:8Od, ý{³\M r-5N)f.6I-, ��,95BCAH�I�:<@� °8±^O_:<s�,�.B./1-. °8b 7<,Ga �

)�:8r-@A,Qü�±�ý{13>�·BI-R3:<46,9H'R3.B:<H�CA,�>E4B5+13>':U�>-CA4B,H�CXWi,�.L,�>-J9,QH�CA5BJ�.L,94BCA©�:<4BCA13>�µ è�ó ð è Ù>íO�)+*-,�CF>�0�I-4�0�:8.B:EOd,94B,�./5q]Â13. OdI-@A4BCAR3./CAH :[5Q:8>PCA4B,�.6:<4BCA78,�5B1E@D7E,�.qCD5dR8CA78,�>`CT>­4N:8r-@A, º µYCF>�0�I-4U�@A, è�ó ð è 'líO� bhc\,§I-56,�:�OdI-@A4BCAR3./CAH ³ta�J9K3J9@A,8µ{CF>-H�CAJ�:<4B,9HlrEK ��¯��Añ ó è ¯1� ó © ��RTR�� ,9g#I�:<@A5°8b M >-I�Odr�,�._4/je1MOd,�:8>-5;:MOdI-@D46CDR-./CAHdc a�J9K-J9@D,PCF>-5B4B,�:[HYbwc\,�I-5B,�1->-,�0�./,Ga�:8>-Hl13>-,0�1E5B4/a�5NOd1-184N*-CF>-R=5B4B,�0h5N02,9J9CÔU�,9H?rEK ðMì óAó �7ð b�)+*-CA5ZCA5Z:<@A5B1'J�:[@D@A,9Hn: ³)��3ë��a�J9K3J9@A,8bc\,�I-5B,�:º WJY 5GOd131E4N*-,�.+:8>-H'13>-,CA4B,�.B:<4BCA13> º WEY :<5q:dJ913:8./56,�R3.LCDHh5B18@A78,�.9b)+*-,qJ913:8./56,+R3./CAH'CA5+:Qü�×=üZR-./CAH;5N02,9J9CXU�,9H�CF> �·îA� ¼ ð ó ñ·� è2è �1� ó bf)+*-CA5�O�,�:8>-5{13>-,�CF>E4B,�./CA13.>-1-H�,Z:8>-H'13>-, H�,9R3./,9, 1[]Y]¸.L,9,9H�13O 13>_4N*-, J913:8.L5B,95B4{R3./CAHYb^ªp,�>-J9,Z4N*-,qJ913:8.L5B, R3./CAH�0�./13r-@A,�OCA5d5B18@A78,9H?,9s�:<J94B@AK r2,;1->-,'5B4B,�0­1<]�4N*-,'I�>Ea�H�:8O�02,9H S�±���CA4B,�.B:[4BCA13>�b\)+*-,_U�>�:<@{5B18@FI-4BCA13>e é�ÐÔÓ/Üf×"*�Ó�Ñëêª��ä9ó9éGß¦Ä p ��æ6ç<áBó ��p

ü�°

Page 26: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

CA5 13r-4N:<CF>-,9HP13>VR3./CAH\>3I�Odr�,�. K À ò�î.î#� © ¼ �Mô ñ ó » ó ñAð Ã9µ�j�*-CAJ9*�O�,�:8>-5 º .L,GU�>-,�Od,�>8465trEKr-CA5B,9J94BCA13>h:8>-H§:�R3./CAH;56CD©9,1<]e° D bfc­,Z*�:�7E,�4B1d5N02,9J9CX]¥K=4N*-, U � èTV2è�W � O�:<4N.LCDs_4®K�0�,p]Â13.!,�:<J²*R3./CAHYµ-4N*-,qR8@A13r�:<@�R3./CAH R�� è#¼ � ÷.è ¯#� ó µ84N*-,qJ913:8./56,95B4^R3.LCDH �#î2� ¼ ð ó © ¼ �Mô R�� è�¼ � ÷.è ¯1� ó :8>-H4N*-,qU�>-,�.pR3./CAH�5+4N*-, 5GOd131E4N*-,�.{1302,�.B:<4B,95+1-> ð�R�î2î è1ç­ó8¼ R�� è#¼ � ÷ è ¯1� ó b

��� } ��úªý9��� ( }i��� �iù>-1�1<]�R3./CAH�@A,978,9@A5 KJ913:8./56, @T:[4B4BCAJ9, ü5Bj{,9,�0-5 > °8µF° FO�:<4N.LCDs=4/K#02, ³\:<4*[�4*± 0r�:<5BCAJQO�,94N*-13H ôªô�QÂ4B,�.0�./,9J913>-H�CA4BCA13>-CF>-R�4/K�0�, [^./,9J�Zp13>-,� °8±�J91->878,�.LR8,�>-J9,ZOd13>-CA4B13.p>�:8Od, ý{³\M r-5N)f.6I-, ��,95BCAH�I�:<@� °8±^O_:<s�,�.B./1-. °8b 7<,Ga �H�13O�:<CF>VH�,9J913O�0�1E5BCA4BCA13>�Od,94G*-13H ³\I-@A4BCAR3./CAHJ9K3J9@A,d4/K#02, R3:8O�O_: °5NOd1-184N*-,�.+O_:<4N./CAs=4®K�0�, ³\:<4*[�4*± 05NOd1-184N*-,�.+r�:[5BCAJQOd,94N*-1-H S�±��J913:8./56, R3./CAH�O�:<4N.LCDs=4/K#02, ³\:<4*[�4*± 0J913:8./56, R3./CAH�r�:<5BCAJQO�,94N*-13H S�±��J913:8./56, R3./CAH�O�:<s=CA4B,�.B:<4BCA13>-5 °

)�:8r-@A, º ±�³`[email protected]>':U�>-CA4B,H�CÔWY,�./,�>-J9,dH�CA5BJ�./,946CD©�:[4BCA13>�µ èÁó ð è '�íJ�)+*-,�CF>�0�I-4dU�@A,_]¥1-. 4N*-,=J913>�·BI-R3:<4B,9H\R-.B:<H�CA,�>84dOd,94N*-1-HnjpCD4G*n:[email protected]�.L,9J913>-H�CA4BCA13>-,�.CA5 R8CA78,�>PCF>P4N:Er-@D, K µ�CF>�0�I-4¤U�@D, è�ó ð è�f íO� b )+*-,�H�CXWY,�./,�>-J9,�4B1=4N*-,�0�./,97-CD1-I-5Z4N:8r-@A,�:8./,d4N*-,@ACF>-,95'j+CA4N* ®�� ð �1� R óMè#ç î2ô :8>-H � ¼ ó �#îMò�ô�� è �Fî·ò �Ãò © è ¯#� ó H�,946,�.BOdCF>-CF>-R?4N*-,�./18@A,�1[]t4N*-,OdI-@A4BCAR3./CAH�:<@AR81-./CA4N*�O 4B1;0-@F:�K8b[�185B56CTr-@A,¤]¸I�./4G*-,�.{,9s#02,�./CFOd,�>E4B5�CF>-J9@FI-H�,8±

¦ :­J913O�0�:8.LCD5613>?1<]QOdI-@A4BCAR3./CAH�:<5':n0�.L,9J913>-H�CA4BCA13>-,�.�:8>-HwOdI-@D46CDR-./CAH :[5§:8> CA4B,�.B:<4BCA78,5B1E@D7E,�.

¦ :dJ91-O�0�:8./CA5B13>V1<]´OdI-@A4BCAR3./CAH\:<5+:�0�./,9J913>-H�CA4BCA13>-,�. :E>-H'184N*-,�.+0�./,9J91->-H�CD46CD1->-,�./5

¦ :dJ91-O�0�:8./CA5B13>V1<]´OdI-@A4BCAR3./CAH\:<5+:dCA4B,�.B:<46CD7E,Z5B18@A78,�.p:8>-H'184N*-,�.{CA4B,�.6:<4BCA78,Z5618@A78,�./5

¦ 4N*-,�>-I�Odr�,�.q1<][email protected]­CD46,�.B:<4BCA13>-5 H�,�02,�>-H�CT>-RP13>V4N*-,dR3./CAHP5BCA©9,�¯'4N*-,�>-I�Odr�,�.q1<]@A,978,9@A5

¦ 4N*-,�>-I�Odr�,�.q1<][email protected]­CD46,�.B:<4BCA13>-5 H�,�02,�>-H�CT>-RP13>V4N*-,d5NOd1-184N*-,�.9µ�r�1E4N*h4N*-,d4®K�0�,1<]^:<@AR813.LCD4G*�O�:8>-H�4G*-, 4B184N:<@Y>-I�Odr�,�.{1<]�5NOd1-184N*-CF>-R�564B,�0-5

¦ 4N*-,{>3I�Odr2,�.Y1<]�OdI-@A4BCAR3./CAHdCA4B,�.B:<46CD1->-5fH�,�02,�>-H�CT>-R13>q4N*-,{>3I�Odr�,�.Ï1<]�5NO�13184G*-CT>-Rq5B4B,�0-5�µ4N*-,!H�CXWi,�.L,�>-J9,�r�,94/j{,9,�>Z0�./,GaY:8>-H�0�18564/a�5NOd1-184N*-CF>-R�À�j+CA4N*Z.L,95N0�,9J94�4B1q4N*-,^,�.B.L13.Y>-13.BO_Ã

¦ 4N*-,q0�,�.�]Â13.BO�:E>-J9,+1<]f>-13>Ea�5BK�O�Od,[email protected]':<5{:Z0�./,9J913>-H�CA4BCA13>-,�.�]Â13.�J913>�·BI-R3:<4B,9HR3.6:<H�CA,�>84B5{:8>-H=184N*-,�.�>-13>Ea�5BK�O�Od,94G./CAJtk ./K-@A1<7�O�,94N*-13H�5!j+CA4N*�>-13>Ea�56K#O�O�,94N./KdH�I-,�461>-13>Ea�56K#O�O�,94N./CAJ'5NOd1-184N*-,�./5'À�,8bÒR�b\S�±�� ÃZ1-. H�CXWY,�./,�>84=>3I�Odr�,�.L5Q1[]t0�./,GaZ:8>-H 02185B4/a5NO�13184G*-CT>-R

ü8ü

Page 27: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

��� } ��úªý9��� ( }i��� �iù>-1�1<]�R3./CAH�@A,978,9@A5 KJ913:8./56, @T:[4B4BCAJ9, ü5Bj{,9,�0-5 > °8µF° FO�:<4N.LCDs=4/K#02, ³\:<4*[�4*± 0r�:<5BCAJQO�,94N*-13H ý{1->�· þ .B:<H0�./,9J913>-H�CA4BCA13>-CF>-R�4/K�0�, [^.L,9Jrôªô� °8±�J91->878,�.LR8,�>-J9,ZOd13>-CA4B13.p>�:8Od, ý{³\M r-5N)f.6I-, ��,95BCAH�I�:<@� °8±^O_:<s�,�.B./1-. °8b 7<,Ga �H�13O�:<CF>VH�,9J913O�0�1E5BCA4BCA13>�Od,94G*-13H ³\I-@A4BCAR3./CAHJ9K3J9@A,d4/K#02, R3:8O�O_: °5NOd1-184N*-,�.+O_:<4N./CAs=4®K�0�, ³\:<4*[�4*± 05NOd1-184N*-,�.+r�:[5BCAJQOd,94N*-1-H S�±��J913:8./56, R3./CAH�O�:<4N.LCDs=4/K#02, ³\:<4*[�4*± 0J913:8./56, R3./CAH�r�:<5BCAJQO�,94N*-13H S�±��J913:8./56, R3./CAH�O�:<s=CA4B,�.B:<4BCA13>-5 °

)f:Er-@D, K ±�ý{13>�·BI-R3:<4B,9H'R-.B:<H�CA,�>8465+j+CA4N*'³\I-@A4BCAR3./CAHP0�./,9J913>-H�CA4BCA13>-,�.9µ è�ó ð è<f íO�¦ 4N*-,H�CXWi,�.L,�>-J9,Qr�,94/j{,9,�>�:[H�H�CD46CD7E,Q:8>-H�OdI-@A4BCF0-@ACDJ�:[4BCA78,;OdI-@A4BCAR3./CAH�:<5q0�./,9J913>-H�CA4BCA13>-,�.¦ 4N*-,H�CXWi,�.L,�>-J9,Qr�,94/j{,9,�>�OdI-@A4BCF0-@ACAJ�:<4BCA78,�:8>-H�>-,9564B,9H�OdI-@D46CDR-./CAH`:[5eCA4B,�.6:<4BCA78,Z5618@A78,�.¦ 4N*-,�>-I�Odr�,�. 1[][email protected]­CD46,�.B:<4BCA13>-5 H�,�02,�>-H�CT>-R­13>'4N*-,dJ913:E./5B,dR3./CAHYµ�4N*-,d5BCA©9,;:8>-H4N*-,d0�./,9J9CA5BCA13>'1[]´4N*-, J91-:8./5B, R3.LCDHh5B18@FI-4BCA13>

¦ 4N*-, J91-O�0�I-4BCF>-Rd4BCFOd,�H�,�02,�>-H�CF>-R'13>�4G*-,�J913:E./5B,+R3.LCDHYµ2H�CÔWY,�./,�>E4{H�CF./,9J94�:8>-H=CA4B,�.B:<4BCA78,5B1E@D7E,�./5+13>h:Z@F:8.LR8,ZJ913:E./5B,qR3./CAH

g È¾É zXh�i8| É fkj;z!   dv�l f+£Gx± >-,d5B,978,�./,d./,95B4N.LCDJ946CD1->']Â13.p4N*-,Q:E0�0-@DCAJ�:<46CD1->P1<]�U�>-CA4B,dH�CXWi,�.L,�>-J9,95ZCA5�4G*-,QRE,913Od,94N./K=1<]^4N*-,H�13O�:[CT>�bSc\,�*�:�78,'H�,978,9@A1302,9H�:PJ913H�,V]¥1-.�:P5B4N.6I-J94NI�./,9HYµ+I�>-CX]¥1-.BO @F:<4B4BCAJ9,PR3./CAHM]Â13.d4N*-,I�>-CA4�56g#I�:8.L,8bfc­,Zj+CA@A@f>-1<j *�:�78,Z:d@A1-13¬;:[4�:H�13O�:<CF>'H�,�.LCD7E,9H�:<5�:�5NI�r-5B,94+1<]�4N*-CA5+5Bg�I�:8./,5N*�:802,9Hn@F:<464BCAJ9,�R3./CAHYb�±p]�J913I�./56,�4N*-,�./,;:E./,;:<@A5B1POd,9J²*�:E>-CD5GOd5d4B1'J9130�,=j+CA4N*\>-1E4Q:<@ACAR3>-,9H:8>-H?,978,�>MJ�I�.L78,9H r213I�>-H�:8./CA,95�µ+Od1-H�CX]¥K-CF>-RnH�CXWY,�./,�>-J9,h5B4B,�>-J9CA@A5'@D1-J�:<@A@AK8µ+r�I-4�4N*-CA5=j+CA@D@ r�,J91<78,�./,9H',9@A5B,9j�*-,�.L,8b)+*-, @ a�5N*�:E0�,9HMH�13O�:<CF>hCA5ZH�,GU�>-,9H?:<5�4G*-,�CF>-CD46CT:[@eI�>-CA4Z5Bg�I�:8./, > 78(�° F � O�CT>-I-5Z1->-,Qg�I�:8./46,�.1<]f4G*-,Z5Bg�I�:8./,8± > 7�(�° F � � > °�Õ8ü�(�° F × > 7�(�°�ÕEü F µ<5B,9,¤U�R3I�./,�I�bc\,Z>-1<j�J913O_0�I-4B,Z:8>':80�0�./1<s-CFO�:<4BCA13>=1<]

:�ͼ� Á � CF>��?Á > 7�(�° F � � > �� (�° F × > 7�( �� F� Á 7 13>����)+*-, J91-H�, ]Â13.{4N*-,Z>-,9j J9@T:[5B5 U�Ø �iôLR f ]¥18@A@A1<j+5�m<bnlä*å4æèç ÑÒÐÒÐ�é�ê¶Õ Ø�Ù<Ó�Û å Õ�Ó�Ñëêª��ä9ó9éGß¦Æ p

ü º

Page 28: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

interior

boundary

¦�CAR3I�./,�I�±�Q�>-H�,9s'5B,94B5p13>=4N*-, @ a�5N*�:E0�,9HPH�13O�:<CF>�b

0 10

1

X−Axis

Y−

Axi

s

0.00729o

0.00

729p

0.00729q

0.00729

r

0.00729s0.00729t

0.00

729u

0.00729

v

0.0146w

0.01

46x

0.0146y

0.0146z0.0146{

0.01

46|

0.0146

}

0.0219~

0.021

9� �0.

0219�

0.0219�

0.02

19�

0.02

92�

0.02

92�

0.0292�

0.0292

0 10

1

¦�CAR3I�./, D ±;S-1E@TI-46CD1->'13>':8> @ a�5N*�:80�,9HPH�13O_:<CF>ÚÜÛLÝßÞ

$.�Pâ "í áBóGò�é�æGó}��ä9ó9éGߦÆr�6è¦�r~ u �G~�� yr� w t Þ tí é�æGó<á/ò¶æ°��ä9ó9éGߦÆr�6è¦�r~ u �G~�� yr� w t Þ tí á/ò¶ï9â6ì�é�æ s ��ä9ó9éGߦÄ�êÉè ¨ï9â9à9î9î°��ä9ó9éGߦÆ�ãïz9ì9í¶â�áNï}��ä9ó9éGß¦Ä prp�� û9î6è¶à4z¶æ� v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æG~/ò�é²æ2� u æGñr� � |�à6ò�é�â9æ � ~�ò�é�æ�� u æGñ � � í¶ô6ì9ò�é�àGç � �qá/ò�ñdò �J�v<áBç9ñGì¶à9â�v�ô�áBé_é�æGó<á/ò¶æG~/ò�é²æ2� u æGñ�~ � |�à6ò�é�â9æ � ~�ò�é�æ�� u æGñ � ��á/ò�ñ�æGç<áNôGç��qá/ò�ñdò��á/ò�ñ;î2�Jà �J��T�í æ6ò�é<áBó

ü K

Page 29: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

)+*-,'J9@F:<5B5 U]Ø �iôLR f CA5�H�,�./CA78,9H?]�./13O 4N*-,�OdI-@A4BCAR3./CAH ]¥13.¤U�>-CA4B,'H�CXWY,�./,�>-J9,95=J9@T:[5B5 U�Ø � ô�R�' bQÂ4 CFO�0-@A,�Od,�>E4B54N*-,�CF>-H�,9sP5B,94B5 H�CXWY,�./,�>846@DK8b�ý�13>-5B,9g�I-,�>84B@AKP4N*-,dCF>-CA4BCF:<@ACD©�:[4BCA13>?0�./1-J9,9H�I�./,95ô ó �7�kò óTõ ò�ô ó�÷ º óMèC� ]¥1-.�4N*-, r�13I�>-H�:E./K;:8>-H ô ó �7�kò ó�õ ò�ô ó�÷ º óMè�õ ]¥13.!4N*-,qCF>846,�./CA13.{>-13H�,95�:E./,1<78,�./@A13:<H�,9HYb!)+*-,�./,Z:8./,Z>-1�]¸I�./4G*-,�.{J²*�:E>-R8,95�>-,9J9,95B5N:E./K8b ÚÜÛLÝßÞ

$.�Pâ oí á/ò¶ï9â6ì�é�æ s ��ä9ó9éGߦÆ�êÉè ¨v�ô�áBé���ä9ó9éGߦÆ�ã9ã�é�æGó<á/ò¶æG~/ò�é�æ�� u æ6ñ�~� |�à6ò�é�â9æ � ~�ò�é²æ2� u æGñ � �_á/ò�é<áJ� á/ò�ñ_ò�� á/ò�ñ§îGñ�æ4z �� ~/ò�é�æ2� u æGñ�~�ò�é<áBï9æ9î��_á/ò�ñ�æGç<áNôGç�� q ò¶æ2¤�~/ò�é�æ2� u æGñ�~/ò9é<áNïGæ9î � Ä ���á/ò�é<á3ê�ç�æ6í8á�ò�é � á/ò�ñ�æGç<áNôGç �¦�á/ò�ñdò¦Ä��_ò p Ä �áBó ��M ò¦Ä � æGç9ç�ôGç2� ��� ül��äGó9éBߦÄ�ãGã�é�æBó<á�ò¶æG~�ò�é²æ2� u æGñ�ü¦�lü�ò ¨ à M ü �J���ô2��~/ò�é<áNï9æNî¤í¶ô2�r�Jà �� ñrv � á/ò�ñ � îGñ�æ4z¶î � Ä �J�îGñ�æ4z¶î��§îGñ�æ4z �í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é����9é � � �9é����9é � üJ��î6ñ�æ4z]��î6ñ�æ4z���òPÄrû9î6ñ�ælz���ò�Ä �r���í¶ô2�r�JÃ8êÂî9æGñ u ñ�æ�z¶î � îGñ9æ4z¶î �J�á/ò�ñ�æGç<áNôGç�ê àGé9é � í¶ô2�r�Jà �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é����9é � � �9é����9é � üJ��î6ñ�æ4z]��ò�Ä���î6ñ�ælz���ò�û9îBñ�æ4z���ò¦ûGîGñ²æ4z �2�J�í¶ô2�r�JÃ8êÂî9æGñ u ñ�æ�z¶î � îGñ9æ4z¶î �J�á/ò�ñ�æGç<áNôGç�ê àGé9é � í¶ô2�r�Jà �J��

v�ô�áBé���ä9ó9éGߦÆ�ã9ã�é�æGó<á/ò¶æG~/ò�é�æ�� u æ6ñr�� |�à6ò�é�â9æ � ~�ò�é²æ2� u æGñ � �_á/ò�éGí�� á/ò�ñ_ò �� ~/ò�é�æ2� u æGñ�~�ò�é<áBï9æ9î��¤í¶ô6ì9ò�é�àGç � � q ò¶æ2¤�~/ò�é�æ2� u æGñ�~�ò�é<áBï9æNî � ÿ �J�á/ò�éGí�ê�ç�æ6í8á�ò�é � í¶ô6ì9ò�é�àGç ���¦�á/ò�ñdò¦Ä��_ò p Ä �áBó ��M ò¦Ä � æGç9ç�ôGç2� ��� ül��äGó9éBߦÄ�ãGã�é�æBó<á�ò¶æG~�ò�é²æ2� u æGñr��üJ��ü/ò ¨ à M ü �J���ô2��~/ò�é<áNï9æNî¤í¶ô2�r�Jà �í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T� � � � �9é��k� � üJ��òPÄ �r���í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é���à � � �9é����9é � ü¦�!ò¦ÄT��ò¦ÄT��ò¦Ä �2�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é����9é � � �9é����9é � üJ��ò¦Ä��JÃ���ò¦ÄT��ò��Fò¦Ä �2�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �9é����9é � � �9é����9é � üJ��ò���ò¦ÄT��ò���ò �r�¦�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � ���J���9é � � �9é����9é � üJ��ò���ò¦û�Ã���ò �r�J�

üJI

Page 30: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J�í¶ô2�r�JÃ8êÂî9ï9à/ò � à6ó�ôGçBß � ü*Ä � �T�xà � � �T���9é � üJ��ò¦û�à �r�J�í¶ô6ì9ò�é�àGç � ê àGé9é � í¶ô��r��à �J��

)+*-,�>-,9j?CF>-H�,9s�56,94B5^:8./,+J�.L,�:<4B,9H�:<5�J913O�0213I�>-H�5{1<]�56,978,�.B:<@�r�1<sdCA4B,�.B:<4B1-./5�bYc­,+J�:8>�>-184^I-5B,4N*-,dH�,GU�>-CA4BCA13>P1<]{4N*-,dH�13O�:<CF>\� :<5 :'r�1-18@A,�:8>POdCF>3I-5�µfr�I-4 j{,dJ�:8>P:<H�HP5B,978,�.B:<@�H�CA5·/18CF>84J913O�0213>-,�>E4B5�b;)p*-,�CF>84B,�.LCD1-.ZCA5CTO_0-@D,�O�,�>84B,9Hl:[5Z4N*-,�H�CA5·/18CF>E4Q5GI�O 1<]{13>-,§.L,9J94N:8>-R8@A,;:8>-H13>-, 5Bg�I�:8./,Z:E>-H�4N*-,Zr213I�>-H�:8./K=CA5t0�I-4+4618R8,94N*-,�.!]�./13O D @ACF>-,95�b)+*-,hCT>�0�I-4P0�:8.B:8O�,94B,�./5d]Â13._4N*-CA5',9s#:8O_0-@D,P1<]QOdI-@A4BCAR3./CAH�1->?4N*-, @ a�5N*�:E0�,9H�H�13O_:<CF> :E./,5BCFOdCA@F:8.q4B1�4G*-,Q0�:8.B:EOd,94B,�./5+I-5B,9HP13>V4N*-,ZI�>-CA4 5Bg#I�:E./, �¶b+)p*-,ZH�CXWY,�./,�>-J9,dCD5q4N*-,d0�:8.B:8Od,946,�.�#î2� ¼ ð ó ñ·� èAè �1� ó µ�5BCF>-J9,�j{,�*�:�78,4B1'I-5B,;:_@T:E./R8,�.qJ913:8.L5B,dR3./CAHP4B1§*�:�78,d5B1-Od,ZH�,9R3.L,9,95Z1<]]�./,9,9H�13O 13>h4N*-,dJ913:8.L5B,95B4qR3./CAHYbQ)p*-,Q1E@DHMü�×\ü_R3./CAHnOdCF>-I-5Z13>-,dg�I�:8./4B,�.pj{13I-@AHn*�:<HP@A,G]¥413>-@AK'r�13I�>-H�:8.LK�>-13H�,95�b

��� } ��úªý9��� ( }i��� �iùJ913:8./56, @T:[4B4BCAJ9, K

)+*-,;.L,95B4 1<]+4N*-,�CF>�0�I-4U�@A,95�.L,�O�:<CF>-5QI�>-J9*�:8>-R8,9H?:8>-H\4N*-,=>-,9j 78,�./56CD1->-5Q:8./,=:<@A5B1�J�:<@A@A,9HèÁó ð è Ù>íJ� À�4N:8r-@A,Pü8Ã]Â13.d4N*-,'J913>�·BI-R3:<46,9HlR3.B:[H�CD,�>E4;Od,94G*-13HYµ èÁó ð è '�íJ� À¥4N:8r-@A, º Ã]Â13.d4N*-,OdI-@A4BCAR3./CAH�CA4B,�.B:[4BCA13>�:8>-H èÁó ð è�f íJ� À�4G:8r-@A, K Ã!]Â13.{4N*-,ZOdI-@A4BCAR3./CAH�0�.L,9J913>-H�CA4BCA13>-,�.9b[�185B56CTr-@A,¤]¸I�./4G*-,�.{,9s#02,�./CFOd,�>E4B5�CF>-J9@FI-H�,8±

¦ :dJ91-O�0�:8./CA5B13>V1<]f4N*-,5B18@A78,�./5�]¥13.!4N*-,dI�>-CA4+5Bg�I�:8./,Q:E>-H'4N*-, @ a�5N*�:802,9HPH�13O�:<CF>�b¦ 4N*-,�>-I�Odr�,�.q1<][email protected]­CD46,�.B:<4BCA13>-5 H�,�02,�>-H�CT>-RP13>V4N*-,dR3./CAHP5BCA©9,�¯'4N*-,�>-I�Odr�,�.q1<]@A,978,9@A5

:8>-Hd1<]�J913I�./5B,{4N*-,{184N*-,�.�,9s�0�,�.LCTO�,�>84B5!1<]�4N*-,{@ACA5B4�]Â13.�4N*-,+I�>-CA4^5Bg�I�:8./,{,9s�:8O�0-@A,8b�)+*-,�0218CF>84*-,�./,dCA5Q>-184 1->-@DKP4N*-,dH�CXWY,�./,�>E4ZR8,913O�,94N./K'1<]{4N*-,�H�13O�:[CT>�µYr�I-44N*-,d@A,95B5Q./,9R-I-@T:E.q5B18@FI-4BCA13>:8>-H'CA4B5+CF>E½�I-,�>-J9,�1->=4N*-, J913>E78,�./R8,�>-J9,8b� uwv¤x¤¢:�²¡¤|�£Bv¤x

Q�>4N*-CA5{./,�0�13.L4fj{,+*�:�78,^H�,�Od1->-5B4N.B:<46,9HQ4G*-,�I-56,e1[]�CA4B,�.B:<46CD7E,�OdI-@A4BCAR3./CAH�,9g�I�:<4BCA13>5B1E@D7E,�./5�]¥1-.4N*-,¤U�>-CD46,�H�CXWi,�.L,�>-J9,�H�CA5BJ�./,94BCA©�:<4BCA13>`1<]e0�:8./4BCF:<@�H�CXWY,�./,�>846CT:[@�,9g#I�:<46CD1->-5�b c\,dCF>84G./13H�I-J9,9HM:�7���A�#�����F� 5BCFOdI-@F:<4B13.�]Â13.!4N*-,�5618@FI-4BCA13>=1<]´:�[�18CA5B5B1->;0�./1-r-@D,�O r�:<5B,9H'13>�U�>-CA4B, H�CXWi,�.L,�>-J9,95�bc\,',9s-4B,�>-H�,9Hl4G*-,�5BCFOdI-@F:<4B13.d461�,9s-,9J�I-4B,P:\OdI-@D46CDR-./CAH 5618@A78,�.9bl)+*-CA5;O�,�:8>84_4B1nO�:8>�:[R8,:8>-H'5B4613./, :Q*-CA,�.B:E./J²*EK=1<]fH�CA5BJ�./,94BCA©�:<46CD1->-5t:8>-H'461QCF>E4N./1-H�I-J9,Z4N.6:8>-5/]Â,�.�1302,�.B:<4B13.L5^]�./13Oå13>-,R3./CAHS4B1\4N*-,P>-,9s34_U�>-,�.':8>-Hl4G*-,�>-,9s-4;J913:E./5B,�.dR3.LCDHYb c\,�CF>E4N./1-H�I-J9,9Hw5B,978,�.B:<@p4®K�0�,95=1<],9g�I�:<4BCA13>'5618@A78,�./5�µ2]¥1-.+4N*-,J913:8./56,95B4+R3./CAHYµ2]¥1-.�:_R3./CAH�CF>h4N*-,d*-CD,�.6:8./J9*8KP:8>-H']Â13.{4N*-,dR8@A13r�:[@,9g�I�:<4BCA13>=5BK-5B4B,�O'b�¦�CF>�:<@A@AK�j{,qje,�.L,Z:8r-@A,Z4B1�I-5B, 4N*-, �A��º2î�ñ¶» ó8¼ ]¸:EOdCA@DK'1[]´OdI-@D46CDR-./CAH ÀÂ:8>-HH�13O�:[CT>VH�,9J913O�02185BCA4BCA13>�à Od,94N*-1-H�5�:�7¶:[CD@F:8r-@A,�CF> �­�±�2�1��� �M� b!c\, J913I-@AH\I-5B,4N*-,QOdI-@A4BCAR3./CAH� é�ÐÔÓ/Üf×"*�Ó�Ñëêª��ä9ó9éGß¦Æ p ��æ6ç<áBó ��p

ü D

Page 31: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

Od,94N*-1-HQr2184N*:<5´:p5B4N:E>-H-a®:<@A13>-,{CD46,�.B:<4BCA78,!5B18@A78,�.�:8>-H�:<5f:q0�./,9J913>-H�CA4BCA13>-,�.�]Â13.�:{J913>�·BI-R3:<4B,9HR3.B:[H�CD,�>E4�Od,94G*-13HnÀÂk .LK3@A1<7§O�,94N*-13H�5qCF>'R8,�>-,�.6:<@FÃ9b? 184N*V4N*-,dH�CD56J�./,94BCA©�:<4BCA13>\:E>-H�4N*-,d4N.B:E>-5/]¥,�.�130�,�.B:[4B13./5qj{,�./,ZCFO�0-@A,�Od,�>E4B,9H\CF>\:�U�>-CA4B,dH�CÔ]�a]Â,�./,�>-J9,Z5B46,�>-J9CD@i]¸:<5G*-CD1->�µ�j�*-CAJ²*h./,9g�I-CT.L,95�13>-@AK�:J913>-5B4G:8>84q:8Od13I�>E4{1<]�O�,�Od13./K;.L,9R3:8./H�@A,95B54N*-,\>-I�OQr2,�.=1<]QI�>�¬->-1¶jq>-5�:8>-H�O�:<4N./CAsS5BCA©9,8b )+*-CA5'CF>wJ913>�>-,9J946CD1->wj+CA4N*w:\564N.BI-J94NI�./,9H:8.B.6:�KEa�@ACF¬<,d5B4B13.B:[R8, 5BJ9*-,�Od,]¥13.q4G*-,QI�>�¬->-1¶jq>-5 J�:8>Pr�,�J913>-56CDH�,�.L,9Hn:<5 :=R3./,�:<4 :<H�7¶:E>84N:[R8,J913O�0�:E./,9H;461Z13./H�CF>�:8.LK;I�>-5B4N.BI-J94GI�./,9H�U�>-CA4B, ,9@A,�Od,�>E4B5�b�c\,�5G*-1¶j{,9H=4N*�:<4!4N*-CA5�:E0�0�./13:<J9*�CA5:<@A5B1;:E0�0-@DCAJ�:8r-@A,�461Q5613Od,Z>-13>Ea®I�>-CA4/a�5Bg�I�:8./,5N*�:802,9H�H�13O�:[CT>-5�b)+*-,hH�CD56J�I-5B5BCA13>S1<] 4N*-,�J91-H�,�:8>-HS5B13Od,�>-I�Od,�./CAJ�:<@ 0�./1302,�./4BCA,95=j�:[5;:<J9J913O�0�:E>-CD,9Hwj+CA4N*>-I�Od,�./CAJ�:<@Y,9s�0�,�./CFOd,�>E4B5 :8>-H�,9s-,�./J9CA5B,4B1;r�, H�1->-,QI-5BCF>-R=4N*-, J913H�,95�b

ü �

Page 32: M^A^2 - Wissenschaftliches Rechnencse.mathe.uni-jena.de/pub/zumbusch/mgFdm.pdf · ` a#bPc5dfefgihMdkjlbmconJhMp5giq#jPbsrktueSv]bwjPayxMekdJcObzhM{}|i~ Chf jPa2eS 5b ]g q#nk c h kjPv

� z;e�z��Yz{xZ¢�z!|> ?A@CBJD¦F MQb�³?b ? .BI�:[5B,94§:E>-H ªb;[�b @ :8>-R84G:8>-R8,�>�b M J913O�0�./,�*-,�>-5BCA78,P5B,94_1<]t461318@A5_]¥1-.

5B18@A73CF>-RZ0�:8.L4BCF:<@3H�CXWY,�./,�>E4BCF:<@�,9g�I�:<4BCA13>-5�Eô CÔWf0�:<J9¬�b]Q�>³ bGô �+*-@A,�>d:8>-H�MQb<){78,9CA4B1�µ,9H�CD4613./5�µ��[�ApLekdJbwnJq#t.�}e�jPa2h>�Mc�q·{����­hf��jPv�qMdeW`7h±h#t¯cZbP{��{����2ckjPdObzq#t.�}q#jPa2eÃp q1jPbwnÃc²b? CF.B¬-*.�:8I-5B,�.¹µf° BEBJD b

> ªq:<J�HJI F c b!ª�:<J9¬-r�I-5BJ²*�b���� tujPbb�( �dOb��O� e±jPa2h�McWq·{���¡�g#g7tÓbznJq#jPbwh·{�c9b S�0�./CF>-R8,�.9µ ? ,�./@ACF>�µ° B HJI�b

> ªq:<J B�K�F c b�ªq:<J9¬3r�I-5BJ9*�b¢�OjzeÃdq#jPbmxMeY�7h#t£�ijPbzhM{ hf�Y¤ q·d¦¥1e��#g qMdOckeY�¨§Mc�jzeÃp�cyh��Y©«ª � q#jPbwh·{�c9bS�0�./CF>-R8,�.9µ ? ,�.L@DCF>�µ�° BJB�K b> @ :E> B�K�F ªZb�[�b @ :8>-RE4N:8>-R8,�>�b þ ,94B46CT>-R?564N:8./4B,9HSj+CA4N*SU�>-CA4B,\,9@A,�Od,�>E4�0�.L18R3.B:8O_OdCF>-RnCF>ô CÔWf0�:<J9¬�b�)Y,9J9*�>-CAJ�:<@8��,�0213./4�S�){¦ ºJº M B�K 7EIJ7�µ S�Q�Z�)¬9�¦»Q�>E]¥13.6O�:<4BCAJ95�µª±q5B@A1�µ

° BJB�K b> @ [ B�K�F ªZbT[fb @ :8>-R84N:8>-RE,�>�:8>-H þ bJ[�,9H�,�./5B,�>�b�S-CTO_0-@D,Z:E>-H�½�,9s3CFr-@A, CF>�0�I-4^H�:<4G:Z*�:8>-H�@ACF>-R

CT>hý +ª+ 0�./18R-.B:8Od5�b�)Y,9J9*�>-CAJ�:<@@�q,�0�13.L4 S�){¦ ºJº M B�K 7JI º µ�S�QkZq)k9�¦ Q�>E]Â13.BO�:<46CDJ95�µ±q5B@A1�µY° BJBLK b> O I�O BJDGFdþ b8c b O I�Odr�I-5BJ9*�bY³\I-@A4BCAR3./CAH�Od,94N*-1-H�5{CF> ô CÔWf0�:<J9¬�bf)Ï,9J²*�>-CAJ�:<@@�q,�0�1-./4cS�){¦ K ü

¦ BJD 7�° D µ�S�QkZq)k9�¦ M 0�0-@DCA,9HP³\:<4N*-,�O�:[4BCAJ95�µ�±q5B@A1�µf° BJBJD b

üJH