Two-level Schwarz Methods Research Articles

Two-Level Schwarz Methods for a Discontinuous Galerkin Approximation of Elliptic Problems with Jump Coefficients

We present two-level nonoverlapping and overlapping Schwarz preconditioners for the linear algebraic system arising from the weighted symmetric interior penalty Galerkin approximation of elliptic problems with highly heterogeneous coefficients. The coarse space is constructed by the local Dirichlet-to-Neumann maps the theoretical results show that the condition number of the preconditioned system is independent of the discontinuous coefficient, the number of subdomains and the mesh size for the nonoverlapping case. For the overlapping case adding an extra assumption of coefficient distribution, the similar conclusion is also obtained. Numerical experiments validate the theoretical results and illustrate the performance and robustness of the proposed two-level methods.

Additive and Hybrid Nonlinear Two-Level Schwarz Methods and Energy Minimizing Coarse Spaces for Unstructured Grids

Nonlinear domain decomposition (DD) methods, such as ASPIN (additive Schwarz preconditioned inexact Newton), RASPEN (restricted additive Schwarz preconditioned inexact Newton), nonlinear FETI-DP (f...

A Parallel Implementation of a Two-Level Overlapping Schwarz Method with Energy-Minimizing Coarse Space Based on Trilinos

We describe a new implementation of a two-level overlapping Schwarz preconditioner with energy-minimizing coarse space (GDSW: generalized Dryja--Smith--Widlund) and show numerical results for an additive and a hybrid additive-multiplicative version. Our parallel implementation makes use of the Trilinos software library and provides a framework for parallel two-level Schwarz methods. We show parallel scalability for two- and three-dimensional scalar second-order elliptic and linear elasticity problems for several thousands of cores. We also discuss techniques for the parallel construction of coarse spaces which are also of interest for other parallel preconditioners and discretization methods using energy minimizing coarse functions. We finally show an application in monolithic fluid-structure interaction, where significant improvements are achieved compared to a standard algebraic, one-level overlapping Schwarz method.

Schwarz Methods for a Preconditioned WOPSIP Method for Elliptic Problems

Abstract We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order H/h, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented.

One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.

Schwarz methods for inequalities with contraction operators

We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier–Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.

Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements

This paper analyses two-level Schwarz methods for matrices arising from the p-version finite-element method on triangular and tetrahedral meshes. The coarse level consists of the lowest-order finite-element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a simple model problem and an elasticity problem on a complex geometry.

Parallel Schwarz methods for convection-dominated semilinear diffusion problems

Parallel two-level Schwarz methods are proposed for the numerical solution of convection-diffusion problems, with the emphasis on convection-dominated problems. Two variants of the methodology are investigated. They differ from each other by the type of boundary conditions (Dirichlet- or Neumann-type) posed on a part of the second-level subdomain interfaces. Convergence properties of the two-level Schwarz methods are experimentally compared with those of a variant of the standard multi-domain Schwarz alternating method. Numerical experiments performed on a distributed memory multiprocessor computer illustrate parallel efficiency of the methods.

Acceleration of the non-symmetrized two-level iteration

In this paper we discuss two-level multiplicative Schwarz methods, which result in unsymmetric systems in their solution spaces. We introduce a new algorithm to accelerate this iteration by applying the conjugate gradient (CG) scheme to a subspace. It will be shown that this algorithm possesses a CG-type convergence rate. Our algorithm has obvious advantages over the existing Schwarz-CG algorithms. Its application to two-level grid methods will be also considered.

Two-level parallel Schwarz methods for singularly perturbed semilinear elliptic problems

Two-level domain decomposition methods based on the Schwarz alternating procedure are proposed for the numerical solution of singularly perturbed semilinear elliptic problems. Convergence properties of the two-level Schwarz methods are established. Numerical experiments for a test problem illustrate the behaviour of the methods with respect to the critical parameters, such as the perturbation parameter, the amount of overlapping and sizes of subdomains. Application of one of the two-level Schwarz methods to a model semiconductor problem is also discussed.

Parallel two-level Schwarz methods for singularly perturbed elliptic problems

Schwarz domain decomposition methods are developed for the numerical solution of singularly perturbed elliptic problems. Three variants of a two-level Schwarz method with interface subproblems are investigated both theoretically and from the point of view of their computer realization on a distributed memory multiprocessor computer. Numerical experiments illustrate their parallel performance as well as their behavior with respect to the critical parameters such as the perturbation parameter, the size of the interface subdomains and the number of parallel processors. Application of one of the methods to a model problem with an interior layer of complex geometry is also discussed.

Boundary Treatments for Multilevel Methods on Unstructured Meshes

In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarse-to-fine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle different types of boundary conditions. We propose two effective ways to adapt the standard coarse-to-fine interpolations to correctly implement boundary conditions for two-dimensional polygonal domains, and we provide some numerical examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as efficient as in the structured case. In addition, we prove that the proposed interpolants possess the local optimal L2 -approximation and H1 -stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for two-level Schwarz methods.

Two-level Schwarz methods for Wilson element approximation of elasticity problem

In this paper, a two-level additive and multiplicative Schwarz methods are developed for the Wilson nonconforming element approximation of elasticity problem. A new intergrid operator for the Wilson element is constructed. It is well known that the two-level additive Schwarz method can be seen as a preconditioning system, so the condition number of the preconditioning system for elasticity problem is proved to be bounded by a constant independent of mesh size and the number of subdomains. For the multiplicative method, the optimal rate of convergence is also obtained.

Domain Decomposition Algorithms with Small Overlap

Numerical experiments have shown that two-level Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.