Preconditioning for boundary element methods in domain decomposition

A comparison is made between the probabilistic domain decomposition (DD) method and a certain deterministic DD method for solving linear elliptic boundary-value problems. Since in the deterministic approach the CPU time is affected by intercommunications among the processors, it turns out that the probabilistic method performs better, especially when the number of subdomains (hence, of processors) is increased. This fact is clearly illustrated by some examples. The probabilistic DD algorithm has been implemented in an MPI environment, in order to exploit distributed computer architectures. Scalability and fault-tolerance of the probabilistic DD algorithm are emphasized.

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 multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

A Domain Decomposition (DD) method for solving time-harmonic aeroacoustic problems is presented. The computational domain is decomposed into subdomains, and the aerodynamic-aeroacoustic boundary-value problem is solved independently for each subdomain. On the artificially introduced subdomain boundaries impedance-type transmission conditions are imposed, which assure the uniqueness of the solution. The Finite Element method is used to discretize the subdomain problems. A direct solver (LU factorization) is used to solve the subdomain systems of equations and an iterative scheme updates the transmission conditions to recover the global solution. This DD method is applied to solve the classical problem of a thin airfoil in a transverse gust. The problem is formulated in terms of the unsteady pressure, and exact non-reflecting conditions are imposed on the outer computational boundary. The accuracy of the numerical scheme is validated by comparison with existing solutions for both the near-field unsteady pressure and the farfiled radiated sound. The convergence as well as the computational time and memory requirements of the present method are investigated. It is shown that by combining the subdomain direct solvers with global iterations, this DD method significantly reduces both the computational time and the memory requirements. The present algorithm is particularly suited for parallel computing.

Study of indentation by using domain decomposition and boundary elements methods on a microcomputer

When solving large size systems of equations by preconditioned iterative solution methods, one normally uses a fixed preconditioner which may be defined by some eigenvalue information, such as in a Chebyshev iteration method. In many problems, however, it may be more effective to use variable preconditioners, in particular when the eigenvalue information is not available.In the present paper, a recursive way of constructing variable‐step of, in general, nonlinear multilevel preconditioners for selfadjoint and coercive second‐order elliptic problems, discretized by the finite element method is proposed. The preconditioner is constructed recursively from the coarsest to finer and finer levels. Each preconditioning step requires only block‐diagonal solvers at all levels except at every k0, k0 ≥ 1 level where we perform a sufficient number ν, ν ≥ 1 of GCG‐type variable‐step iterations that involve the use again of a variable‐step preconditioning for that level.It turns out that for any sufficiently large value of k0 and, asymptotically, for ν sufficiently large, but not too large, the method has both an optimal rate of convergence and an optimal order of computational complexity, both for two and three space dimensional problem domains.The method requires no parameter estimates and the convergence results do not depend on the regularity of the elliptic problem.

In this work, we develop an efficient machine-learning–assisted framework for the approximation of partial differential equations arising in multiphysics science and engineering. Building on the Galerkin Neural Network (GNN) formulation for symmetric, positive-definite problems and leveraging the well-established efficiency of domain decomposition methods (DDM), we introduce a coupled GNN–DDM strategy designed to enhance scalability and computational performance. The proposed framework integrates data-driven trial spaces with physics-based subdomain solvers, enabling scalable training and solution procedures for large-scale multiphysics systems. We establish convergence of the coupled method using previous results, demonstrate its performance on a suite of benchmark multiphysics problems, and provide a flexible computational infrastructure built on modern Python libraries for reproducible and extensible implementation. Results show that the GNN–DDM approach achieves accurate, scalable, and efficient solutions, highlighting its potential as a practical tool for next-generation multiphysics simulation.

9R1. Natural Boundary Integral Method and Its Applications. - De-hao Yu (Inst of Comput Math and Sci/Eng Comput, Chinese Acad of Sci, Beijing, ROC). Kluwer Acad Publ, Dordrecht, Netherlands. Distributed in USA by Kluwer Acad Publ, Norwell MA. 2002. 539 pp. ISBN 1-4020-0457-5. $155.00.Reviewed by D Givoli (Dept of Aerospace Eng, Technion-Israel, Haifa, 32000, Israel).Kang Feng (1923-1993) was a well-known Chinese mathematician, and until his death was the President of the Chinese Society of Computational Mathematics. De-hao Yu, the author of this book, was his PhD student. This is an English version of a previous monograph by Yu that appeared in Chinese ten years ago. The book summarizes Yu’s research work based on the mathematical foundation laid by Feng. Most of Feng’s and Yu’s publications have appeared in Chinese and have not been generally accessible to western readers. This monograph, published by Kluwer, is especially welcome since it allows general access to the interesting work of both Feng and Yu. The book focuses on the concept of Natural Boundary Reduction (NBR) in the context of two-dimensional (2D) elliptic problems, and in particular problems governed by Laplace’s equation, the biharmonic equation, plane elasticity equations, and Stokes’ equations. The idea is to replace the differential equation in the given 2D domain Ω by an integral equation on its boundary Γ, and to solve this integral equation numerically on Γ via a variational formulation. This sounds like the basic idea of the Boundary Element Method (BEM); however, in NBR the reduction of the differential equation in Ω to an integral equation on Γ is performed differently, using the so-called natural integral operator, which is also known as the Dirichlet-to-Neumann (DtN) map. The book is written beautifully. It is very clear, interesting, and not dry despite being mathematically rigorous. Although the author is a mathematician, he wrote this book in a way that makes it accessible to mathematically-oriented graduate students and researchers in computational and applied mechanics and engineering. The basic ideas are described clearly and in a well-organized way. In addition, one can learn quite a lot from this book about areas as diverse as singular integrals, harmonic functions, and domain decomposition. The summary on singular integrals in Section 1.4, for example, is superb. The book, with 540 pages, is divided into seven chapters. Chapter 1 introduces the main ideas of NBR, and is followed by Chapters 2–5, which apply the method to the four differential equations mentioned above. Chapter 6 discusses the coupling of NBR and Finite Elements. This is similar to the DtN method, which has been devised independently in the west primarily for wave problems (which are not discussed at all in this book). The last chapter shows how to solve unbounded domain problems using NBR combined with iterative Domain Decomposition. The book also includes a preface (telling about Prof Feng), a list of more than 200 references (about half of which are Chinese publications), and an index. The NBR method is described on the back cover as a competitor to the standard BEM. Such a comparison is not totally appropriate. The BEM is a general method that can handle problems in irregular geometries. It makes use of the full-space Green’s function associated with the operator involved, which does not depend on the given geometry. On the other hand, NBR makes use of the natural integral operator, which depends on the specific geometry under consideration. Finding this operator on Γ (analytically) is equivalent to solving the problem in Ω analytically. Thus, in its simplest form, the NBR method can be applied in practice only in cases where the analytic solution is already known! Of course, there is no point in using NBR in such a manner. However, this does not mean that the idea underlying NBR is not useful. It is definitely useful when the domain of the problem under consideration involves a region where the problem is analytically-solvable and a region where it is not. Such problems are discussed in the last two chapters, where the practical usefulness of the method is demonstrated. In addition, understanding the various ways in which a problem can be represented (via partial differential equations, as a variational problem, by an integral equation, etc) is always beneficial and may lead to interesting insights regarding the properties of solutions and to new computational methods. Natural Boundary Integral Method and Its Applications is neither a course textbook nor a state-of-the-art research book on a wide scientific area, but it is a satisfying self-contained summary of a very interesting piece of work that has been hidden from the western reader so far. This monograph is highly recommended as an enjoyable and eye-opening reading for the mathematically-oriented researcher and practitioner of applied mechanics.

Some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset. One of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods. For domain decomposition and multigrid methods, the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems. In order to analyse the convergence rate, we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints. A special nonlinear interpolation operator is introduced for decomposing the functions.

In this paper, we study the bilateral or unilateral contact with Coulomb friction between two elastic solids, using a domain decomposition method coupled with the boundary element method. The decomposition method we have selected is the Schur complement method, a non-overlapping technique. It enables to reduce the solution of the global problem to the solution of a problem defined only on the contact surface. Moreover, its principal advantage is that computing is done separately on each solid. We have chosen to associate it with the boundary element method. Indeed, it only requires the discretization of the boundaries of solids. This technique of coupling reduces the number of unknowns and the time of computing. We have applied it to the study of indentation of an elastic foundation by an elastic flat punch and a sphere. In this last case, our results are in conformity with the Hertz theory and the analytical solution of Spence. Moreover, we have shown the influence of friction on the size of the contact radius and on the normal pressure at centre. Copyright © 1999 John Wiley & Sons, Ltd.

A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

Domain decomposition methods were first developed for elliptic problems, taking advantage of the strong regularity of their solutions. In the last two decades, many investigations have been devoted to improve the performance of these methods for elliptic and parabolic problems. The situation is less clear for hyperbolic problems with possible singular solutions. In this paper, we will discuss a nonoverlapping domain decomposition method for nonlinear hyperbolic problems. We use the finite volume method and an implicit version of the Roe approximate Riemann solver, and propose a new interface variable inspired by Dolean and Lanteri [1]. The new variable makes the Schur complement approach simpler and allows the treatment of diffusion terms. Numerical results for the compressible Navier-Stokes equations in various 2D and 3D configurations such as the Sod shock tube problem or the lid driven cavity problem show that our method is robust and efficient. Comparisons of performances on parallel computers with up to 512 processors are also reported.

In the presence of strong heterogeneities, it is well known that the use of explicit schemes for the transport of species in a porous medium suffers from severe restrictions on the time step. This has led to the development of implicit schemes that are increasingly favored by practitioners for their computational efficiency. The transport equation requires knowledge of the velocity field, which results from an elliptic problem (Darcy problem) that is the most expensive part of the computation. When considering large reservoirs, a cost-effective way of approximating the Darcy problems is using multiscale domain decomposition (MDD) methods. They allow for the pressure and velocity fields to be computed on coarse meshes (large scale), while detailed basis functions are defined locally, usually in parallel, in a much finer grid (small scale). In this work we adopt the Multiscale Robin Coupled Method (MRCM, Guiraldello et al., 2018; Rocha et al., 2020), which is a generalization of previous MDD methods that allows for great flexibility in the choice of interface spaces. In this article we investigate the combination of the MRCM with implicit transport schemes. A sequentially implicit strategy is proposed, with different trust-region algorithms ensuring the convergence of the transport solver. The method is assessed on several very stringent 2D two-phase problems, demonstrating its stability even for large time steps. It is also shown that the best accuracy is achieved by considering recently introduced non-polynomial interface spaces, since polynomial spaces are not optimal for high-contrast channelized permeability fields.

Domain decomposition for finite elements and their implementation on the IBM 3090/600 VF

A new preconditioned conjugate gradient (PCG)-based domain decomposition method is given for the solution of linear equations arising in the finite element method applied to the elliptic Neumann problem. The novelty of the proposed method is in the recommended preconditioner which is constructed by using cyclic matrix. The resulting preconditioned algorithms are well suited to parallel computation.

The unsteady heat equation is considered in thin structures. The asymptotic expansion of the solution constructed earlier is used to evaluate partial derivatives of the solution. The method of partial asymptotic domain decomposition is applied to the unsteady heat equation. It reduces the original 2D model to a hybrid dimension one, partially 2D, partially 1D with some special interface conditions between the 2D and 1D parts. The finite volume method is applied to numerically solve the hybrid dimension model. The error estimate is established. A numerical experiment confirms the theoretical error evaluation.