A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

Summary. We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems. Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements. Received March 17, 1998 / Revised version received June 7, 1999 / Published online January 27, 2000     

A PARALLEL ITERATIVE DOMAIN DECOMPOSITION ALGORITHM FOR ELLIPTIC PROBLEMS

1.IntroductionNolloverlappillgdomaindecolllpositionnletllodshavereceivedalotofattentionlenlsilllldallowefficielltparallelisnl.F'Orarecentdevelopmelltofthesemethods,werefertot…     

Fast iterative solver for convection‐diffusion systems with spectral elements

We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection–diffusion equation. The method is based on a Robin–Robin interface preconditioner coupled to a fast diagonalization solver which is used to efficiently eliminate the interior degrees of freedom and perform subsidiary subdomain solves. Using a spectral element discretization, we first apply our technique to constant wind problems, and then propose a means for applying the technique as a preconditioner for variable wind problems. We demonstrate that iteration counts are mildly dependent on changes in mesh size and convection strength. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems

An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

Convergence and comparison analysis of some numerical Schwarz methods

Summary Domain decomposition methods are a natural means for solving partial differential equations on multi-processors. The spatial domain of the equation is expressed as a collection of overlapping subdomains and the solution of an associated equation is solved on each of these subdomains. The global solution is then obtained by piecing together the subsolutions in some manner. For elliptic equations, the global solution is obtained by iterating on the subdomains in a fashion that resembles the classical Schwarz alternating method. In this paper, we examine the convergence behavior of different subdomain iteration procedures as well as different subdomain approximations. For elliptic equations, it is shown that certain iterative procedures are equivalent to block Gauss-Siedel and Jacobi methods. Using different subdomain approximations, an inner-outer iterative procedure is defined.M-matrix analysis yields a comparison of different inner-outer iterations.Dedicated to the memory of Peter HenriciThis work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48     

On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs

Summary The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology. This work was in part supported by a grant from the RGC of HKSAR, China (HKUST6171/99P)     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.     

Trace Averaging Domain Decomposition Method with Nonconforming Finite Elements

1.IntroductionDomaindec0mpositionreferstonumericaJmethodsf0robtainingsoluti0nsofsci-entificandengineeringproblemsbycombiningsoluti0nstoproblemspo8ed0nphysica1subdomains,or,moregeneraJly,byc0mbiningsoluti0nst0appropriatelyconstructedsubproblems.IthasbeenasubjectofintenseinterestreceDtlybecause0fitssultabil-ityforimplementationonhighperformancecomputerarchitectures.Somepapersarelistedinthereferencesherein,wlilchindicatethatmuchprogresshasbeenmadeinthestudyofnonoverlaPdomaindecompositionmethods…     

Iterative solution of a singular convection-diffusion perturbation problem

An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.Received: February 12, 2004     

Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method

A new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method. It applies to problems involving either positive semi-definite or complex indefinite local matrices. The main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices. We prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down. Each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and, in contrast to a strict domain decomposition method, is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes.     

Synchronous exploration for the control of real-time external noise suppression in a three-dimensional subdomain

A mathematical (difference) model is proposed for a real-time active shielding device that shields an acoustic field in a given subdomain from the influence of sound sources located in an additional subdomain. An algorithm for computing the current control ensuring a prescribed process is based on information produced by the author’s technique of synchronous weak noise exploration. This information can be measured in real time. Active control problems for nonstationary solutions of linear difference equations in a three-dimensional domain consisting of two subdomains are studied using the difference potential method. The shape of the domain and the boundary conditions may depend on time, while the coefficients may depend on time and spatial coordinates. If the difference problem is a mathematical model of sound propagation, the goal of control is to change the acoustic field in the given subdomains, for example, to shield the acoustic field in one subdomain from the undesirable influence (noise) of sources located in the other subdomain.     

A Dual-Primal FETI method for incompressible Stokes equations

In this paper, a dual-primal FETI method is developed for incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures. The domain of the problem is decomposed into nonoverlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, the solution of an indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by the conjugate gradient method with a Dirichlet preconditioner. In each iteration step, both subdomain problems and a coarse level problem are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf-sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains. Numerical experiments demonstrate the scalability of this new method. This work is based on a doctoral dissertation completed at Courant Institute of Mathematical Sciences, New York University. This work was supported in part by the National Science Foundation under Grants NSF-CCR-9732208, and in part by the U.S. Department of Energy under contract DE-FG02-92ER25127.     

An almost fourth order uniformly convergent domain decomposition method for a coupled system of singularly perturbed reaction-diffusion equations

We consider a system of M(≥2) singularly perturbed equations of reaction-diffusion type coupled through the reaction term. A high order Schwarz domain decomposition method is developed to solve the system numerically. The method splits the original domain into three overlapping subdomains. On two boundary layer subdomains we use a compact fourth order difference scheme on a uniform mesh while on the interior subdomain we use a hybrid scheme on a uniform mesh. We prove that the method is almost fourth order ε-uniformly convergent. Furthermore, we prove that when ε is small, one iteration is sufficient to get almost fourth order ε-uniform convergence. Numerical experiments are performed to support the theoretical results.     

A regularized domain decomposition method with Lagrange multiplier

In this paper, we are concerned with the nonoverlapping domain decomposition method with Lagrange multiplier for three-dimensional second-order elliptic problems with no zeroth-order term. It is known that the methods result in a singular subproblem on each internal (floating) subdomain. To handle the singularity, we propose a regularization technique which transforms the corresponding singular problems into approximate positive definite problems. For the regularized method, one can build the interface equation of the multiplier directly. We first derive an optimal error estimate of the regularized approximation, and then develop a cheap preconditioned iterative method for solving the interface equation. For the new method, the cost of computation will not be increased comparing the case without any floating subdomain. The effectiveness of the new method will be confirmed by both theoretical analyzes and numerical experiments. The work is supported by Natural Science Foundation of China G10371129.     

The Uzawa–HSS method for saddle-point problems

Based on the Hermitian and skew-Hermitian splitting iteration scheme, we propose a Uzawa-type iteration method for solving a class of saddle-point problems whose coefficient matrix has non-Hermitian positive definite (1, 1)-block. The convergence properties of this novel method are analyzed, which show that the Uzawa-type iteration method is convergent if the iteration parameters satisfy suitable restrictions.     

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSS) preconditioner as well as the MGSS iteration method is derived in this paper, which generalize the modified shift-splitting (MSS) preconditioner and the MSS iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 317:535–546, 2017), respectively. The convergent and semi-convergent analyses of the MGSS iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. Meanwhile, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSS preconditioner and the MGSS iteration method and also illustrate that the MGSS iteration method outperforms the generalized shift-splitting (GSS) and the generalized modified shift-splitting (GMSS) iteration methods, and the MGSS preconditioner is superior to the shift-splitting (SS), GSS, modified SS (M-SS), GMSS and MSS preconditioners for the generalized minimal residual (GMRES) method for solving the nonsymmetric saddle point problems.     

A preconditioner for the h-p version of the finite element method in two dimensions

Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the - version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order , where is the diameter of the -th subdomain, and are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the -version (i.e. ) and the -version (i.e. ) as the special cases of the - version. Received August 15, 1995 / Revised version received November 13, 1995     

Convergence analysis of a balancing domain decomposition method for solving a class of indefinite linear systems

A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite systems of linear equations of the form (K?σ²M)u=f, which arise from solving eigenvalue problems when an inverse shifted method is used and also from the finite element discretization of Helmholtz equations. Here, both K and M are symmetric positive definite. The proposed BDDC method is closely related to the previous dual–primal finite element tearing and interconnecting method (FETI‐DP) for solving this type of problems (Appl. Numer. Math. 2005; 54 :150–166), where a coarse level problem containing certain free‐space solutions of the inherent homogeneous partial differential equation is used in the algorithm to accelerate the convergence. Under the condition that the diameters of the subdomains are small enough, the convergence rate of the proposed algorithm is established, which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with a decrease of the subdomain diameters. These results are supported by numerical experiments of solving a two‐dimensional problem. Copyright © 2009 John Wiley & Sons, Ltd.