Mesh Adaptivity and Domain Decomposition for Finite Element Method Pavel K˚us Institute of Mathematics, Czech Academy of Sciences, Prague Max Planck Computing and Data Facility, MPI, M¨unchen,Germany THEMATIC MA S of E T U T I T S N I Czech Academy of Sciences In collaboration with Jakub S´ıstekˇ Institute of Mathematics, Czech Academy of Sciences, Prague School of Mathematics, The University of Manchester, UK PANM 2016 Improvements of the finite element calculations Many ways how to improve performance of finite element codes, two very different strategies are: parallel, domain decomposition, the use of supercomputers, very large linear systems adaptivity, higher order, the goal is to have smaller linear systems solvable on PC, still with good accuracy It would be nice to combine both strategies Parallel calculations take advantage from similarity of structure over the domain most of the research done in linear solver part Exascale brings many challenges. It could be good alternative to get more from petascale instead. Adaptivity different treatment of different areas, based on solution behavior a lot of effort in assembly part, trying to minimize number of DOFs There are certain limits for single PC, no matter how smart the algorithm is. P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 2 / 53 Improvements of the finite element calculations Many ways how to improve performance of finite element codes, two very different strategies are: parallel, domain decomposition, the use of supercomputers, very large linear systems adaptivity, higher order, the goal is to have smaller linear systems solvable on PC, still with good accuracy It would be nice to combine both strategies Parallel calculations take advantage from similarity of structure over the domain most of the research done in linear solver part Exascale brings many challenges. It could be good alternative to get more from petascale instead. Adaptivity different treatment of different areas, based on solution behavior a lot of effort in assembly part, trying to minimize number of DOFs There are certain limits for single PC, no matter how smart the algorithm is. P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 2 / 53 Contents Adaptivity and Higher Order Elements Parallel mesh handler BDDC method Parallel implementation Numerical Results P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 3 / 53 Contents Adaptivity and Higher Order Elements Parallel mesh handler BDDC method Parallel implementation Numerical Results P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 4 / 53 Mesh adaptivity Finite element method divides domain to elements On each element polynomial approximation used Finer mesh and higher-order polynomials should lead to better solution It also increases the problem size The goal is to improve the solution where needed benchmark Common: start with coarse grid and successively improve it based on analysis of the solution obtained in each step Different approaches complete re-meshing change of vertices positions, mesh topology is kept (r-adaptivity) uniform mesh element refinements (h-adaptivity) different polynomial orders (p-adaptivity) combination of both (hp-adaptivity) We need information about the error a-posteriori error estimators exact solution for benchmarks hp mesh P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 5 / 53 Mesh adaptivity Each approach has its advantages and drawbacks Complete re-meshing leaves the adaptivity to mesh generator, the solver is not aware of it Meshing of complex domains is very expensive r-adaptivity keeps topology of the mesh, which is good, but might not be flexible enough benchmark For both previous interpolation between completely different meshes p-adaptivity not flexible enough h-adaptivity does not improve the order of convergence hp-adaptivity very challenging for implementation hanging nodes are issue for both h- and uniform mesh hp-adaptivity DG-FEM simplifies hanging nodes treatment, but more complicated weak formulation and more expensive We will use h-adaptivity in the following hp mesh P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 6 / 53 Benchmark no. 8 { Schr¨odingerequation COMSOL COMSOL 106496 elements, p = 2 213761 DOFs, error 171 % Hermes Hermes mesh after hp-adapt. 1908 DOFs, error 0.0089 % P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 7 / 53 Simplification of the problem Assumptions on the mesh only level-1 hanging nodes allowed equal order shape functions, i.e. no hp, but higher p fine O.K. not O.K. We have to keep problem reasonably simple to be able to gain from parallelization Under such circumstances, dealing with hanging nodes is surprisingly simple and straightforward We will go through the process in detail now P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 8 / 53 Hanging nodes (L1;L2;L3;L4) $ (g2; g3; g5; g6) 20 1 0 0 0 03 0 0 1 0 0 0 G = 6 7 K 40 0 0 0 1 05 0 0 0 0 0 1 2 3 a(L1;L1) : : : a(L1;L4) g6 L4 6 . 7 g5 L3 AK = 6 . 7 g4 4 . 5 a(L4;L1) : : : a(L4;L4) X T L 2 A = GK AK GK L1 g 3 K2T g2 k g1 Regular element Matrix GK represents the relationship between local and global DOFs Local stiffness matrix distributed to the Global one Similarly for right-hand side P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 9 / 53 Hanging nodes (L1;L2;L3;L4) $ (g2; g7; g11; g8) X T A = GK AK GK K2Tk A A~ = C g g g9 6 g 5 4 g2 + g5 g11 = g10 L g8 2 g11 L3 4 C = 0; 1 ; 0; 0; 1 ; 0; 0; 0; 0; 0; −1 L2 2 2 L1 g3 g7 g2 g1 First option { constraints added to global matrix There are global degrees of freedom assigned to hanging nodes Matrix A assembled as in the regular case Solution would be discontinuous ! the global system has to be extended by constraints C The matrix A~ is rectangular, has to be modified before actual solution { various techniques P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 9 / 53 Hanging nodes T T (L1;L2;L3;L4) = TK (g2; g7; g5; g8) 2 1 0 0 03 0 1 0 0 T = 6 7 K 41/2 0 1/2 05 0 0 0 1 g g g9 6 g 5 ¯ T 4 AK = TK AK TK g10 g8 X T L L4 ¯ 3 A = GK AK GK K2T L2 k L1 g3 g7 g2 g1 Second option { local change of basis No global degrees of freedom assigned to hanging nodes Matrix TK used to modify the local stiffness matrix and RHS Corresponds to the construction of globally continuous basis function (depicted the one corresponding to g5) P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 9 / 53 Hanging nodes T T (L1;L2;L3;L4) = TK (g2; g8; g5; g9) 21/2 0 1/2 03 0 1 0 0 T = 6 7 K 4 0 0 1 05 0 0 0 1 g g9 6 g L L4 g 5 3 ¯ T 4 AK = TK AK TK L2 L1 g10 g8 X T ¯ A = GK AK GK K2Tk g3 g7 g2 g1 Second option { local change of basis No global degrees of freedom assigned to hanging nodes Matrix TK used to modify the local stiffness matrix and RHS Corresponds to the construction of globally continuous basis function (depicted the one corresponding to g5) P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 9 / 53 Hanging nodes 2·· v(L1) · :::3 6·· v(L2) · :::7 6·· v(L ) · :::7 6 3 7 TK = 6·· v(L ) · :::7 6 4 7 4. 5 . L4 L3 ¯ T AK = TK AK TK L2 L 1 X T ¯ A = GK AK GK K2Tk Second option { local change of basis Works for higher-order basis functions as well Works for 2D and 3D for 1-irregular mesh and elements of same order Not numbering hanging nodes natural for p4est ! we use this approach P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 9 / 53 Contents Adaptivity and Higher Order Elements Parallel mesh handler BDDC method Parallel implementation Numerical Results P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 10 / 53 Refinements without balancing Without balancing has no sense in parallel Most of the refinements would concentrate in those domains, where singularities, boundary or internal layers are present It might be just few subdomains P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 11 / 53 Refinements without balancing Without balancing has no sense in parallel Most of the refinements would concentrate in those domains, where singularities, boundary or internal layers are present It might be just few subdomains P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 11 / 53 Refinements without balancing Without balancing has no sense in parallel Most of the refinements would concentrate in those domains, where singularities, boundary or internal layers are present It might be just few subdomains P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 11 / 53 Refinements without balancing Without balancing has no sense in parallel Most of the refinements would concentrate in those domains, where singularities, boundary or internal layers are present It might be just few subdomains P. K˚us Mesh Adaptivity and Domain Decomposition for Finite Element Method 11 / 53 Refinements without balancing Without balancing has no sense in parallel Most of the refinements would concentrate in those domains, where singularities, boundary or internal layers are present It might be just few subdomains P.