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Primal Formulation 43 DGFEM Primal Formulation Find u h V h such that A h (u h , v h )= Ω fv h dx v h V h , where A h (u h , v h ) = κT h κ u h · v h dx + FF h F ([[ˆ u - u h ]] ·{{h v h }} - {{ ˆ s}}· [[v h ]])ds + FF I h F ({{ˆ u - u h }}[[h v h ]] - [[ˆ s]]{{v h }})ds, where ˆ u u(u h ) and ˆ s s(u h , h u h ).

Primal Formulation - lebesgue.fr · AAAD/XicbVLZattAFJXjLqm7JGkf+3JpVHDAMXb60AUC6UKTh0JSGicByxXj0cgaPJpRZklshKBf06fS135K6d

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Primal Formulation

43

DGFEM Primal Formulation

Find uh � Vh such that

Ah(uh, vh) =

�fvhdx �vh � Vh,

where

Ah(uh, vh) =�

��Th

��uh · �vhdx

+�

F�Fh

F([[u � uh]] · {{�hvh}} � {{s}} · [[vh]])ds

+�

F�FIh

F({{u � uh}}[[�hvh]] � [[s]]{{vh}})ds,

where u = u(uh) and s = s(uh, �huh).<latexit sha1_base64="wVZqoKavWqxK3TMXRb/RKwGWQRI=">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</latexit><latexit sha1_base64="wVZqoKavWqxK3TMXRb/RKwGWQRI=">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</latexit><latexit sha1_base64="wVZqoKavWqxK3TMXRb/RKwGWQRI=">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</latexit><latexit sha1_base64="wVZqoKavWqxK3TMXRb/RKwGWQRI=">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</latexit>

Numerical Fluxes

44

• Consistency:u(v) = v, s(v, �v) = �v

for all smooth v which satisfy the Dirichlet boundary conditions.<latexit sha1_base64="7mTirOqeePOmsqMToE+NtTSNiSM=">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</latexit><latexit sha1_base64="7mTirOqeePOmsqMToE+NtTSNiSM=">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</latexit><latexit sha1_base64="7mTirOqeePOmsqMToE+NtTSNiSM=">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</latexit><latexit sha1_base64="7mTirOqeePOmsqMToE+NtTSNiSM=">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</latexit>

• Galerkin Orthogonality: For u � H2(�), we have

Ah(u � uh, vh) = 0 �vh � Vh.

[This automatically follows from the consistency of u and s.]<latexit sha1_base64="herwaWxuHdbvh623G5KspH/Rjpo=">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</latexit><latexit sha1_base64="herwaWxuHdbvh623G5KspH/Rjpo=">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</latexit><latexit sha1_base64="herwaWxuHdbvh623G5KspH/Rjpo=">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</latexit><latexit sha1_base64="herwaWxuHdbvh623G5KspH/Rjpo=">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</latexit>

• Conservation: The numerical fluxes are conservative if they are single valued, i.e.,

[[u]] = 0, [[s]] = 0.<latexit sha1_base64="gg4Eyg909B5swL32FQQo3elZhEQ=">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</latexit><latexit sha1_base64="gg4Eyg909B5swL32FQQo3elZhEQ=">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</latexit><latexit sha1_base64="gg4Eyg909B5swL32FQQo3elZhEQ=">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</latexit><latexit sha1_base64="gg4Eyg909B5swL32FQQo3elZhEQ=">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</latexit>

Interior Penalty Methods

45

• Given � � [�1, 1], we define

u = u(uh) =

�{{uh}} + 1+�

2 n+ · [[uh]] on F � FIh ,

(1 + �)uh on F � FBh ,

s = s(uh, �huh) =

�{{�huh}} � �[[uh]] on F � FI

h ,�uh � �uhn on F � FB

h .<latexit sha1_base64="RlZfoPRv7CHgO/OGqjDliNj1uMQ=">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 d2mr9+Ax5+xeg=</latexit><latexit sha1_base64="RlZfoPRv7CHgO/OGqjDliNj1uMQ=">AAAFP3icnVTLbtNAFHVDgBIebWGDxOaKFpSqbhSXBbBAqgBRFiyKILRSJlTj8XUyZDy2ZsYlkWX4GrbwD/wFf8AKsWXH2E6rOi0CMdJIZ859nqtr+4ng2nS73xYa55rnL1xcvNS6fOXqtaXlletvdJwqhj0Wi1jt+1Sj4BJ7hhuB+4lCGvkC9/zxk8K+d4hK81i+NtMEBxEdSh5yRo2lDlYaN4mMuQxQGshI4MMa8VMh0KzlsMMPUVrCjNBQIFxCf9NzvcGaC+8RAgxtzRbxkbbIiJoszeERzFA7PRitF0+BoSEZFG5DLjOqFJ3mmRB5i0SHIrNuOWwACRVlmbdRlcqzrRxImmYyf2ttLIgNkHdRUnnfBRL58SSDWEIOH54VfdlrQspQuwBEyhgIabWP0q2DjTs7zA/UtIprEZTBrL0WUXw4Mh3bNRbijhUWPen8WObsWWh1iaS+oLZOKfwflJ8MyGETiObDiP6HzipRqfE4S/GoBvgX3Z0/6z5YXu12uuWB08CbgVVndnbtLi2RIGZpZHeJCap13+smZmCTG84E2vSpxoSyMR1i30JJI9SDrNzhHO5YJoAwVvbaXSzZkxEZjbSeRr71jKgZ6XlbQZ5l66cmfDDIuExSg5JVhcJUgImh+CAg4AqZEVMLKFPc9gpsRO0+GvvZ1KpIzrBY1JqSbKhoMuJsUmcp5md2WCMZFeyUWymyRibaKM7G2rVgU8YBulAggxNToXtBPaAcQoJzuSepFWCj51hhJkZRS2o0EeWyGH+2w4WAV1TqI94mLAztp3zIjXZf2N+HdHcU4ni95lzO2TbAVKxzu0Le/MKcBr2tzsOO93JrdfvxbJcWnVvObafteM59Z9t57uw6PYc1PjY+NT43vjS/Nr83fzR/Vq6NhVnMDa d2mr9+Ax5+xeg=</latexit><latexit sha1_base64="RlZfoPRv7CHgO/OGqjDliNj1uMQ=">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 d2mr9+Ax5+xeg=</latexit><latexit sha1_base64="RlZfoPRv7CHgO/OGqjDliNj1uMQ=">AAAFP3icnVTLbtNAFHVDgBIebWGDxOaKFpSqbhSXBbBAqgBRFiyKILRSJlTj8XUyZDy2ZsYlkWX4GrbwD/wFf8AKsWXH2E6rOi0CMdJIZ859nqtr+4ng2nS73xYa55rnL1xcvNS6fOXqtaXlletvdJwqhj0Wi1jt+1Sj4BJ7hhuB+4lCGvkC9/zxk8K+d4hK81i+NtMEBxEdSh5yRo2lDlYaN4mMuQxQGshI4MMa8VMh0KzlsMMPUVrCjNBQIFxCf9NzvcGaC+8RAgxtzRbxkbbIiJoszeERzFA7PRitF0+BoSEZFG5DLjOqFJ3mmRB5i0SHIrNuOWwACRVlmbdRlcqzrRxImmYyf2ttLIgNkHdRUnnfBRL58SSDWEIOH54VfdlrQspQuwBEyhgIabWP0q2DjTs7zA/UtIprEZTBrL0WUXw4Mh3bNRbijhUWPen8WObsWWh1iaS+oLZOKfwflJ8MyGETiObDiP6HzipRqfE4S/GoBvgX3Z0/6z5YXu12uuWB08CbgVVndnbtLi2RIGZpZHeJCap13+smZmCTG84E2vSpxoSyMR1i30JJI9SDrNzhHO5YJoAwVvbaXSzZkxEZjbSeRr71jKgZ6XlbQZ5l66cmfDDIuExSg5JVhcJUgImh+CAg4AqZEVMLKFPc9gpsRO0+GvvZ1KpIzrBY1JqSbKhoMuJsUmcp5md2WCMZFeyUWymyRibaKM7G2rVgU8YBulAggxNToXtBPaAcQoJzuSepFWCj51hhJkZRS2o0EeWyGH+2w4WAV1TqI94mLAztp3zIjXZf2N+HdHcU4ni95lzO2TbAVKxzu0Le/MKcBr2tzsOO93JrdfvxbJcWnVvObafteM59Z9t57uw6PYc1PjY+NT43vjS/Nr83fzR/Vq6NhVnMDa d2mr9+Ax5+xeg=</latexit>

• Here, � denotes the interior penalty parameter:

� = C�p2

h, C� > 0.

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Interior Penalty Methods

46

Interior Penalty Methods

Given � � [�1, 1] , find uh � Vh such that�

��huh · �hvhdx �

F�Fh

F{{�huh}} · [[vh]]ds

+��

F�Fh

F{{�hvh}} · [[uh]]ds +

F�Fh

F�[[uh]] · [[uh]]ds =

�fvhdx

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sha1_base64="ovTcFu3iX3LZhMa9Pb8t+ZymJZ0=">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</latexit><latexit sha1_base64="ovTcFu3iX3LZhMa9Pb8t+ZymJZ0=">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</latexit>

• � = �1 - Symmetric Interior Penalty Method (SIP).

• � = 1 - Nonsymmetric Interior Penalty Method (NIP).

• � = 0 - Incomplete Interior Penalty Method (IIP).<latexit sha1_base64="/0wkMTik5KWKqulNbvHgY6u9EMk=">AAAEAnichVNLbxMxEHa6PEp5NIUjF4sUKZWSaLc9AAekqiCVSFAFhdBKSRQ53klixWsv9ixKWOXIr+GEuPJHkPgxeDcBsUkQc/r0zXzjeXkYS2HR93+Udrxr12/c3L21d/vO3Xv75YP7761ODIcO11KbqyGzIIWCDgqUcBUbYNFQwuVw+iLzX34EY4VW73AeQz9iYyVGgjN01KD8szeEsVCpQIjEJ1js9TJED3s4AWT0Oa0Hh7RO2/MoAjSC06ZCMEIb2gLFJM7pG8CJDmm13WwdNTbkufpCK/v/BBdbE/hZgqbiOoolIPxb3lzKQYV/mhmUK37Dz41ugmAFKmRlrcHBzn4v1DyJQCGXzNpu4MfYT5lBwWU2ncRCzPiUjaHroGIR2H6ar2FBHzsmpCNX20grpDn7tyJlkXVzGLrIiOHErvsycpuvm+DoaT8VKk4QFF8+NEokRU2zndJQGOAo5w4wboSrlfIJM4y7URVfUYLDyDkKnaRjw+KJ4LMiy2CxtcICyZnkG2F5kwUyttnup7bmQF3pEGo0QwgzXKKTsCjIhxDDWu5Z4hpw6jVW4gwNc6QFjJhQ2fjTcyElbTNlf/MuYeaovhRjgbb22v0AVTs3ANOjQnA+Z1cAN9pmJxSsH8wm6Bw3njWCt8eV07PVLe2Sh+QRqZKAPCGn5BVpkQ7hpbPSpPShZLzP3hfvq/dtGbpTWmkekIJ5338BaOpS9A==</latexit><latexit sha1_base64="/0wkMTik5KWKqulNbvHgY6u9EMk=">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</latexit><latexit sha1_base64="/0wkMTik5KWKqulNbvHgY6u9EMk=">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</latexit><latexit sha1_base64="/0wkMTik5KWKqulNbvHgY6u9EMk=">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</latexit>

Arnold, Brezzi, Cockburn, & Marini 2002

Adjoint Consistency

47

Adjoint Consistency

Given � � L2(�), let z � H2(�) be the analytical solution of the dual/adjoint problem

��z = � in �, z = 0 on ��.

Then the discretization is adjoint consistent if and only if

Ah(v, z) =

��vdx �v � H2(Th),

where H2(Th) =�v � L2(�) : v|� � H2(�), � � Th

�.

<latexit sha1_base64="OlqcpoQIiAY+3dUY8KvtLi/Qk4M=">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</latexit><latexit sha1_base64="OlqcpoQIiAY+3dUY8KvtLi/Qk4M=">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</latexit><latexit sha1_base64="OlqcpoQIiAY+3dUY8KvtLi/Qk4M=">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</latexit><latexit sha1_base64="OlqcpoQIiAY+3dUY8KvtLi/Qk4M=">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</latexit>

Lemma 1

The above discretization is adjoint consistent, if and only if, the numericalfluxes u and s are conservative.

<latexit sha1_base64="So62aYR3KEw7fh3r1U6BuScSBJc=">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</latexit><latexit sha1_base64="So62aYR3KEw7fh3r1U6BuScSBJc=">AAADt3icbVLLbtNAFJ0kPEp4NIUlQhqRAkUyUVIWPFYVIJUFiyIaWimOwmR8nQwZz5iZ6yjBykfweXwJW66dIOGkI1k6Pvfc9x2nWnnsdn/X6o1r12/c3LvVvH3n7r391sH9r95mTkJfWm3d5Vh40MpAHxVquEwdiGSs4WI8e1/YL+bgvLLmHJcpDBMxMSpWUiBRo9av0FhlIjDIz6fAxdjOgUfKSweofpYirjwX0XeSIZfWeCqK5AFXMRcm4tboJeGAI/mbLAFHwTUPQx7rbAGeH4ZTgXm2Oizl678wy3K/KigHZVBwc0o2h05z1Gp3O93y8V3Q24A227yz0UF9P4yspMwGpRbeD3rdFIe5cKikhlUzzDykQs7EBAYEjUjAD/Nydiv+hJiIx9bRVzZI7P8euUi8XyZjUiYCp37bVpBX2QYZxq+HuTJpRuOS60RxpjlaXiyChuxAIs0uUkI6RbVyORVOSKR1VbIYJSEmQ6WTfOJEOlVyUWUFrK6ssELSfuSOrGyyQqYeaZczHxB4YWwEAS8QwgLX6GVUdSiHkMJW7EVGDZD3FqtxgU4Q6QEToUwx/vxUac2/COP/8RSwMBx9UBOFPvhER2KCUwcwe14Rl3OmAqSzfkUn1Ns+mF3QP+686fQ+H7dP3m1uaY89ZI/ZEeuxV+yEfWRnrM8k+1N7VHtae9Z42/jWiBvTtbRe2/g8YJXX+PEXyQY7zQ==</latexit><latexit sha1_base64="So62aYR3KEw7fh3r1U6BuScSBJc=">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</latexit><latexit sha1_base64="So62aYR3KEw7fh3r1U6BuScSBJc=">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</latexit>

Adjoint Consistency

48

Arnold, Brezzi, Cockburn & Marini 2002, Prudhomme, Pascal, Oden & Romkes 2000, Harriman, H., Senior & Suli 2003

Stability �u � uh�H1(�) �u � uh�L2(�) |J(u) � J(uh)|SIP C� > Cmin

� > 0 O(hp) O(hp+1) O(h2p)

IIP C� > Cmin� > 0 O(hp) O(hp+1) (p odd) O(hp+1) (p odd)

O(hp) (p even) O(hp) (p even)

NIP C� > 0 O(hp) O(hp+1) (p odd) O(hp+1) (p odd)O(hp) (p even) O(hp) (p even)

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Adjoint Consistency

49

Symmetric IP Scheme Nonsymmetric IP Scheme

10−1

100

10−15

10−10

10−5

1

8

1

6

1

4

12

|J(u)−J(u

h)|

h

p = 1

p = 2p = 3

p = 4

10−1

100

10−15

10−10

10−5

1

6

1

4

12

|J(u)−J(u

h)|

h

p = 1

p = 2p = 3

p = 4

p = 5, 6

DGFEM for Diffusion

50

Babuska & Zlamal (1973)(Symmetric) Interior Penalty Method; Douglas \& Dupont (1976)Bassi & Rebay (1997)Bassi, Rebay, Mariotti, Pedinotti & Savini (1997)Local DG Method; Cockburn & Shu (1998)Brezzi, Manzini, Marini, Pietra & Russo (1999)Baumann & Oden (1999)(Non-Symmetric) IP Method; Riviere, Wheeler & Girault (1999)Brezzi, Manzini, Marini, Pietra & Russo (2000)WOPSIP; Brenner, Owens, & Sung (2008)

Arnold, Brezzi, Cockburn & Marini (2002)Prudhomme, Pascal, Oden & Romkes (2000)

Applications/Extensions

51

• ApplicationsLinear elliptic/parabolic/hyperbolic PDEs, Fokker-Planck equations,Incompressible/Compressible fluid flows, Turbulent flows, Non-Newtonian flows,Time and frequency domain Maxwell's equations, Acoustics, MHD, Fully nonlinear PDEs.

• Overlapping/Fictitious Domain MethodsHansbo & Hansbo 2002, Becker, Hansbo, & Stenberg 2003, Hansbo, Hansbo & Larson 2003,Becker, Burman, & Hansbo 2009, Johansson & Larson 2011 …

• Hybridizable DG MethodsCockburn & Gopalakrishnan 2004, 2005, Cockburn, Dong, & Guzman 2008, Cockburn, Gopalakrishnan, & Lazarov 2009, Cockburn, Nguyen, & Peraire 2010 …

• Linear SolversFeng & Karakashian 2001, Gopalakrishnan & Kanschat 2003, Brenner & Wang 2005, Antonietti & Ayuso 2007, 2008, 2009, Antonietti & H. 2011, Antonietti, Ayuso, Brenner, & Sung 2011

Outline

Polytopic MeshesMesh Construction

52

Mesh Construction

53

• Polygonal mesh generator, e.g., Polymesher,cf. Talischi et al. 2012.

• Agglomeration of fine geometry-conforming mesh

➡ Graph Partitioning, e.g., METIS

➡ Overlapping Refined Meshes.

(Composite FEMs)

• This allows for the construction of very coarse finite element meshes, even on complicated domains containing microstructures.

• Mesh can then be automatically refined on the basis of solution accuracy.

Hackbusch & Sauter 1997, Antonietti, Giani, & H 2013

Georgy Voronoi 1868-1908Voronoi diagrams date back to Descartes in 1644

Composite FEM: Overlapping Refined Meshes

54

� /I]�-HIE�MW�XS�IQTPS]�X[S�QIWLIW�

�� TL MW�E�½RI�QIWL�[LMGL�EGGYVEXIP]�VITVIWIRXW ��

�� TCFE MW�E�GSEVWI EKKPSQIVEXIH QIWL�GSRWMWXMRK�SJ�TSP]KSRW�

� TCFE TVSZMHIW�ER EJJSVHEFPI GSQTYXEXMSREP�QIWL�

� 'SQTSWMXI�*)1 QIWL�GSRWXVYGXMSR�

Hackbusch & Sauter 1997→

Construction of the FEM meshes

55

�� 7IX TL� = T,� ERH�XLI�QIWL�GSYRXIV � = ��

�� 7IX TL�+� = ��

�� *SV�EPP � � TL� HS

�E -J � � � XLIR TL�+� = TL�+�

�{�}�

�F 3XLIV[MWI VI½RI � =�R�

M=� �M� *SV M = �, . . . , R�� MJ �M � � �= � XLIR�WIXTL�+� = TL�+�

�{�M}�

�� -J�XLI�VIJIVIRGI�QIWL TL� MW WYJ½GMIRXP] ½RI� MR�XLI�WIRWI�XLEX�MX�TVSZMHIW�E�KSSHVITVIWIRXEXMSR�SJ�XLI�FSYRHEV]�SJ �� XLIR ��3XLIV[MWI� WIX � = � + �� ERH

��

Refinement Algorithm (Construction of the Reference Meshes):

;I�HI½RI�ER SZIVPETTMRK WLETI�VIKYPEV�GSEVWI�QIWL T,� GSRWMWXMRK�SJ�WXERHEVH�HMWNSMRXIPIQIRXW�

� � �, =

��

��T,

��

ERH �� � � �= � �� � T,.

Composite FEM: Overlapping Refined Meshes

56

Composite FEM: Overlapping Refined Meshes

57

6IJIVIRGI�1IWLIW TLM � M = �, . . . , ��

4L]WMGEP��EKKPSQIVEXIH �1IWLIW TLM � M = �, . . . , �� �

TCFE

Composite FEM: Overlapping Refined Meshes

58

Graph Partitioning Techniques

59

Triangular Meshes

Polygonal Meshes