The rate of convergence of conjugate gradients
Numerische Mathematik
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation
Mathematics of Computation
Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
Generalized Schwarz splittings
SIAM Journal on Scientific and Statistical Computing
Balancing domain decomposition for problems with large jumps in coefficients
Mathematics of Computation
A domain decomposition method for the Helmholtz equation and related optimal control problems
Journal of Computational Physics
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Absorbing boundary conditions for domain decomposition
Applied Numerical Mathematics - Special issue on absorbing boundary conditions
Journal of Computational Physics
A Deflated Version of the Conjugate Gradient Algorithm
SIAM Journal on Scientific Computing
The optimized order 2 method: application to convection-diffusion problems
Future Generation Computer Systems - I. High Performance Numerical Methods and Applications. II. Performance Data Mining: Automated Diagnosis, Adaption, and Optimization
Optimized Schwarz Methods without Overlap for the Helmholtz Equation
SIAM Journal on Scientific Computing
GMRES with Deflated Restarting
SIAM Journal on Scientific Computing
Dual-Primal FETI Methods for Three-Dimensional Elliptic Problems with Heterogeneous Coefficients
SIAM Journal on Numerical Analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Journal of Computational Physics
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
Optimized Domain Decomposition Methods for the Spherical Laplacian
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
An asymptotic solution approach for elliptic equations with discontinuous coefficients
Journal of Computational Physics
| Hi-index | 7.30 |
When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods (DDM) suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the ''bad'' eigenvalues. In the framework of DDM, we propose a way to design interface conditions so that convergence is fast and does not have any plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on van der Sluis' result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous and anisotropic media. Numerical results are given and compared with other approaches.