The coupling of boundary integral and finite element methods for the exterior steady Oseen problem

In this article, we present a new numerical method for solving the steady Oseen equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one. The solution in exterior region is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocity-pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. Finally, the optimal error estimates of the numerical solution are derived.Computer results will be discussed in a forthcoming paper.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

On the initialization methods of an exterior point algorithm for the assignment problem

In this paper, we present a theoretical investigation and an extensive computational study of exterior point simplex algorithm (EPSA) initialization methods for the assignment problem (AP). We describe the exterior point algorithm using three different initialization methods. Effective implementations are explored for each initialization method. Then we perform an experimental evaluation on a large set of benchmark problems from the TSPLib 95 and OR Library collections. The results obtained demonstrate the advantages of the three initialization methods. Finally, we give a theoretical justification of the initialization methods efficiency. We explain theoretically the computational ranking for these methods.     

Monotone methods for a discrete boundary problem

This paper is motivated by recent interests in space discrete Nagumo equations and is concerned with the existence of solutions of a nonlinear discrete boundary value problem. Monotone methods are used to derive the existence theorems. These methods, as is well known, provide constructive schemes for calculating the solutions.     

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.     

Analysis of finite element methods for the Brinkman problem

The parameter dependent Brinkman problem, covering a field of problems from the Darcy equations to the Stokes problem, is studied. A mathematical framework is introduced for analyzing the problem. Using this uniform a priori and a posteriori estimates for two families of finite element methods are proved. Nitsche’s method for imposing boundary conditions is discussed.     

An efficient method for solving elliptic boundary element problems with application to the tokamak vacuum problem

A method for regularizing ill-posed Neumann Poisson-type problems based on applying operator transformations is presented. This method can be implemented in the context of the finite element method to compute the solution to inhomogeneous Neumann boundary conditions; it allows to overcome cases where the Neumann problem otherwise admits an infinite number of solutions. As a test application, we solve the Grad–Shafranov boundary problem in a toroidally symmetric geometry. Solving the regularized Neumann response problem is found to be several orders of magnitudes more efficient than solving the Dirichlet problem, which makes the approach competitive with the boundary element method without the need to derive a Green function. In the context of the boundary element method, the operator transformation technique can also be applied to obtain the response of the Grad–Shafranov equation from the toroidal Laplace n=1 response matrix using a simple matrix transformation.     

An easy boundary element discretisation for microprocessor

An easy data input facility for two-dimensional boundary element analysis using microprocessors was presented in this paper. The coordinates of a few extreme points of the piecewise straight line approximation of the given geometry were digitised in computer-readable format through TABLET. Therefore, we can avoid errors occurring from manual input of the coordinates through KEYBOARD. A simple problem with steady fluid flow was shown as an example, for which inexperienced students could prepare a whole input data for the calculation within a few minutes.     

A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem

In this paper we propose a new a posteriori error estimator for a boundary element solution related to a Dirichlet problem with a second order elliptic partial differential operator. The method is based on an approximate solution of a boundary integral equation of the second kind by a Neumann series to estimate the error of a previously computed boundary element solution. For this one may use an arbitrary boundary element method, for example, a Galerkin, collocation or qualocation scheme, to solve an appropriate boundary integral equation. Due to the approximate solution of the error equation the proposed estimator provides high accuracy. A numerical example supports the theoretical results. Received: June 1999 / Accepted: September 1999     

An Internet-based computing platform for the boundary element method

Computers combined with the Internet are dramatically changing the engineering practices in design and analysis. More and more engineering software applications are moving to the Internet, prompted by the promising advantages, such as easy access for users everywhere with an Internet connection, easy upgrade of the software, easy control of the software. All these advantages will contribute to faster and cost-effective engineering software development and applications. Successful applications of the Internet computing depend largely on the speed of data transfer on the network. The boundary element method (BEM) has the inherent advantages over other domain-based numerical methods, because the size of the BEM model files are always considerably smaller, leading to faster data transfer on the current network. In this paper, a successful investigation in implementing the BEM on the Internet is presented. A BEM code for 2D elastostatic problems is used as the first solver in this work. A graphical-user interface (GUI) for the pre- and post-processing using Java, which provides platform-independent applications, is developed and implemented on the Internet. Demonstration problems using the developed platform clearly show the feasibility of the Internet-based computing and the potentials of this platform for future BEM development with further research.     

A Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary

In this paper, we investigate a Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary. Based on natural boundary reduction, the algorithm is constructed and its convergence is discussed. The finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain. The convergence rate is analyzed in detail for a typical domain. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.     

Iterative methods of solution for the elliptic fourth boundary value problem

A special form of the elliptic fourth boundary value problem has been investigated by Hockney (1965). The techniques used, however, are valid only for a Laplacian operator on a square mesh. In this paper, a more general solution process is introduced for a wide range of problems.     

Finite element methods for the Stokes problem on complicated domains

It is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process. In case of complicated domains which appear in many applications such as ground water flows this requirement sometimes becomes a bottleneck. The resolution condition links the computational complexity to the number (and size) of geometric details although the accuracy requirements, possibly, are moderate and would allow a (locally) coarse mesh width. Therefore even the coarsest available discretization can lead to a huge number of unknowns. The composite mini element is a remedy to this dilemma because the degrees of freedom are not linked to the number of geometric details. The basic concept for the Stokes problem with uniform no-slip boundary conditions has been introduced and analyzed in [D. Peterseim, S. Sauter, The composite mini element – coarse mesh computation of Stokes flows on complicated domains, SINUM, 46(6) (2008) 3181–3206]. Here, we generalize the composite mini element to slip, leak and Neumann boundary conditions so that it becomes applicable to this much larger and more important problem class. The main results are (a) the algorithmic concept remains unchanged and the new boundary conditions can be implemented as a weighted quadrature rule, (b) the stability and convergence can be proved under very mild assumption on the domain geometries, (c) the analysis is far from trivial and requires the development of substantially new tools compared to the simple case of uniform no-slip boundary conditions.     

A comparison of the boundary element and superposition methods

For some time now, the Boundary Integral Equation (BIE) Method or as it is alternately called the Boundary Element Method (BEM) has been hailed as the technique best suited to problems in elasticity and related fields, both for accuracy and efficiency. The authors demonstrate by example that this is not the case. A much simpler and more versatile technique, the Superposition Method (SUP), is introduced and is shown to outperform BEM in both areas. Followed by a discussion of the merits and drawbacks of each method, compact computer programs for both BEM and SUP and numerical results for nine different example problems are presented to support the authors' claim.     

Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem

We present the numerical implementation of a boundary element approximation for the plane mixed boundary value problem of the Laplacian. The performed Galerkin procedure is based on the direct boundary integral method. Its accuracy is improved by using appropriate singularity functions as additional test and trial functions besides quadratic splines. We analize the consistency error of the used numerical integrations and present asymptotic error estimates for the fully discretized numerical scheme which are of the same optimal orders as the Galerkin errors.     

A new fast multipole boundary element method for two dimensional acoustic problems

A new fast multipole boundary element method (BEM) is presented in this paper for solving large-scale two dimensional (2D) acoustic problems based on the improved Burton–Miller formulation. This algorithm has several important improvements. The fast multipole BEM employs the improved Burton–Miller formulation, and successfully overcomes the non-uniqueness difficulty associated with the conventional BEM for exterior acoustic problems. The improved Burton–Miller formulation contains only weakly singular integrals, and avoids the numerical difficulties associated to the evaluation of the hypersingular integral, it leads to the numerical implementations more efficient and straightforward. Furthermore, the fast multipole method (FMM) and the approximate inverse preconditioned generalized minimum residual method (GMRES) iterative solver are adopted to greatly improve the overall computational efficiency. The numerical examples with Neumann boundary conditions are presented that clearly demonstrate the accuracy and efficiency of the developed fast multipole BEM for solving large-scale 2D acoustic problems in a wide range of frequencies.     

Comparison of boundary element and finite element methods for dynamic analysis of elastoplastic plates

Boundary and finite element methodologies for the determination of the response of inelastic plates are compared and critically discussed. Flexural dynamic plate bending problems are considered and a hardening elastoplastic constitutive model is used to describe material behaviour. The domain/boundary element methodology using linear boundary and quadratic interior elements and the finite element method with quadratic Mindlin plate elements are used in this work. The discretized equations of motion in both methodologies are solved by an efficient step-by-step time integration algorithm. Numerical results obtained are presented and compared in order to access the accuracy and computational efficiency of the two methods. In order to make the comparison as meaningful as possible, boundary and finite element computer codes developed by the author are used in this paper. In general, boundary elements appear to be a better choice than finite elements with respect to computational efficiency for the same level of accuracy.     

An efficient solution algorithm for boundary element equations

Boundary element techniques result in the solution of a linear system of equations of the type HU = GQ + B, which can be transformed into a system of equations of the type AX = F. The coefficient matrix A requires the storage of a full matrix on the computer. This storage requirement, of the order of n^*n memory positions (n = number of equations), for a very large n is often considered negative for the boundary element method. Here, two algorithms are presented where the memory requirements to solve the system are only n^*(n - 1)/2 and n^*n/4 respectively. The algorithms do not necessitate any external storage devices nor do they increase the computational efforts.