Overlapping Schwarz Algorithms Research Articles

A new coarse space for overlapping Schwarz algorithms for H(curl) problems in three dimensions with irregular subdomains

A new coarse space for a two-level overlapping Schwarz algorithm is presented for problems posed in three dimensions in the space H(curl, Ω). Previous studies for these methods are very restrictive about the geometry of the subdomains while this new space is well defined for general subdomains. The coarse space is based on energy minimization and its dimension equals the number of interior subdomain edges. Local direct solvers are used on the overlapping subdomains. The algorithm can be defined for any subdomain geometry and works for highly discontinuous coefficient distributions. Numerical experiments with irregular subdomains and different coefficient distributions are presented. The algorithm appears very promising even for random and discontinuous values of the coefficients.

An overlapping Schwarz method for virtual element discretizations in two dimensions

A new coarse space for domain decomposition methods is presented for nodal elliptic problems in two dimensions. The coarse space is derived from the novel virtual element methods and therefore can accommodate quite irregular polygonal subdomains. It has the advantage with respect to previous studies that no discrete harmonic extensions are required. The virtual element method allows us to handle polygonal meshes and the algorithm can then be used as a preconditioner for linear systems that arise from a discretization with such triangulations. A bound is obtained for the condition number of the preconditioned system by using a two-level overlapping Schwarz algorithm, but the coarse space can also be used for different substructuring methods. This bound is independent of jumps in the coefficient across the interface between the subdomains. Numerical experiments that verify the result are shown, including some with triangular, square, hexagonal and irregular elements and with irregular subdomains obtained by a mesh partitioner.

On the approximation of a virtual coarse space for domain decomposition methods in two dimensions

A new extension operator for a virtual coarse space is presented which can be used in domain decomposition methods for nodal elliptic problems in two dimensions. In particular, a two-level overlapping Schwarz algorithm is considered and a bound for the condition number of the preconditioned system is obtained. This bound is independent of discontinuities across the interface. The extension operator saves computational time compared to previous studies where discrete harmonic extensions are required and it is suitable for general polygonal meshes and irregular subdomains. Numerical experiments that verify the result are shown, including some with regular and irregular polygonal elements and with subdomains obtained by a mesh partitioner.

On the Design of Small Coarse Spaces for Domain Decomposition Algorithms

Methods are presented for automatically constructing coarse spaces of low dimension for domain decomposition algorithms. These constructions use equivalence classes of nodes on the interface between the subdomains into which the domain of a given elliptic problem has been subdivided, e.g., by a mesh partitioner such as METIS; these equivalence classes already play a central role in the design, analysis, and programming of many domain decomposition algorithms. The coarse space elements are well defined even for irregular subdomains, are continuous, and are well suited for use in two-level or multilevel preconditioners such as overlapping Schwarz algorithms. An analysis for scalar elliptic and linear elasticity problems reveals that significant reductions in the coarse space dimension can be achieved while not sacrificing the favorable condition number estimates for larger coarse spaces previously developed. These estimates depend primarily on the Lipschitz parameters of the subdomains. Numerical examples f...

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

AbstractThe goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.

A Highly Scalable Multilevel Schwarz Method with Boundary Geometry Preserving Coarse Spaces for 3D Elasticity Problems on Domains with Complex Geometry

We consider overlapping Schwarz algorithms for solving linear and nonlinear systems of equations arising from the finite element discretization of elasticity problems on unstructured meshes in three dimensions. The parallel scalability of Schwarz methods is determined almost completely by how the coarse space is constructed. In this paper, we introduce a low cost, boundary geometry preserving coarse mesh that shares the same boundary geometry with the fine mesh but has a better scalability than the coarse mesh obtained by uniformly coarsening the fine mesh. A new coarsening algorithm and a partitioning strategy are developed. We numerically show that a multilevel Schwarz method with the new coarse spaces is highly scalable, in terms of the total compute time, for solving some three-dimensional linear and nonlinear elasticity equations discretized on unstructured meshes with several hundreds of millions of unknowns and on a supercomputer with over 10,000 processor cores.

Two-Level Overlapping Schwarz Algorithms for a Staggered Discontinuous Galerkin Method

Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [SIAM J. Numer. Anal., 47 (2009), pp. 3820--3848] for the acoustic wave equation. Two types of coarse problems are introduced for the two-level Schwarz algorithms. The first is built on a nonoverlapping subdomain partition, which allows quite general subdomain partitions, and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation. Condition number bounds are established and numerical results are presented.

An Overlapping Schwarz Algorithm for Raviart--Thomas Vector Fields with Discontinuous Coefficients

Overlapping Schwarz methods form one of two major families of domain decomposition methods. In this paper, a two-level overlapping Schwarz method for Raviart--Thomas vector fields is developed and analyzed. The global component of the preconditioner is based on local energy-minimizing discrete harmonic extensions of coarse interface values, and the local components use local direct solvers on overlapping subdomains. The algorithm can be implemented algebraically using submatrices of a fully assembled stiffness matrix of the entire finite element problem. Moreover, it gives a condition number bound which depends only on the size of the local subproblems. The condition number of our method is shown to be bounded by $C ( 1 + \log (H/h)) ( 1 + H/\delta)$, where $H$ is the diameter of the subdomain and $h$ and $\delta$ are the sizes of the elements and the overlap between subdomains, respectively. The bound of the condition number is independent of the values and jumps of the coefficients across the interface between the subdomains. Numerical results, which support the theory, are also presented.

An Alternative Coarse Space for Irregular Subdomains and an Overlapping Schwarz Algorithm for Scalar Elliptic Problems in the Plane

In earlier work on domain decomposition methods for elliptic problems in the plane, an assumption that each subdomain is triangular or a union of a few coarse triangles has often been made. This is similar to what is required in geometric multigrid theory and is unrealistic if the subdomains are produced by a mesh partitioner. In an earlier paper, coauthored with Axel Klawonn, the authors introduced a coarse subspace for an overlapping Schwarz method with one degree of freedom for each subdomain vertex and one for each subdomain edge. A condition number bound proportional to $(1+\log(H/h))^2(1+H/\delta)$ was established assuming only that the subdomains are John domains; here $H/\delta$ measures the relative overlap between neighboring subdomains and $H/h$ the maximum number of elements across individual subdomains. We were also able to relate the rate of convergence to a parameter in an isoperimetric inequality for the subdomains into which the domain of the problem has been partitioned. In this paper, the dimension of the coarse subspace is decreased by using only one degree of freedom for each subdomain vertex; if all subdomains have three edges, this leads to a reduction of the dimension of the coarse subspace by approximately a factor four. In addition, the condition number bound is shown to be proportional to $(1+\log(H/h))(1+H/\delta)$ under a quite mild assumption on the relative length of adjacent subdomain edges. In this study, the subdomains are assumed to be uniform in the sense of Peter Jones. As in our earlier work, the results are insensitive to arbitrary large jumps in the coefficients of the elliptic problem across the interface between the subdomains. Numerical results are presented which confirm the theory and demonstrate the usefulness of the algorithm for a variety of mesh decompositions and distributions of material properties. It is also shown that the new algorithm often converges faster than the older one in spite of the fact that the dimension of the coarse space has been decreased considerably.

Weak Approximation Properties of Elliptic Projections with Functional Constraints

This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems. We develop an abstract framework for such constructions, akin to an abstract Bramble–Hilbert-type lemma, and then apply it in the design of coarse spaces for discretizations of PDEs with highly varying coefficients. The stability and approximation bounds of the constructed interpolant are in the weighted norm and are independent of the variations in the coefficients. Such spaces can be used, for example, in two-level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid or for numerical upscaling purposes. Some numerical illustration is provided.

An Overlapping Schwarz Algorithm for Almost Incompressible Elasticity

Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a low-dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and, essentially, to the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings.

Overlapping Schwarz Methods for Fekete and Gauss–Lobatto Spectral Elements

The classical overlapping Schwarz algorithm is here extended to the triangular/tetrahedral spectral element (TSEM) discretization of elliptic problems. This discretization, based on Fekete nodes, is a generalization to nontensorial elements of the tensorial Gauss-Lobatto-Legendre quadrilateral spectral elements (QSEM). The overlapping Schwarz preconditioners are based on partitioning the domain of the problem into overlapping subdomains, solving local problems on these subdomains, and solving an additional coarse problem associated with either the subdomain mesh or the spectral element mesh. The overlap size is generous, i.e., one element wide, in the TSEM case, while it is minimal or variable in the QSEM case. The results of several numerical experiments show that the convergence rate of the proposed preconditioning algorithm is independent of the number of subdomains $N$ and the spectral degree $p$ in case of generous overlap; otherwise it depends inversely on the overlap size. The proposed preconditioners are also robust with respect to arbitrary jumps of the coefficients of the elliptic operator across subdomains.

Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves

The classical overlapping Schwarz algorithm is here extended to the spectral element discretization of linear elastic problems, for both homogeneous and heterogeneous compressible materials. The algorithm solves iteratively the resulting preconditioned system of linear equations by the conjugate gradient or GMRES methods. The overlapping Schwarz preconditioned technique is then applied to the numerical approximation of elastic waves with spectral elements methods in space and implicit Newmark time advancing schemes. The results of several numerical experiments, for both elastostatic and elastodynamic problems, show that the convergence rate of the proposed preconditioning algorithm is independent of the number of spectral elements (scalability), is independent of the spectral degree in case of generous overlap, otherwise it depends inversely on the overlap size. Some results on the convergence properties of the spectral element approximation combined with Newmark schemes for elastic waves are also presented.

A convergence theory for an overlapping Schwarz algorithm using discontinuous iterates

A new type of overlapping Schwarz methods, using discontinuous iterates, is constructed by modifying the classical overlapping Schwarz algorithm. This new algorithm allows for discontinuous iterates across the artificial interface. For Poisson’s equation, this algorithm can be considered as an overlapping version of Lions’ Robin iteration method for which little is known concerning the rate of convergence. Since overlap improves the performance of the classical algorithms considerably, the existence of a uniform convergence factor is the fundamental question for our algorithm. A new theory using Lagrange multipliers is developed and conditions are found for the existence of an almost uniform convergence factor for the dual variables, which implies rapid convergence of the primal variables, in the two overlapping subdomain case. Our result also shows a relation between the boundary conditions of the given problem and the artificial interface condition. Numerical results for the general case with cross points are also presented. They indicate possible extensions of our results to this more general case.

Overlapping Schwarz preconditioners for spectral element discretizations of convection‐diffusion problems

AbstractThe classical overlapping Schwarz algorithm is here extended to stabilized spectral element discretizations of convection‐diffusion problems. The algorithm solves iteratively the resulting non‐symmetric system of linear equations by a preconditioned GMRES method. The preconditioner is built from local convection‐diffusion solvers on overlapping subdomains and from a coarse convection‐diffusion solver on a coarse mesh defined by the subdomain boundaries. Several numerical experiments on test problems in the plane indicate that this algorithm retains the fast convergence rate and optimal scalability properties of classical overlapping methods for diffusion dominated problems. Fast convergence is also obtained for convection dominated problems without closed streamlines and with a moderate number of subdomains. Copyright © 2001 John Wiley & Sons, Ltd.

Parallel Newton--Krylov--Schwarz Algorithms for the Transonic Full Potential Equation

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton--Krylov--Schwarz (NKS), employs an inexact finite difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization

The spectral element method is used to discretize self-adjoint elliptic equations in three-dimensional domains. The domain is decomposed into hexahedral elements, and in each of the elements the discretization space is the set of polynomials of degree N in each variable. A conforming Galerkin formulation is used, the corresponding integrals are computed approximately with Gauss--Lobatto--Legendre (GLL) quadrature rules of order N, and a Lagrange interpolation basis associated with the GLL nodes is used. Fast methods are developed for solving the resulting linear system by the preconditioned conjugate gradient method. The conforming finite element space on the GLL mesh, consisting of piecewise Q1 or P1 functions, produces a stiffness matrix Kh that is known to be spectrally equivalent to the spectral element stiffness matrix KN. Kh is replaced by a preconditioner $\tilde{K}_h$ which is well adapted to parallel computer architectures. The preconditioned operator is then $\tilde{K}_h^{-1} K_N$. Techniques for nonregular meshes are developed, which make it possible to estimate the condition number of $\tilde{K}_h^{-1} K_N$, where $\tilde{K}_h$ is a standard finite element preconditioner of Kh , based on the GLL mesh. Two finite element--based preconditioners: the wirebasket method of Smith and the overlapping Schwarz algorithm for the spectral element method are given as examples of the use of these tools. Numerical experiments performed by Pahl are briefly discussed to illustrate the efficiency of these methods in two dimensions.

Convergence estimates for product iterative methods with applications to domain decomposition

In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space % which are defined in terms of products of operators defined with respect to a number of subspaces. The simplest algorithm of this sort has an error-reducing operator which is the product of orthogonal projections onto the complement of the subspaces. New normreduction estimates for these iterative techniques will be presented in an abstract setting. Applications are given for overlapping Schwarz algorithms with many subregions for finite element approximation of second-order elliptic problems.