Overlapping Schwarz Methods Research Articles

Monolithic parallel overlapping Schwarz methods in fully-coupled nonlinear chemo-mechanics problems

We consider the swelling of hydrogels as an example of a chemo-mechanical problem with strong coupling between the mechanical balance relations and the mass diffusion. The problem is cast into a minimization formulation using a time-explicit approach for the dependency of the dissipation potential on the deformation and the swelling volume fraction to obtain symmetric matrices, which are typically better suited for iterative solvers. The MPI-parallel implementation uses the software libraries deal.II, p4est and FROSch (Fast of Robust Overlapping Schwarz). FROSch is part of the Trilinos library and is used in fully algebraic mode, i.e., the preconditioner is constructed from the monolithic system matrix without making explicit use of the problem structure. Strong and weak parallel scalability is studied using up to 512 cores, considering the standard GDSW (Generalized Dryja-Smith-Widlund) coarse space and the newer coarse space with reduced dimension. The FROSch solver is applicable to the coupled problems within in the range of processor cores considered here, although numerical scalablity cannot be expected (and is not observed) for the fully algebraic mode. In our strong scalability study, the average number of Krylov iterations per Newton iteration is higher by a factor of up to six compared to a linear elasticity problem. However, making mild use of the problem structure in the preconditioner, this number can be reduced to a factor of two and, importantly, also numerical scalability can then be achieved experimentally. Nevertheless, the fully algebraic mode is still preferable since a faster time to solution is achieved.

Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems

Abstract Generalized eigenvalue problems on the overlap(GenEO) 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 elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers that are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection–diffusion–reaction problems, which may be nonself-adjoint and indefinite, and whose discretizations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalized eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts that are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the nonself-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus, we obtain an iterative solver that is efficient in parallel and very effective for a wide range of convection–diffusion–reaction problems.

Scalable semismooth Newton methods with multilevel domain decomposition for subsurface flow and reactive transport in porous media

Large-scale modeling and predictive simulations of the subsurface flow and reactive transport system in porous media is significantly challenging, due to the high nonlinearity of the governing equations and the strong heterogeneity of material coefficients. The design of novel mathematical models and state-of-the-art methods for the flow simulation through porous media typically needs to satisfy the so-called bound-preserving property, i.e., the computed solution should stay inside a physically meaningful range. This paper presents a robust, scalable numerical framework based on the variational inequality formulation and the semismooth Newton method in a fully implicit manner, to model and simulate highly nonlinear flows without violating the boundedness requirement of the solution. Rigorous theoretical analysis for the variational inequality formulation of the problem is provided for facilitating the design of algorithms. Specifically, our approach further enhances the numerical formulation by utilizing a family of multilevel monolithic overlapping Schwarz methods for efficiently preconditioning, and the parallel implementation of the bound-preserving solver is based on the fast and robust domain decomposition technique. Numerical experiments are presented to demonstrate the efficiency and parallel scalability of the solution strategy for both standard benchmarks as well as realistic flow problems involving strong heterogeneity and high nonlinearity. We also show that the proposed framework is more robust and efficient than the commonly used inexact Newton algorithm in terms of the bound-preserving property.

Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods

A new reduced-dimension adaptive generalized Dryja--Smith--Widlund (GDSW) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition, the reduced-dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced-dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented.

Two-level Overlapping Schwarz Methods Based on Local Generalized Eigenproblems for Hermitian Variational Problems

The research of the two-level overlapping Schwarz (TL-OS) methods based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g., mainly for a second order elliptic problem and too heavy computational cost of coarse space construction. In this paper, by introducing appropriate assumptions, we propose more concise coarse basis functions for general Hermitian positive and definite discrete systems, and establish the algorithmic and theoretical frameworks of the corresponding TL-OS methods. Furthermore, to enhance the practicability of the algorithm, we design two economical TL-OS preconditioners and prove the condition number estimate. As the first application of the frameworks, we prove that the assumptions hold for the linear finite element discretization of a second order elliptic problem with high contrast and oscillatory coefficient and the condition number of the TL-OS preconditioned system is robust with respect to the model and mesh parameters. In particular, we also prove that the condition number of the economically preconditioned system is independent of the jump range under a certain jump distribution. Experimental results show that the first kind of economical preconditioner is more efficient and stable than the existing one. Second, we construct TL-OS and the economical TL-OS preconditioners for the plane wave least squares discrete system of the Helmholtz equation by using the frameworks. The numerical results for homogeneous and nonhomogeneous cases illustrate that the preconditioned conjugate gradient method based on the proposed preconditioners has good stability in terms of the angular frequency, mesh parameters, and the number of degrees of freedom in each element.

A General Case for the Maximum Norm Analysis of an Overlapping Schwarz Methods of Evolutionary HJB Equation with Nonlinear Source Terms with the Mixed Boundary Conditions

In this paper we provide a maximum norm analysis of an overlapping Schwarz method on non-matching grids for evolutionary HJB equation with nonlinear source terms with the mixed boundary conditions and a general elliptic operator. Moreover, an asymptotic behavior in uniform norm is established.

Overlapping Schwarz methods for isogeometric analysis based on generalized B-splines

Generalized B-splines (GB-splines) are a special class of Tchebycheff B-splines that are smooth piecewise function with sections in more general spaces. GB-splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation. In this paper, we present overlapping Schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on GB-splines. An h-analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes. Numerical results in two- and three-dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters, the domain deformation and the jumping coefficients.

Adaptive GDSW Coarse Spaces for Overlapping Schwarz Methods in Three Dimensions

A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend ...

Overlapping Schwarz methods with adaptive coarse spaces for multiscale problems in 3D

We propose two variants of the overlapping additive Schwarz method for the finite element discretization of scalar elliptic problems in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces. In the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges. The functions that constitute the coarse spaces are chosen adaptively, and they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space. Numerical results are given to support the theory.

Multiscale coarse spaces for overlapping Schwarz methods based on the ACMS space in 2D

Two-level overlapping Schwarz domain decomposition methods for second-order elliptic problems in two dimensions are proposed using coarse spaces constructed from the Approximate Component Mode Synthesis (ACMS) multiscale discretization approach. These coarse spaces are based on eigenvalue problems using Schur complements on subdomain edges. It is then shown that the convergence of the resulting preconditioned Krylov method can be controlled by a user-specified tolerance and thus can be made independent of heterogeneities in the coefficient of the partial differential equation. The relations of this new approach to other known adaptive coarse space approaches for overlapping Schwarz methods are also discussed. Compared to one of the competing adaptive approaches, the new coarse space can be significantly smaller. Compared to other competing approaches, the eigenvalue problems are significantly cheaper to solve, i.e., the dimension of the eigenvalue problems is minimal among the competing adaptive approaches under consideration. Our local eigenvalue problems can be solved using one iteration of LobPCG for essentially the same cost as a Cholesky-decomposition of a Schur complement on a subdomain edge.

An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell’s equations

In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell’s equations in waveguide configurations. The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell’s equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.

Robust multigrid for high-order discontinuous Galerkin methods: A fast Poisson solver suitable for high-aspect ratio Cartesian grids

We present a polynomial multigrid method for nodal interior penalty and local discontinuous Galerkin formulations of the Poisson equation on Cartesian grids. For smoothing we propose two classes of overlapping Schwarz methods. The first class comprises element-centered and the second face-centered methods. Within both classes we identify methods that achieve superior convergence rates, prove robust with respect to the mesh spacing and the polynomial order, at least up to P=32. Consequent structure exploitation yields a computational complexity of O(PN), where N is the number of unknowns. Further we demonstrate the suitability of the face-centered method for element aspect ratios up to 32.

Overlapping Schwarz Methods with a Standard Coarse Space for Almost Incompressible Linear Elasticity

Low-order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill-conditioning, when the material approaches the incompressible limit, if only the displacement variables are used. Mixed finite elements using both displacement and pressure variables provide a well-known remedy, but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging. Two-level overlapping Schwarz preconditioners for the almost incompressible system of linear elasticity, discretized by mixed finite elements with discontinuous pressures, are constructed and analyzed. The preconditioned systems are accelerated either by a GMRES (generalized minimum residual) method applied to the resulting discrete saddle point problem or by a PCG (preconditioned conjugate gradient) method applied to a positive definite, although extremely ill-conditioned, reformulation of the problem obtained by eliminating all pressure variables on the element level. A novel theoretical analysis of the algorithm for the positive definite reformulation is given by extending some earlier results by Dohrmann and Widlund. The main result of the paper is a bound on the condition number of the algorithm which is cubic in the relative overlap and grows logarithmically with the number of elements across individual subdomains but is otherwise independent of the number of subdomains, their diameters and mesh sizes, the incompressibility of the material, and possible discontinuities of the material parameters across the subdomain interfaces. Numerical results in the plane confirm the theory and also indicate that an analogous result should hold for the saddle point formulation, as well as for spectral element discretizations.

Implicit Space-Time Domain Decomposition Methods for Stochastic Parabolic Partial Differential Equations

We introduce and study parallel space-time domain decomposition methods for solving deterministic and stochastic parabolic equations. Traditional parallel algorithms solve parabolic problems time step by time step. The parallelism is restricted to each time step, and the algorithms are purely sequential in time. In this paper, we develop some overlapping Schwarz methods whose subdomains cover both space and time variables, and we show that the methods work well for stochastic parabolic equations discretized with an implicit stochastic Galerkin method. One- and two-level Schwarz preconditioned recycling GMRES methods are carefully investigated such that many components of the methods are reused when a large number of linear systems are solved. The key elements of this approach include an ordering algorithm and two grouping algorithms. We present some experimental results obtained on a parallel computer with more than one thousand processors.

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.

Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. An assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincare-type inequalities. Numerical experiments are...

Overlapping Schwarz Methods for Isogeometric Analysis

We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an $h$-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and „overlap sizes” for fixed polynomial degree $p$ and regularity $k$ of the basis functions. Numerical results in two- and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters $h, p, k$, number of subdomains $N$, overlap size, and also jumps in the coefficients of the elliptic operator.

Parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed-sphere

High resolution and scalable parallel algorithms for the shallow water equations on the sphere are very important for modeling the global climate. In this paper, we introduce and study some highly scalable multilevel domain decomposition methods for the fully implicit solution of the nonlinear shallow water equations discretized with a second-order well-balanced finite volume method on the cubed-sphere. With the fully implicit approach, the time step size is no longer limited by the stability condition, and with the multilevel preconditioners, good scalabilities are obtained on computers with a large number of processors. The investigation focuses on the use of semismooth inexact Newton method for the case with nonsmooth topography and the use of two- and three-level overlapping Schwarz methods with different order of discretizations for the preconditioning of the Jacobian systems. We test the proposed algorithm for several benchmark cases and show numerically that this approach converges well with smooth and nonsmooth bottom topography, and scales perfectly in terms of the strong scalability and reasonably well in terms of the weak scalability on machines with thousands and tens of thousands of processors.

Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity

AbstractOverlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains.In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive‐definite problems and the standard preconditioned conjugate gradient method.Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings. Copyright © 2009 John Wiley & Sons, Ltd.

Energy‐minimizing coarse spaces for two‐level Schwarz methods for multiscale PDEs

AbstractTwo‐level overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully choose the coarse approximation. Recent theoretical results by the authors have shown that bases for such robust coarse spaces should be constructed such that the energy of the basis functions is minimized. We give a simple derivation of a method that finds such a minimum energy basis using one local solve per coarse space basis function and one global solve to enforce a partition of unity constraint. Although this global solve may seem prohibitively expensive, we demonstrate that a one‐level overlapping Schwarz method is an effective and scalable preconditioner and we show that such a preconditioner can be implemented efficiently using the Sherman–Morrison–Woodbury formula. The result is an elegant, scalable, algebraic method for constructing a robust coarse space given only the supports of the coarse space basis functions. Numerical experiments on a simple two‐dimensional model problem with a variety of binary and multiscale coefficients confirm this. Numerical experiments also show that, when used in a two‐level preconditioner, the energy‐minimizing coarse space gives better results than other coarse space constructions, such as the multiscale finite element approach. Copyright © 2009 John Wiley & Sons, Ltd.