Domain decomposition preconditioning for elliptic problems with jumps in coefficients. (English) Zbl 1361.35057
Summary: We propose an effective iterative preconditioning method to solve elliptic problems with jumps in coefficients. The algorithm is based on the additive Schwarz method (ASM). First, we consider a domain decomposition method without cross points on the interfaces between subdomains and next, we treat the cross points case. In both cases the main computational cost is an implementation of preconditioners for the Laplace operator in whole domain and in subdomains. Iterative convergence is independent of jumps in coefficients and mesh size. Several numerical examples are given to demonstrate the performance of proposed algorithm and theory developed in the paper.
MSC:
| 35J30 | Higher-order elliptic equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
References:
| [1] | Nepomnyaschikh, S. V., Preconditioning operators for elliptic problems with bad parameters, (Lai, C.-H.; etal., Domain Decomposition Methods in Sciences and Engineering (1999), Domain Decomposition Press: Domain Decomposition Press Bergen), 81-87 |
| [2] | Dryja, M.; Sarkis, M.; Widlund, O. B., Multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math., 72, 313-348 (1996) · Zbl 0857.65131 |
| [3] | Oswald, P., On the robustness of the BPX-preconditioner with respect to jumps in the coefficients, Math. Comp., 68, 633-650 (1999) · Zbl 1043.65121 |
| [4] | Wang, J.; Xie, R., Domain decomposition for elliptic problems with large jumps in coefficients, (Proceedings of Conference on Scientific and Engineering Computing (1994), National Defense Industry Press: National Defense Industry Press Beijing, China), 74-86 |
| [5] | Nepomnyaschikh, S. V., Application of domain decomposition to elliptic problems on with discontinuous coefficients, (Glowinski, Roland; Kuznetsov, Yuri A.; Meurant, Gérard A.; Périaux, Jacques; Widlund, Olof, Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations (1991), SIAM: SIAM Philadelphia, PA), 242-251 · Zbl 0766.65106 |
| [6] | Bramble, J. H.; Pasciak, J. E.; Schatz, A. H., The construction of preconditioners for elliptic problems by substructuring I, Math. Comp., 47, 103-134 (1986) · Zbl 0615.65112 |
| [7] | Xu, J.; Zhu, Y., Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients, Math. Models Methods Appl. Sci., 18, 1, 77-105 (2008) · Zbl 1151.65097 |
| [8] | Zhu, Y., Domain decomposition preconditioners for elliptic equations with jump coefficients, Numer. Linear Algebra Appl., 15, 2-3, 271-289 (2008) · Zbl 1212.65502 |
| [9] | Nepomnyaschikh, S. V., Fictitious space method on unstructured meshes, East-West J. Numer. Math., 4, 71-79 (1995) · Zbl 0831.65116 |
| [10] | Oswald, P., Multilevel Finite element Approximation Theory and Applications, (Teubner Skripten zur Numerik (1994), B. G. Teubner: B. G. Teubner Stuttgart) · Zbl 0830.65107 |
| [11] | Xu, J., The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56, 215-235 (1996) · Zbl 0857.65129 |
| [12] | Lions, P.-L., On the Schwarz alternating method I, (Glowinski, Roland; Golub, Gene H.; Meurant, Gérard A.; Périaux, Jacques, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (1988), SIAM: SIAM Philadelphia, PA), 1-42 · Zbl 0658.65090 |
| [13] | Matsokin, A. M.; Nepomnyaschikh, S. V., A Schwarz alternating method in a subspace, Sov. Math., 29, 10, 78-84 (1985) · Zbl 0611.35017 |
| [14] | Mathew, T. P., (Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 61 (2008), Springer: Springer Berlin) · Zbl 1147.65101 |
| [15] | Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differntial Equations (1999), Oxford Science Publications · Zbl 0931.65118 |
| [16] | Toselli, A.; Widlund, O., (Domain Decomposition Methods. Domain Decomposition Methods, Springer Ser. Comput. Math., vol. 34 (2005), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1069.65138 |
| [17] | Nepomnyaschikh, S. V., Method of splitting into subspaces for solving elliptic boundary-value problems in complex-form domains, Soviet J. Numer. Anal. Math. Modelling, 6, 2, 151-168 (1991) · Zbl 0816.65096 |
| [18] | Haase, G.; Nepomnyaschikh, S. V., Explicit extension operators on hierarchical grids, East-West J. Numer. Math., 5, 4, 231-248 (1998) · Zbl 0894.65061 |
| [19] | Yakovlev, G. N., On traces on piecewise smooth surfaces of functions from the space \(W_p^l\), Math. Notes, 74, 526-543 (1967) · Zbl 0162.44802 |
| [21] | Zhu, Y., Robust preconditioners for H(grad), H(curl) and H(div) systems with strongly discontinuous coefficients, 177 (2008), The Pennsylvania State University, (Ph.D thesis) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
