Wolfgang L. Wendland

A finite element method for some integral equations of the first kind

Abstract This paper discusses a finite element approximation for a class of singular integral equations of the first kind. These integral equations are deduced from Dirichlet problems for strongly elliptic differential equations in two independent variables. By a variation of technique due to Aubin, it is shown that the Galerkin method with finite elements as trial functions leads to an optimal rate of convergence.

On the asymptotic convergence of collocation methods

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.

A Galerkin collocation method for some integral equations of the first kind

Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkins method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.ZusammenfassungWir entwickeln hier ein modifiziertes Kollokationsverfahren als vollständige numerische Diskretisierung des Galerkin-Verfahrens zur Lösung von Fredholmschen Integralgleichungen erster Art mit logarithmischem Kern als Hauptteil. Diese Methode verknüpft hohe Genauigkeit der Galerkin-Verfahren mit der Schnelligkeit von Kolookationsmethoden. Die asymptotische Fehleranalysis liefert optimale Konvergenzordnung im Sinne finiter Element-Approximation. Das Verfahren gehört zu den verbesserten Randelement-Methoden, die zur Lösung einer großen Klasse ebener Randwertprobleme unter Verwendung finiter Elemente auf der Randkurve verwendet werden können. Die numerischen Experimente bestätigen hohe Genauigkeit bei kurzen Rechenzeiten.

The convergence of spline collocation for strongly elliptic equations on curves

SummaryMost boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.

On numerical cubatures of singular surface integrals in boundary element methods

SummaryWe present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in ℝ3.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 Duffys 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 ℂ2.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.

The construction of some efficient preconditioners in the boundary element method

The discretization of first kind boundary integral equations leads in general to a dense system of linear equations, whose spectral condition number depends on the discretization used. Here we describe a general preconditioning technique based on a boundary integral operator of opposite order. The corresponding spectral equivalence inequalities are independent of the special discretization used, i.e., independent of the triangulations and of the trial functions. Since the proposed preconditioning form involves a (pseudo)inverse operator, one needs for its discretization only a stability condition for obtaining a spectrally equivalent approximation.

On the Asymptotic Convergence of Collocation Methods With Spline Functions of Even Degree

We investigate the collocation of linear one-dimensional strongly elliptic integro-dif- ferential or, more generally, pseudo-differential equations on closed curves by even-degree polynomial splines. The equations are collocated at the respective midpoints subject to uniform nodal grids of the even-degree s-splines. We prove quasioptimal and optimal order asymptotic error estimates in a scale of Sobolev spaces. The results apply, in particular, to boundary element methods used for numerical computations in engineering applications. The equations considered include Fredholm integral equations of the second and the first kind, singular integral equations involving Cauchy kernels, and integro-differential equations having convolutional or constant coefficient principal parts, respectively. The error analysis is based on an equivalence between the collocation and certain varia- tional methods with different degree splines as trial and as test functions. We further need to restrict our operators essentially to pseudo-differential operators having convolutional prin- cipal part. This allows an explicit Fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing Babuskas stability condition based on strong ellipticity. Our asymptotic error estimates extend partly those obtained by D. N. Arnold and W. L. Wendland from the case of odd-degree splines to the case of even-degree splines. 1. Introduction. In this paper we investigate the asymptotic convergence of the collocation method using even-degree polynomial splines applied to strongly elliptic systems of pseudo-differential equations on closed curves with convolutional prin- cipal part. The collocation here employs the Gauss points of one-point integration as collocation points which corresponds to the usual boundary element collocation in applications (5), (12), (17), (19), (26), (31), (34). This is in contrast to the method investigated by G. Schmidt in (29) where one collocates at the break points of even-order splines. The asymptotic convergence properties for the standard Galerkin method with splines of arbitrary orders are well-known (22). For the collocation method, however, asymptotic convergence, up to now, has been shown for strongly elliptic systems only in the case of odd-order splines by D. N. Arnold and W. L. Wendland (7).

A hypersingular boundary integral method for two-dimensional screen and crack problems

We analyze hypersingular integral equations on a curved open smooth arc in R 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen. By using the Mellin transformation we obtain sharp regularity results for the solution of these equations in Sobolev spaces in the form of singular expansions. In particular we show that the expansions do not contain logarithmic singularities

Boundary Element Methods and Their Asymptotic Convergence

Nowadays the most popular numerical methods for solving elliptic boundary value problems are finite differences, finite elements and, more recently, boundary element methods. The latter are numerical methods for solving integral equations (or their generalizations) on the boundary Γ of the given domain. The reduction of interior or exterior stationary boundary value problems as well as transmission problems to equivalent boundary integral equations is by no means unique, the two most popular reductions are the “direct method” and the “method of potentials”. In all these cases one needs a fundamental solution of the differential equations explicitly since it will be used in numerical computations. This restricts the boundary integral methods to cases of simple computability of a fundamental solution, i.e. essentially to differential equations with constant coefficients. The formulation on the boundary surface F reduces the dimensions of the original problem by one. For the computational treatment the boundary surface is decomposed into a finite number of segments and the boundary functions are approximated by corresponding finite elements, the boundary elements. The appropriately discretized version of the boundary integral equation then provides a finite system of linear approximate equations whose coefficient matrix, the influence matrix is fully distributed.

An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems

Here we present a new solution procedure for Helm-holtz and Laplacian Dirichlet screen and crack problems in IR2 via boundary integral equations of the first kind having as an unknown the jump of the normal derivative across the screen or a crack curve T. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problem. Via the method of local Mellin transform in [5]-[lo] and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behaviour near the screen or crack tips.With our integral equations we set up a Galerkin scheme on T and obtain high quasi-optimal convergence rates by using special singular elements besides regular splines as test and trial functions.