Primal Formulation. Find u h V h such that. A h (u h, v h )= fv h dx v h V h, where. u h v h dx. A h (u h, v h ) = DGFEM Primal Formulation.

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sha1_base64="gg4eyg909b5swl32fqqo3elzheq=">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</latexit> <latexit sha1_base64="gg4eyg909b5swl32fqqo3elzheq=">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</latexit> Numerical Fluxes Consistency: û(v) =v, ŝ(v, v) = v for all smooth v which satisfy the Dirichlet boundary conditions. Galerkin Orthogonality: For u H 2 ( ), we have A h (u u h, v h )=0 v h V h. [This automatically follows from the consistency of û and ŝ.] Conservation: The numerical fluxes are conservative if they are single valued, i.e., [[û]] = 0, [[ŝ]] = 0. 44

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sha1_base64="pvf8rh9e2xvbexlenyobr8sdpv4=">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</latexit> <latexit sha1_base64="pvf8rh9e2xvbexlenyobr8sdpv4=">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</latexit> <latexit sha1_base64="pvf8rh9e2xvbexlenyobr8sdpv4=">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</latexit> Interior Penalty Methods Given [ 1, 1], we define û = û(u h )= { u h } n+ [[u h ]] on F F I h, (1 + )u h on F F B h, Here, ŝ = ŝ(u h, hu h )= { h u h } [[u h ]] on F F I h, u h u h n on F F B h. denotes the interior penalty parameter: = C p2 h, C > 0. 45

4 Interior Penalty Methods Interior Penalty Methods Given [ 1, 1], find uh h uh F Fh <latexit sha1_base64="ovtcfu3ix3lzhma9pb8t+zymjz0=">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 sha1_base64="l1aezaswuzut0w+kublrtiiogze=">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</latexit> sha1_base64="vrovc22ap+vimy26+3s24abti5m=">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</latexit> sha1_base64="i0vzt0h3dwpiodc3u1795cbbbtm=">aaaezniclvnnb9naehvdomvap+dizuvdvdq0sgajobrvgfooslsctjxikbqvx/gs9dryrk0qy+lkp+e38w8yuymn00qipt2/mxk7m/vsxljo0+3+xmrcat6+s7xy1753/8hq2vrgwxmdpqnhpo9kljy5ofekhx0jjmszoeeixymn7ur9gt/nmneiul/neyzdemzk+ikdiwq0sftluzfqhirddkwgiruce6ab5gjfbru9dm/yym3muw5rpaog4k9gqyvplafmbgbsx0wwt7yoykbo5xdhwjijwjva+vun9yjzrwul5xlsau8yr6fhkd8ovr2q0geoumcvvh5qke6yyxxerlhu+xbgevz9k5qmdruxx6e+dtjlcp+lns1op3+17r32by0txihmv96otdcrotuvzl/arnw+7sat0o8srjewvja37c5ofbpb6vahxqe9gdi0zudotnfyc7yipye9lzeg9adxjc0wh8qilrgwnvrjdhwcyxwqvbcihuavpqr2lbipla34evmjyucrcgi1pg9dygzbbhoxvpi3xqap8v8pc6hi1kdifxf5qwqmyqu/mscs5eaeewcecoqv8qas4izcxltfcy4+bwqt5ome4kdwaz0flg7ssezykpxawjvkjyy1sqsf6dabxrv52gylmjg1f+ifvy+olhdjgvy0pqgoeogvzmosifkjcugocv35osa/fagll3kslapbh8rygn3+rh+zah8mijnnterqz9qatyjdkiv6i4a5dvrpo286vepu5v67mzdwrmfwe2vb6lmvrh3ro3vk9s3ewg28bow13japm9+brfphrwpjavbzykqd5s8/sy+zaw==</latexit> sha1_base64="ovtcfu3ix3lzhma9pb8t+zymjz0=">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</latexit> h vh dx + F Vh such that F Fh {{ h vh }} F {{ [[uh ]]ds + h uh }} F Fh [[vh ]]ds F [[uh ]] [[uh ]]ds = for all vh Vh. = = 1 - Nonsymmetric Interior Penalty Method (NIP). = 0 - Incomplete Interior Penalty Method (IIP). fvh dx 1 - Symmetric Interior Penalty Method (SIP). Arnold, Brezzi, Cockburn, & Marini

5 <latexit sha1_base64="olqcpoqiiay+3duy8kvtli/qk4m=">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</latexit> <latexit sha1_base64="olqcpoqiiay+3duy8kvtli/qk4m=">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</latexit> <latexit 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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 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sha1_base64="so62ayr3kew7fh3r1u6buscsbjc=">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</latexit> <latexit sha1_base64="so62ayr3kew7fh3r1u6buscsbjc=">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</latexit> Adjoint Consistency Adjoint Consistency Given L 2 ( ), let z H 2 ( ) be the analytical solution of the dual/adjoint problem z = in, z = 0 on. Then the discretization is adjoint consistent if and only if A h (v, z) = vdx v H 2 (T h ), where H 2 (T h )= v L 2 ( ):v H 2 ( ), T h. Lemma 1 The above discretization is adjoint consistent, if and only if, the numerical fluxes û and ŝ are conservative. 47

6 <latexit sha1_base64="ztdqgxinhzkur6q10udpose572e=">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</latexit> <latexit sha1_base64="ztdqgxinhzkur6q10udpose572e=">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</latexit> <latexit sha1_base64="ztdqgxinhzkur6q10udpose572e=">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</latexit> <latexit sha1_base64="ztdqgxinhzkur6q10udpose572e=">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</latexit> Adjoint Consistency Stability u u h H 1 ( ) u u h L 2 ( ) J(u) J(u h ) SIP C > C min > 0 O(h p ) O(h p+1 ) O(h 2p ) IIP C > C min > 0 O(h p ) O(h p+1 ) (p odd) O(h p+1 ) (p odd) O(h p ) (p even) O(h p ) (p even) NIP C > 0 O(h p ) O(h p+1 ) (p odd) O(h p+1 ) (p odd) O(h p ) (p even) O(h p ) (p even) Arnold, Brezzi, Cockburn & Marini 2002, Prudhomme, Pascal, Oden & Romkes 2000, Harriman, H., Senior & Suli

7 Adjoint Consistency p =1 p =2 p = p =1 p =2 p =3 p =4 J(u) J(uh) p =4 J(u) J(uh) p =5, h h Symmetric IP Scheme Nonsymmetric IP Scheme 49

8 DGFEM for Diffusion 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) 50

9 Applications/Extensions Applications Linear 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 Methods Hansbo & Hansbo 2002, Becker, Hansbo, & Stenberg 2003, Hansbo, Hansbo & Larson 2003, Becker, Burman, & Hansbo 2009, Johansson & Larson 2011 Hybridizable DG Methods Cockburn & Gopalakrishnan 2004, 2005, Cockburn, Dong, & Guzman 2008, Cockburn, Gopalakrishnan, & Lazarov 2009, Cockburn, Nguyen, & Peraire 2010 Linear Solvers Feng & Karakashian 2001, Gopalakrishnan & Kanschat 2003, Brenner & Wang 2005, Antonietti & Ayuso 2007, 2008, 2009, Antonietti & H. 2011, Antonietti, Ayuso, Brenner, & Sung

10 Outline Polytopic Meshes Mesh Construction 52

11 Mesh Construction Polygonal mesh generator, e.g., Polymesher,cf. Talischi et al Georgy Voronoi Voronoi diagrams date back to Descartes in 1644 Agglomeration of fine geometry-conforming mesh Overlapping Refined Meshes. (Composite FEMs) Hackbusch & Sauter 1997, Antonietti, Giani, & H 2013 Graph Partitioning, e.g., METIS 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. 53

12 Composite FEM: Overlapping Refined Meshes T T CFE T CFE Hackbusch & Sauter

13 Construction of the FEM meshes ˆT ˆT = ˆT = ˆT + = ˆ ˆT = ˆ ˆT ˆ ˆ = ˆ ˆT. Refinement Algorithm (Construction of the Reference Meshes): ˆ ˆT + = ˆT + {ˆ} ˆ = ˆ = ˆ =,..., ˆ ˆ = ˆT + = ˆT + {ˆ } ˆT = + 55

14 Composite FEM: Overlapping Refined Meshes 56