Optimal preconditioning for the symmetric and nonsymmetric coupling of adaptive finite elements and boundary elements

We analyze a multilevel diagonal additive Schwarz preconditioner for the adaptive coupling of FEM and BEM for a linear 2D Laplace transmission problem. We rigorously prove that the condition number of the preconditioned system stays uniformly bounded, independently of the refinement level and the local mesh‐size of the underlying adaptively refined triangulations. Although the focus is on the nonsymmetric Johnson–Nédélec one‐equation coupling, the principle ideas also apply to other formulations like the symmetric FEM‐BEM coupling. Numerical experiments underline our theoretical findings. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 603–632, 2017     

The coupling method of finite elements and boundary elements for radiation problem

The paper presents the variational formulation and well posedness of the coupling method of finite elements and boundary elements for radiation problem. The convergence and optimal error estimate for the approximate solution and numerical experiment are provided.This research was supported in part by the Institute for Mathematics and its applications with funds provided by NSF, USA.     

A nodal coupling of finite and boundary elements

This article presents and analyzes a simple method for the exterior Laplace equation through the coupling of finite and boundary element methods. The main novelty is the use of a smooth parametric artificial boundary where boundary elements fit without effort together with a straight approximate triangulation in the bounded area, with the coupling done only in nodes. A numerically integrated version of the algorithm is also analyzed. Finally, an isoparametric variant with higher order is proposed. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 555–570, 2003     

On the coupling of boundary integral and finite element methods with nonlinear transmission conditions

We apply the coupling of boundary integral and finite element methods to study the weak solvability of certain exterior boundary value problems with nonlinear transmission conditions. As a model we consider a nonlinear second order elliptic equation in divergence form in a bounded inner region of the plane coupled with the Laplace equation in the corresponding exterior domain. The flux jump across the common nonlinear-linear interface is unknown and assumed to depend nonlinearly on the Dirichlet data. We establish the associated variational formulation in an operator equation setting and provide existence, uniqueness and approximation results.     

Implicit residual error estimators for the coupling of finite elements and boundary elements

We consider a symmetric Galerkin method for the coupling of finite elements and boundary elements for elliptic problems with a monotone operator in the finite element domain. We derive an a posteriori error estimator which involves the solution of equilibrated local Neumann problems in the finite element domain and requires computation of a residual term on the coupling interface. Finally, we discuss a similar approach for a coupling with Signorini contact conditions on the interface. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity

The coupling of finite elements and boundary elements is analyzed, where in the FEM domain we assume an incompressible elastic material governed by a uniformly monotone operator and use a Stokes‐type mixed FEM. In the BEM domain, linear elasticity is considered. We prove existence and uniqueness of the solution and quasi‐optimal convergence of a Galerkin method. We derive an a posteriori error estimator of explicit residual type. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 79–92, 2001     

Unified symmetric finite element and boundary integral variational coupling methods for linear fluid–structure interaction

This article concerns the development of energy-based variational formulations and their corresponding finite element–boundary element Rayleigh–Ritz approximations for solving the time-harmonic vibration and scattering problem of an inhomogeneous penetrable fluid or solid object immersed in a compressible, inviscid, homogeneous fluid. The resulting coupled finite element and boundary integral methods (FEM-BEM) have the following attractive features: (1) Separate direct and complementary variational principles lead naturally to several alternative structure variable and fluid variable methodologies. (2) The solution in the exterior region is represented by a combined single- and double-layer potential which ensures the validity of the methods for all wave numbers; even though this representation introduces hypersingular integrals, for actual computations the hypersingular operator may be rewritten in terms of single-layer potentials, which can be integrated by standard techniques. (3) Since the discretized equations for the interior region and for the boundary are derived from the first variation of bilinear functionals the resulting algebraic systems of equations are always symmetric. In addition, the transition conditions across the interface are natural. This allows one to approximate the solutions within the interior and exterior regions independently, without imposing any boundary constraints.     

Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics

Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included. Received February 21, 1995 / Revised version received December 21, 1995     

An approximation of incompressible miscible displacement in porous media by mixed finite elements and symmetric finite volume element method of characteristics

A nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium. A sequential implicit time‐stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated by a mixed finite element method and the concentration is approximated by a combination of a modified symmetric finite volume element method and the method of characteristics. Optimal order convergence in H¹ and in L² are proved for full discrete schemes. Finally, some numerical experiments are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Symmetric coupling of finite and boundary elements for exterior magnetic field problems

We consider a coupled finite element (fe)–boundary element (be) approach for three‐dimensional magnetic field problems. The formulation is based on a vector potential in a bounded domain (fe) and a scalar potential in an unbounded domain (be). We describe a coupled variational problem yielding a unique solution where the constraints in the trial spaces are replaced by appropriate side conditions. Then we discuss a Galerkin discretization of the coupled problem and prove a quasi‐optimal error estimate. Finally we discuss an efficient preconditioned iterative solution strategy for the resulting linear system. Copyright © 2002 John Wiley & Sons, Ltd.     

On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems

In this paper we extend some recent results on the stability of the Johnson–Nédelec coupling of finite and boundary element methods in the case of boundary value problems. In Of and Steinbach (Z Angew Math Mech 93:476–484, 2013), Sayas (SIAM J Numer Anal 47:3451–3463, 2009) and Steinbach (SIAM J Numer Anal 49:1521–1531, 2011), the case of a free-space transmission problem was considered, and sufficient and necessary conditions are stated which ensure the ellipticity of the bilinear form for the coupled problem. The proof was based on considering the energies which are related to both the interior and exterior problem. In the case of boundary value problems for either interior or exterior problems, additional estimates are required to bound the energy for the solutions of related subproblems. Moreover, several techniques for the stabilization of the coupled formulations are analysed. Applications involve boundary value problems with either hard or soft inclusions, exterior boundary value problems, and macro-element techniques.     

The coupling of boundary integral and finite element methods for the nonstationary exterior flow

In this article, we represent a new numerical method for solving the nonstationary Navier–Stokes equations in an unbounded domain. The technique consists of coupling the boundary integral and the finite element method. The variational formulation and the well-posedness of the coupling method are obtained. The convergence and optimal error estimates for the approximate solution are provided. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 549–565, 1998     

Analysis of free-edge effects by boundary finite element method

The scaled boundary finite element method is a novel semi-analytical analysis technique that combines the advantages of the finite element method and the boundary element method. Only a part of the boundary of the considered domain has to be discretized but nevertheless the method is solely finite element based. The governing equations are solved in the so-called scaling direction analytically, whereas a finite element approximation of the solution is performed in the circumferential directions, which form the boundary of the considered domain. Thus, the numerical effort can be reduced considerably when handling stress concentration problems such as e.g. the free-edge effect in laminated plates. In order to analyze the free-edge effect in a semi-infinite half plane, some kinematic coupling equations have to be introduced, that not only couple the degrees of freedom on the boundary, but also within the non-discretized domain. The implementation of kinematic coupling equations within the method is presented. Finally, the efficiency of the new approach is shown in some benchmark examples. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling of mixed finite elements and boundary elements

The symmetric coupling of mixed finite element and boundaryelement methods is analysed for a model interface problem withthe Laplacian. The coupling involves a further continuous ansatzfunction on the interface to link the discontinuous displacementfield to the necessarily continuous boundary ansatz function.Quasi-optimal a priori error estimates and sharp a posteriorierror estimates are established which justify adaptive mesh-refiningalgorithms. Numerical experiments prove the adaptive couplingas an efficient tool for the numerical treatment of transmissionproblems.     

Optimizations of a fast multipole symmetric Galerkin boundary element method code

Numerical Algorithms - This paper presents some optimizations of a fast multipole symmetric Galerkin boundary element method code. Except general optimizations, the code is specially sped up for...     

On a hybrid boundary element method

Summary. In this paper we study a symmetric boundary element method based on a hybrid discretization of the Steklov–Poincaré operator well suited for a symmetric coupling of finite and boundary elements. The representation used involves only single and double layer potentials and does not require the discretization of the hypersingular integral operator as in the symmetric formulation. The stability of the hybrid Galerkin discretization is based on a BBL–like stability condition for the trial spaces. Numerical examples confirm the theoretical results. Received December 15, 1997 / Revised version received December 21, 1998/ Published online November 17, 1999