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2 � � 2 *�� � # � 2 � 2 *�� � � � A�C1P H @ K @MC a � &[��' �°* % �.- � �.4�576I6 ���� � 4 � � , � � # � ;c2 � 2 *�� � � P_O�O =1G@? @BqFP_C1@ GLW � &[��' � * % �.- � �.4�576 , �8:9<;>=?;�� @BA°A G q ; R GLW C1@ W±U�GLU A°@ W C ;>= O GLK P_C1@ GLW GI= O =?G�o ; RXC1@ GLW�G O ;>= P_C GI=�U�W C G C 9<; A U<H A�OQPIR ; A]P W ^@BA =1;«K P_C ; ^ C G 4 E 5 k Y 5:4�576 , @BA R KMG A ;«KMa =1;«K P_C ; ^ C G C 9�; A�C1P W ^�P = ^ � @ W @BC ;¢k KM; q ;«W C®C 9�;>GI=1a GIS@ W C ;>= O GLK P_C1@ W<Z O GLKMa#W<G qF@BP K A S�GI=°K @ W�; P =�;«KM; q ;«W C1A!rut , w 8�9 @ KM;�k Y 5�4�576I6 @BA°P)A�C1P_C ; q ;«W C =1;>Z P = ^�@ W<ZC 9<; A1q G±G C 9�W�; A1A GIS C 9<; @ W C ;>= O GLK P_C1@ GLWlG O ;>= P_C GI= 5 3 S�8f; R GLW A1@B^ ;>= C 9<;)U A ; GIS P89 ;>=1G C 9 ^ ;>ZI=1;>;@ W C ;>= O GLK P W C � @ W�o�; RXC1@ GLW � n_k Y 5x4�576I6 @eA N @ GLK P_C ; ^°A1@ W R ; @MC�^ G�; A W<G C K @ ; @ W ( * % 5 ��U<= C 9<;>= R GLW A1@B^ ;>= P_C1@ GLWGIS3k Y 5�4�576 , @eq�O K @ ; A�C 9 P_CpC 9<; @ W C ;>= O GLK P_C1@ W<Z]G O ;>= P_C GI=34 E 9 PIAcC G�HJ; P_C KB; PIA�C K @ W<; P = 5 3 Sd8f; qFP XI;C 9 @BA�q�@ W @BqFP K PIA1A U q�O�C1@ GLW P HJGLU C�C 9<;pGI= ^ ;>=�GIS 4 E n C 9<;«W O 9<;>GI=1; q 4�5 4 @eA�P U C G qFP_C1@BR>P KBKBa A1P_C1@eA \�; ^S�GI= P W±a R 9<G @BR ;fGIS @ W C ;>= O GLK P_C1@ GLWsGI= ^ ;>=3S�GI= � E 5 � ;«W R ;In O 9<;>GI=?; q 4�5 4 @BA3P�q GI=1; ZI;«W�;>= P K A�C1P_C ; q ;«W CGIS C 9<; P_O�O =1G�? @Bq�P_C1@ GLW P W ^�A�C1P H @ K @MC a O =1G O ;>= C1@ ; A 5 ��GI= P!q GI=1; ^ ; C1PI@ KM; ^ R GLW±NI;>=1ZI;«W R ; C 9<;>GI=?aIn[8f;=1;>S�;>= C G r �@d � 4�� n t 0 � U a w 5h O =1G qF@BA1@ W<Z C ; R 9QW @BY U<; S�GI= C 9<; R GLW A�C =?U RXC1@ GLW{GIS C 9<; A ; @ W C ;>= O GLK P_C1@ GLW°G O ;>= P_C GI= A 8:9 @BR 9 A1P_C1@eA SjaC 9<;®=?U�KM; A GLU C K @ W<; ^+P HJG�NI; @BA H PIA ; ^ GLW C 9<; P ZIZLKMG q ;>= P_C1@ GLW C ; R 9�W @BY U<; r U�� n UI4 w 8:9 @BR 9+G O ;>= P_C ; AH�a+SVU A1@ W<Z¢W<; @ ZL9±HJGI= @ W<Z)\[W<;�ZI= @B^ ;«W C1@MC1@ ; A�C G¢S�GI= q R G P = A ;®ZI= @B^ qFPIR =1G�;«W C1@MC1@ ; A 5 O 9 @eA°O =?G�N @B^ ; AP W P_C U<= P K P W ^iP U C G qFP_C1@BR 8 P a¢SjGI= R G P = A ; A1OQPIR ; R GLW A�C =?U RXC1@ GLW 5 O 9�; P ZIZLKMG q ;>= P_C1@ GLW C ; R 9�W @BY U<;^ ;>\QW�; ^ GLW C 9�;1� @ W @MC ;]k�KM; q ;«W C�A1OQPIR ; R>P W�HJ;sNI;>= C ; ? H PIA ; ^ � W<G ^�P K �sr 6 U � ,�� w GI=D;«KM; q ;«W C H PIA ; ^r 6 t n ,I4 w 5�� U<= R 9�G @BR ;°GIS`;«KM; q ;«W C1P K P ZIZLKBG q ;>= P_C1@ GLW @BAmq G C1@ N P_C ; ^ H±a C 9<;{W<;>; ^ C G PI^Q^ =1; A1A 9 @ ZL9<;>=GI= ^ ;>= PIR>R U<= PIR a SjGI= C 9<; @ W C ;>= O GLK P_C1@ GLW G O ;>= P_C GI= A�P W ^ 9 PIA HJ;>;«W C 9<; A U<H#o�; RXC GIST=1; R ;«W C =1; A ; P = R 9r �@U n v U n 6 t n ,�6 n U vxw 5

O 9�; A�C1P W ^QP = ^ U A ;°GIS�W�G ^�P K P ZIZLKMG q ;>= P_C1@ GLW r 6 U � ,�� w�C G R GLW A1C =?U RXCDC 9<; @ W C ;>= O GLK P_C1@ GLW)G O ;>= P C GI= A =1; A U�K C1AD@ W C 9<; ^ ;>\QW @MC1@ GLW)GIS C 9�;{=?; A�C = @BRXC1@ GLW PIA�P W @ W�o�; RXC1@ GLW¢G O ;>= P_C GI= 5 O 9 @BA�O =1; A ;«W C1A�C 8fGSVU�W ^�PIq ;«W C1P K O =1GIHQKB; qFA S�GI= ^�@BA1R =?; C1@:9«P_C1@ GLW A GIS 9 @ ZL9<;>=�GI= ^ ;>= ^�@ 8 ;>=1;«W C1@BP K3G O ;>= P_C GI= A 5 � @ = A�C n C 9 @BA

��� ��� � � � ���� N� ��� � � �+�W ���1��� � � ����� 6 tG O ;>= P_C GI=pN @ GLK P_C ; AfC 9<; A�C1P H @ K @BC a O =1G O ;>= C a 5 E�; R GLW ^ KBaIn % � P W ^ %

� P =1;�U�W @BC a P W ^ S�GI= C 9�; � P_O K P R>@BP W G O ;>= P_C GI= �.4 % $ 4 � n C 9<; PIR>R U<= PIR a R GLW ^�@MC1@ GLW � k Y 5�4�576 vL��@eA N @ GLK P_C ; ^ r 6 0 w 5 O 9 @eA =1; A U�K C1A@ W A U<HJG O�C1@BqFP K R GLW�NI;>=1ZI;«W R ;®= P_C ; A{^ U<; C G q ; A 9 ^ ; O ;«W ^ ;«W C°A1R>P K @ W<Z);>=1=1GI= A]@ W C 9<; R G P = A ;®ZI= @B^R GI=1=1; RXC1@ GLW A 5� G P = A ;�VD= @B^ � G P = A ;�V�= @B^ � O 5 0 ; A�C = @BRXC1@ GLW < =1GLKMGLW<Z P_C1@ GLW � GLW�NI;>=?ZI;«W R ;� P�� 3 W ^ ; O ;«W ^ ;«W C 0 ; ^�@BA?R =1; C1@:9«P_C1@ GLW � @ W<; P = � @ W<; P = a 5�� ama� H � O = @BP W<ZLUQK P_C ; ^ E�;>; ^)OQC1A 0 ; ^�@BA?R =1; C1@:9«P_C1@ GLW � @ W<; P = � @ W<; P = a 5��«4m,� R�� h ZIZLKMG q ;>= P_C ; ^ V � h 3 W�o ; RXC1@ GLW 3 W�o�; RXC1@ GLW a 5-,��«4� ^[� h ZIZLKMG q ;>= P_C ; ^ E R>P KM; ^ V � h 3 W�o ; RXC1@ GLW 3 W�o�; RXC1@ GLW a 5 4m,�6

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O P HQKM; 4�5�� @eA�P =1; O =1G ^ U RXC1@ GLW GIS C 9<;{=1; A U�K C1A H�a [ P N±= @MO K @BA]r7v 0 w S�GI= C 9<; C 8 G�ZI= @e^�A GLKeU C1@ GLW)GISC 9<; @BA G C =?G OQ@BR � P_O K PIR ;c; Y U P_C1@ GLW�U A1@ W<Z P�q U�K C1@ ZI= @e^ R R a R KM;c8 @BC 9 C 9�=1;>; O =1; P W ^{O G A1C A1q G�G C 9 @ W<ZSLPIR GIH @ A 8f;>; OQA GLW C 9<;®\QW<;�ZI= @B^ n P W ^ 4 a A 8f;>; OQA GLW C 9<; R G P = A ;�ZI= @B^ 5�h R G q�OQP = @eA GLW+S�GI= C 9<;q U�K C1@ ZI= @e^{R GLW±NI;>=1ZI;«W R ;D= P_C ; Af@BATqFPI^ ;mSjGI= � P���P�C 8 G]ZI= @B^ GxNI;>= A ; Ccq ; A 9 P_O�O =1G PIR 9!U A1@ W<Z P W @ W ^ ; O ;«W ^ ;«W C KMa®ZI;«W<;>= P_C ; ^gR G P = A ; q ; A 9 n � H �:P�R G P = A ; C = @BP W�ZLU�K P = q ; A 9 ZI;«W<;>= P_C ; ^ H�a C = @BP W�ZLU�K P_C1@ W<ZC 9<; A ;>; ^�O G @ W C1A GIS C 9�; P ZIZLKMG q ;>= P_C ; ^�R G P = A ;�ZI= @B^ n � R��fP W�G ^�P K P ZIZLKMG q ;>= P_C1@ GLW @BqFO KM; q ;«W C1P_C1@ GLWGIS C 9<;�V � h S�GI= q U�K P_C1@ GLW P W ^ � ^��fP�A1R>P KM; ^ W<G ^�P K P ZIZLKMG q ;>= P_C1@ GLW @Bq�O KB; q ;«W C1P_C1@ GLW!GIS C 9<;1V � hS�GI= q UQK P_C1@ GLWs8�9<;>=1; C 9<; R G P = A ; A�OQPIR ; qFP_C = @ ? @BA�A1R>P KM; ^]C G C1P XI; C 9<; @ W C ;>= O GLK P_C1@ GLW�GI= ^ ;>= ^ ;>\ R>@ ;«W R a@ W C G PIR>R GLU�W C 5fh ApR>P W)H�;DGIH A ;>=?NI; ^)@ W O P H[KM; 4�5�� n C 9<;>=1; @BAcP_O�O =?; R>@BP HQKM; ^ ;>ZI= PI^�P_C1@ GLW @ W C 9<; R GLW� NI;>=1ZI;«W R ;�= P_C ;�8:9<;«W C 9<; @ W C ;>= O GLK P_C1@ GLW�G O ;>= P_C GI= AfP =1;�GIS 9 ;>=?G C 9 ^ ;>ZI=1;>; 5Th OQO ;«W ^�@ ?3: GLU C K @ W�; A�PH�= @ ;>S3GLW<; ^�@Bq ;«W A?@ GLW P K O =1G�GIS`GIS C 9<; @ W PI^ ; Y U PIR a!GISTW<G ^�P K P ZIZLKBG q ;>= P_C1@ GLW�8:9<;«W P_OQO K @ ; ^ C G C 9<;A GLKBU C1@ GLW�GIS P�A1@Bq�O KB; < G @BA?A GLW ; Y U P_C1@ GLW 5����; )+$3 �!0��=<��� #> �=? �3�&�E$&, �.�! /����$ %h A{^Q@BA1R U A1A ; ^ n C 9�; A1C1P W ^�P = ^lP_O�O K @eR>P_C1@ GLW+GIS�W<G ^QP K P ZIZLKMG q ;>= P_C1@ GLW C ; R 9�W @BY U�; A =?; A U�K C1A{@ W A U<H� G O�C1@Bq�P K R GLW±NI;>=1ZI;«W R ; = P_C ; A 8:9�;«W P_O�O K @ ; ^iC Gg9 @ ZL9<;>={GI= ^ ;>= ^�@ 8 ;>=1;«W C1@BP KT; Y U P_C1@ GLW A]^ U�; C G C 9<;N @ GLK P_C1@ GLW GIS C 9<; @ W C ;>= O GLK P_C1@ GLW)GI= ^ ;>=:=¥U�KM; � k Y 5J4�576 vL� 5�h W P K C ;>=?W P_C1@ NI; C G C 9 @BA�@BAmC 9<;];«KM; q ;«W C1P KP ZIZLKMG q ;>= P_C1@ GLW C ; R 9�W @BY U<;m8�9 @BR 9�;«W P HQKM; A U ATC G]; PIA1@ KMa R GLW A�C =¥U RXC 9 @ ZL9<;>=TGI= ^ ;>= @ W C ;>= O GLK P W C1ATC 9 P_CA1P_C1@BA Sja C 9<; @ W C ;>= O GLK P_C1@ GLW�GI= ^ ;>=m=?U�KM; 5O 9�; O =1G O G A ; ^gP KMZIGI= @MC 9 q @BA H PIA ; ^ GLW C 9<;�S�U A?@ GLW GISf;«KM; q ;«W C1A�@ W C G qFPIR =1G�;«KM; q ;«W C1A 8 @MC 9 P

6 U ����� ���"���I� � � ����� ���������"��� � �� m �������� � ��� ��� � �W����A U<H A ; Y U�;«W C�^ ;>\QW @MC1@ GLW�GIS C 9�; R G P = A ;fZI= @B^�C G O GLKMGIZIa P W ^ H PIA1@eA SVU�W RXC1@ GLW A 5 O 9 @BA�q ; C 9<G ^°@BAdP_OQO K @ ; ^=1; R U<= A1@ NI;«KMa C G{ZI;«W<;>= P_C ; C 9<;�9 @ ;>= P = R 9±a�GIS R G P = A ; A�O[PIR ; A 5�h =1;>N @ ;>8 GISd;«KM; q ;«W C1P K P ZIZLKMG q ;>= P_C1@ GLW@BA Z @ NI;«W�H�a � 9 P W #� �6 r7v U w 8:9<G P K A G O =?G O G A ; P K C ;>=¥W P_C1@ NI;p;«KM; q ;«W C1P K P ZIZLKMG q ;>= P_C1@ GLW P KMZIGI= @BC 9 qFA 5

3 W�GLU<= P_O�O =1G PIR 9 C G°;«KB; q ;«W C1P K P ZIZLKMG q ;>= P_C1@ GLW n C 9�; ^ = @ N @ W<ZDS�GI= R ;:HJ;«9 @ W ^]C 9�; P ZIZLKMG q ;>= P_C1@ GLW@BApC 9�;s=1; ^ U RXC1@ GLW�GIS q ; A 9 P W @eA G C =1G O a!8:9 @BR 9 HJ; R G q ; A�@Bq�O GI= C1P W C K P_C ;>= @ W C 9<; ^�@BA?R =1; C1@:9«P_C1@ GLW GISR GLW±NI; RXC1@ GLW� ^�@ 8 U A1@ GLW C a O ;�; Y U P_C1@ GLW A�r 0 n t � w 5 O 9 @BA°@eA 9<Gx8 ;>NI;>=«n KM; A1A{@Bq�O GI= C1P W C S�GI=��`k [ ^�@BA R =1; C1@:9«P_C1@ GLW A GIS @BA G C =1G OQ@BR ;«KBK @MO�C1@BRfO =1GIH[KM; qFA 5 � W<; @eq�O GI= C1P W CT^Q@BA�C1@ W RXC1@ GLW�H�; C 8 ;>;«W C 9 @BATO =1G O G A ; ^q ; C 9<G ^ P W ^lC 9 P_CF^ ; A?R = @ HJ; ^ H±a � 9 P W �� �6 r7v U w:@BA�C 9 P_C�C 9<; R G P = A ; q ; A 9�;«KB; q ;«W C1A�P =1;)W<G CR GLW±NI;>= C ; ^ @ W C G A�C1P W ^�P = ^ ;«KB; q ;«W C1A H�a P =?; C = @BP W<ZLU�K P_C1@ GLW�H[U CfP =?;�ZI;«W<;>= P K @:9 ; ^�O GLKMa�ZIGLW A SjGI= q ; ^H�a C 9�; P ZIZLKMG q ;>= P_C ; ^ \QW<; q ; A 9);«KM; q ;«W C1A 5 O 9 @BAp@BA ; A1O ; R>@eP KBKMa P_C�C = PIRXC1@ NI; @ W v b�H�; R>P U A ;sGIS C 9<;R G q�O K @BR>P_C ; ^ =?U�KB; A 8:9 @BR 9 qFP a HJ; @ W±NIGLKMNI; ^ S�GI= C 9<; =1; C = @BP W<ZLUQK P_C1@ GLW r7v U w 5 � W<; ^ = P 8mH PIR XlGISC 9 @BA S�GI= q U�K P_C1@ GLW @BAfC 9 P_CcC 9<; A U OQO GI= C S�GI= C 9<;�H PIA1@BA SVU�W RXC1@ GLW Af^ ;>\QW�; ^ GLW C 9�; A ; qFPIR =1G�;«KM; q ;«W C1A@BA K P =?ZI;>= C 9 P W A1C1P W ^�P = ^gC = @eP W<ZLU�K P =�;«KM; q ;«W C1A 5 O 9 @BA�P KMZIGI= @MC 9 q P K A G)9 PIA�C 9<;FS�; P_C U<=1; C 9 P_CsC 9<;=1; A UQK C1@ W<Z R G P = A ; A�OQPIR ; @BA W�; A�C ; ^ 5 O 9 @BA�@eq�O K @ ; A3C 9 P_CTC 9�; @ W C ;>= O GLK P W C1A�P U C G qFP_C1@BR>P KBKMa A1P_C1@BA Sja C 9<;A�C1P H @ K @MC a R GLW ^�@MC1@ GLW � k Y 5[4�576I6 �sr �@U n v U n ,I4 w 5������� � ��� �� � � � �� ��� ����� ��� ���

Macroedge

Coarse Grid Node

Macroelement

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Macroelement Macroedge

Fine Grid nodeSupport Coarse Grid Node

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CoarseNode

Φ = 0 0

Φ = 1/3 0Φ = 2/3 0

Φ = 1 0

X 0

Φ = 0 0

Φ = 0 0

Φ = 0 0Φ = 0 0

Φ = 1/2 0

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:F�TE �sP KMZIGI= @MC 9 q 8:9 @BR 9 @BA ; A1A ;«W C1@BP KeKMa P VD=1;>; ^ a P KMZIGI= @MC 9 qS�GI=mZI= P_O 9 C = P NI;>= A1P K 5 O 9<; ^ ;>\[W @MC1@ GLW®S�GI= C 9<; :F�TE P KMZIGI= @MC 9 q r Um, w@eAK �±~`�|j�x|��[� $ �����*��� A&���� � ����� ��� � �� � ������� � �7� �# ������ �6 ;/$�� ����� 3 � ��� �>��$�9V����& ���� 9 �� ;�� �� $�����$ <� ������� � � ���% �� � � � $�@V�0; $���9/; &#;% ��$ <������� � ������ � � � &/ ������ $��(��9N���V!�# ������ � ���< $�� > �9?=$�@V�0; $��79/;4�&#;/��($ <����(������� � � @V� � ��� ��P �A��� � "��> �� �� �� � � & 6 ���� +

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0 10 20 30 40 50 6010

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