The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian

Summary In this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for Laplace equation in a smooth plane domain is given. The method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations. We prove the convergence of this method, provide asymptotic error estimates for the BEM-solution and give some numerical examples.     

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the Hk norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

An improved boundary element method for the charge density of a thin electrified plate in ℝ3

A weakly singular integral equation of the first kind on a plane surface piece Γ is solved approximately via the Galerkin method. The determination of the solution of this integral equation (with the single-layer potential) is a classical problem in physics, since its solution represents the charge density of a thin, electrified plate Γ loaded with some given potential. Using piecewise constant or piecewise bilinear boundary elements we derive asymptotic estimates for the Galerkin error in the energy norm and analyse the effect of graded meshes. Estimates in lower order Sobolev norms are obtained via the Aubin–Nitsche trick. We describe in detail the numerical implementation of the Galerkin method with both piecewise-constant and piecewise-linear boundary elements. Numerical experiments show experimental rates of convergence that confirm our theoretical, asymptotic results.     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

Wavelet approximations for first kind boundary integral equations on polygons

Summary. An elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation. Its Galerkin discretization with degrees of freedom on the boundary with spline wavelets as basis functions is analyzed. A truncation strategy is presented which allows to reduce the number of nonzero elements in the stiffness matrix from to entries. The condition numbers are bounded independently of the meshwidth. It is proved that the compressed scheme thus obtained yields in operations approximate solutions with the same asymptotic convergence rates as the full Galerkin scheme in the boundary energy norm as well as in interior points. Numerical examples show the asymptotic error analysis to be valid already for moderate values of . Received March 12, 1994 / Revised version received January 9, 1995     

Fully discrete Galerkin methods for systems of boundary integral equations

In this paper we analyze a family of full discretizations of spline Galerkin methods for a class of systems of boundary integral equations of the first kind with logarithmic principal part. We prove the existence of an asymptotic expansion of the error of the Galerkin and the optimal order Galerkin collocation method. We finally derive asymptotic expansions for some common postprocessings of the solutions, both exactly and under the effect of additional discretization. Some examples where these techniques apply are provided.     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

On numerical cubatures of singular surface integrals in boundary element methods

Summary We present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in ³.The class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three-dimensional domains by the boundary integral equation method. The possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point. This rules out weighted quadrature approaches with a fixed singular weight.For weakly singular integrals it is shown that Duffy's triangular coordinates leadalways to a removal of the kernel singularity. Also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero. These are based on the Rabinowitz-Richter estimates in connection with an asymptotic estimate of domains of analyticity in ².It is further shown that the modified extrapolation approach due to Lyness is in the weakly singular case always applicable. Corresponding error and asymptotic work estimates are presented.For the weakly singular as well as for Cauchy and hypersingular integrals which e.g. arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates. These rules are hierarchically constructed from finite part integration formulas in radial and Gaussian formulas in angular direction. Again, we show how the Rabinowitz-Richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size.Partially supported by the Priority Research Programme Boundary Element Methods of the German Research Foundation DFG under Grant No. We 659/16-1 (guest programme) and under AFOSR-grant 89-0252.     

Artificial boundary conditions for viscoelastic flows

The steady three-dimensional exterior flow of a viscoelastic non-Newtonian fluid is approximated by reducing the corresponding nonlinear elliptic–hyperbolic system to a bounded domain. On the truncation surface with a large radius R, nonlinear, local second-order artificial boundary conditions are constructed and a new concept of an artificial transport equation is introduced. Although the asymptotic structure of solutions at infinity is known, certain attributes cannot be found explicitly so that the artificial boundary conditions must be constructed with incomplete information on asymptotics. To show the existence of a solution to the approximation problem and to estimate the asymptotic precision, a general abstract scheme, adapted to the analysis of coupled systems of elliptic–hyperbolic type, is proposed. The error estimates, obtained in weighted Sobolev norms with arbitrarily large smoothness indices, prove an approximation of order O(R^−2+ε), with any ε>0. Our approach, in contrast to other papers on artificial boundary conditions, does not use the standard assumptions on compactly supported right-hand side f, leads, in particular, to pointwise estimates and provides error bounds with constants independent of both R and f. Copyright © 2007 John Wiley & Sons, Ltd.     

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

In this paper, we represent a new numerical method for solving the steady-state Stokes 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 one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type

The approximation of solutions to boundary value problems on unbounded domains by those on bounded domains is one of the main applications for artificial boundary conditions. Based on asymptotic analysis, here a new method is presented to construct local artificial boundary conditions for a very general class of elliptic problems where the main asymptotic term is not known explicitly. Existence and uniqueness of approximating solutions are proved together with asymptotically precise error estimates. One class of important examples includes boundary value problems for anisotropic elasticity and piezoelectricity. Copyright © 2004 John Wiley & Sons, Ltd.     

$C^1$ error estimation on the boundary for an exterior Neumann problem in $\mathbb R^3$

Summary.   In this paper we establish a error estimation on the boundary for the solution of an exterior Neumann problem in . To solve this problem we consider an integral representation which depends from the solution of a boundary integral equation. We use a full piecewise linear discretisation which on one hand leads to a simple numerical algorithm but on the other hand the error analysis becomes more difficult due to the singularity of the integral kernel. We construct a particular approximation for the solution of the boundary integral equation, for the solution of the Neumann problem and its gradient on the boundary and estimate their error. Received May 11, 1998 / Revised version received July 7, 1999 / Published online August 24, 2000     

Asymptotics near the boundary of spectral functions of elliptic self-adjoint boundary problems

We obtain the asymptotic behavior of the spectral function of an elliptic self-adjoint boundary problem in a closed domain and give a uniform estimate for the remainder. The estimate for the remainder is the same as the interior estimates of Agmon-Kannai and Hormander.     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.