Related papers: GenEO spectral coarse spaces in SPD domain decompo…
GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for…
GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from the family of spectral coarse spaces that can efficiently rely on local eigensolves in order to build a robust parallel domain decomposition preconditioner for…
Coarse spaces are essential to ensure robustness w.r.t. the number of subdomains in two-level overlapping Schwarz methods. Robustness with respect to the coefficients of the underlying partial differential equation (PDE) can be achieved by…
Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. The GenEO coarse space has been shown to lead to a fully robust two-level Schwarz preconditioner which scales well over…
Motivated by recent work on coarse spaces for Helmholtz problems, we provide in this paper a comparative study on the use of spectral coarse spaces of GenEO type for heterogeneous indefinite elliptic problems within an additive overlapping…
Two-level domain decomposition (DD) methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level…
Two-level domain decomposition preconditioners lead to fast convergence and scalability of iterative solvers. However, for highly heterogeneous problems, where the coefficient function is varying rapidly on several possibly non-separated…
Domain decomposition (DD) methods are a natural way to take advantage of parallel computers when solving large scale linear systems. Their scalability depends on the design of the coarse space used in the two-level method. The analysis of…
Wave propagation problems governed by the Helmholtz equation remain among the most challenging in scientific computing, due to their indefinite nature. Domain decomposition methods with spectral coarse spaces have emerged as some of the…
This paper introduces a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space…
Nonlinear Schwarz methods are a type of nonlinear domain decomposition method used as an alternative to Newton's method for solving discretized nonlinear partial differential equations. In this article, the first parallel implementation of…
Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in…
Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable…
In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming…
Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for…
An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this…
In this article a new family of preconditioners is introduced for symmetric positive definite linear systems. The new preconditioners, called the AWG preconditioners (for Algebraic-Woodbury-GenEO) are constructed algebraically. By this, we…
We present a two-level overlapping Schwarz preconditioner for three-dimensional problems discretized with the Virtual Element Method. Our approach handles general polyhedral meshes and irregular subdomains, extending the applicability of…
We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite-element discretisations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned…
Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational…