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Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. (English) Zbl 1224.65292
The authors compare different choices of coarse spaces for two-level overlapping Schwarz-type domain decomposition methods, applied to second-order elliptic boundary problems with highly varying coefficients. A construction of the coarse space based on energy minimization is proposed and its efficient implementation is described on a model scalar problem of stationary heat conduction type. Numerical experiments on a 2D model problem show that the proposed coarse space outperforms in scalability and robustness other investigated choices of coarse spaces. However, some examples are presented for which the proposed method is not robust with respect to coefficient variation.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
65Y05 Parallel numerical computation
Software:
GAUSSIAN
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[1] Toselli, Domain Decomposition Methods-Algorithms and Theory (2005) · Zbl 1069.65138
[2] Chan, Acta Numerica pp 61– (1994)
[3] Sarkis M. Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements. Ph.D. Thesis, Courant Institute of Mathematical Sciences, Department of Computer Science, New York University, 1994. TR-671. · Zbl 0884.65119
[4] Graham, Domain decomposition for multiscale PDEs, Numerische Mathematik 106 (4) pp 589– (2007) · Zbl 1141.65084
[5] Scheichl, Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients, Computing 80 (4) pp 319– (2007) · Zbl 1171.65372
[6] Graham, Robust domain decomposition algorithms for multiscale PDEs, Numerical Methods in Partial Differential Equations 23 (4) pp 859– (2007) · Zbl 1141.65085
[7] Graham, Coefficient-explicit Condition Number Bounds for Overlapping Additive Schwarz pp 365– (2008)
[8] Xu, Iterative methods by space decomposition and subspace correction, SIAM Reviews 34 (4) pp 581– (1992) · Zbl 0788.65037
[9] Wan, An energy-minimizing interpolation for robust multigrid methods, SIAM Journal of Scientific Computing 21 (4) pp 1632– (1999)
[10] Mandel, Energy optimization of algebraic multigrid bases, Computing 62 (3) pp 205– (1999) · Zbl 0942.65034
[11] Xu, On an energy minimizing basis for algebraic multigrid methods, Computing and Visualization in Science 7 (3-4) pp 121– (2004) · Zbl 1077.65130
[12] Brannick, Algebraic Multigrid Methods Based on Compatible Relaxation and Energy Minimization pp 15– (2007)
[13] Ruge, Algebraic Multigrid. Multigrid Methods, Frontiers in Applied Mathematics 3 pp 73– (1987)
[14] Vaněk, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing 56 (3) pp 179– (1996)
[15] Brandt, General highly accurate algebraic coarsening, Electronic Transactions on Numerical Analysis 10 pp 1– (2000) · Zbl 0951.65096
[16] Brezina, Algebraic multigrid based on element interpolation (AMGe), SIAM Journal on Scientific Computing 22 (5) pp 1570– (2000) · Zbl 0991.65133
[17] Jones, AMGe based on element agglomeration, SIAM Journal on Scientific Computing 23 (1) pp 109– (2001) · Zbl 0992.65140
[18] Vaněk, Convergence of algebraic multigrid based on smoothed aggregation, Numerische Mathematik 88 (3) pp 559– (2001)
[19] Livne, Coarsening by compatible relaxation, Numerical Linear Algebra with Applications 11 (2-3) pp 205– (2004) · Zbl 1164.65353
[20] Brezina, Adaptive smoothed aggregation ({\(\alpha\)}SA) multigrid, SIAM Reviews 47 (2) pp 317– (2005) · Zbl 1075.65042
[21] Brannick, An energy-based AMG coarsening strategy, Numerical Linear Algebra with Applications 13 (2-3) pp 133– (2006) · Zbl 1174.65542
[22] MacLachlan, A greedy strategy for coarse-grid selection, SIAM Journal on Scientific Computing 29 (5) pp 1825– (2007) · Zbl 1154.65016
[23] Hou, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics 134 (1) pp 169– (1997) · Zbl 0880.73065
[24] Hou, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation 68 (227) pp 913– (1999) · Zbl 0922.65071
[25] Sarkis, Partition of Unity Coarse Spaces and Schwarz Methods with Harmonic Overlap pp 77– (2002) · Zbl 1009.65075
[26] Sarkis, Partition of Unity Coarse Spaces: Enhanced Versions, Discontinuous Coefficients and Applications to Elasticity pp 149– (2003)
[27] Dohrmann, A Family of Energy Minimizing Coarse Spaces for Overlapping Schwarz Preconditioners (2008) · Zbl 1359.65289
[28] Brezina, A black-box iterative solver based on a two-level Schwarz method, Computing 63 (3) pp 233– (1999) · Zbl 0951.65133
[29] Musy, Compatible coarse nodal and edge elements through energy functionals, SIAM Journal on Scientific Computing 29 (3) pp 1315– (2007) · Zbl 1149.65023
[30] MacLachlan, Multilevel upscaling through variational coarsening, Water Resources Research 42 (2) pp W02418– (2006)
[31] Golub, Matrix Computations (1996)
[32] Kozintsev B. Gaussian user’s manual. http://www.math.umd.edu/bnk/bak/SOURCE/manual.pdf 1999.
[33] Cliffe, Parallel computation of flow in heterogeneous media modelled by mixed finite elements, Journal of Computational Physics 164 (2) pp 258– (2000) · Zbl 0995.76044
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