Related papers: A two-level overlapping Schwarz method with energy…
BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the…
We present a two-level preconditioner for solving linear systems arising from the discretization of the elliptic, linear-elastic deformation equation, in displacement unknowns, over domains that have arbitrary geometric and topological…
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…
The Schwarz domain decomposition method can be used for approximately solving a Laplace equation on a domain formed by the union of two overlapping discs. We consider an inexact variant of this method in which the subproblems on the discs…
We investigate the parallel one-level overlapping Schwarz method for solving finite element discretization of high-frequency Helmholtz equations. The resulting linear systems are large, indefinite, ill-conditioned, and complex-valued. We…
The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the contrast and linearly decreases with respect to mesh size if oversampling size is…
Owing to the ability of nonlinear domain decomposition methods to improve the nonlinear convergence behavior of Newton's method, they have experienced a rise in popularity recently in the context of problems for which Newton's method…
In this paper we compare numerically two different coarse space definitions for two-level domain decomposition preconditioners for the Helmholtz equation, both in two and three dimensions. While we solve the pure Helmholtz problem without…
We present an algebraic method for constructing a highly effective coarse grid correction to accelerate domain decomposition. The coarse problem is constructed from the original matrix and a small set of input vectors that span a low-degree…
We explore and analyze the use of multiprecision arithmetic for several classes of Schwarz methods and preconditioners, where the approximate solution of the local problems is performed at a lower precision, i.e., with fewer digits of…
This paper introduces a novel approach to analyzing overlapping Schwarz methods for N\'{e}d\'{e}lec and Raviart--Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the…
We analyse two-level Schwarz domain-decomposition GMRES preconditioners -- both the classic additive Schwarz preconditioner and a hybrid variant -- for finite-element discretisations of the Helmholtz equation with wavenumber $k$, where the…
Randomized neural networks (RaNNs), in which hidden layers remain fixed after random initialization, provide an efficient alternative for parameter optimization compared to fully parameterized networks. In this paper, RaNNs are integrated…
A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric…
This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding…
We study overlapping Schwarz methods for the Helmholtz equation posed in any dimension with large, real wavenumber and smooth variable wave speed. The radiation condition is approximated by a Cartesian perfectly-matched layer (PML). The…
In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous…
A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner are proposed for the solution of a nonsymmetric system of algebraic equations arising from a general finite volume element discretization of symmetric elliptic…
We analyze the convergence of the (algebraic) multiplicative Schwarz method applied to linear algebraic systems with matrices having a special block structure that arises, for example, when a (partial) differential equation is posed and…
We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follow the…