Error analysis of boundary integral methods

Abstract

Many of the boundary value problems traditionally cast as partial differential equations can be reformulated as integral equations over the boundary. After an introduction to boundary integral equations, this review describes some of the methods which have been proposed for their approximate solution. It discusses, as simply as possible, some of the techniques used in their error analysis, and points to areas in which the theory is still unsatisfactory.

The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

“…If L is the operator associated with Symm's equation on a smooth closed curve, then Lo is given by (2.1) and (2.1a) with a = -1/2, and Lx is bounded from Hp to H'p for any s, t eR (see e.g. [12]). The condition of Theorem 3.1 on L» is obviously satisfied.…”

“…This kind of integral equation is of importance in solving interior or exterior boundary value problems of potential theory. The most common example is Symm's first-kind integral equation with logarithmic kernel (see [6,7,12]). Other applications are hypersingular integral equations and singular integral equations of Cauchy type, which occur, e.g., in elasticity (see [18]).…”

Numerical solution of boundary integral equations by means of attenuation factors

“…A theoretical study of the method for C ∞ -curves was performed in McLean (1986) and McLean et al (1989). Starting with Hsiao et al (1980), the splitting (1.4) was subsequently used by several authors for the theoretical study of equation (1.1) (see, e.g., (Graham & Yan, 1990;Sloan, 1992, and the literature cited therein, and for example Yan & Sloan, 1988, for polygonal curves). In Hsiao et al (1980), the existence of a Fredholm equation of the second kind derived from (1.1) was first noticed.…”

“…, where c k (h) denotes the (k, )th two-dimensional Fourier coefficient of h. By analysing the Fourier series of Qx, which can be given explicitly, one can easily see that Q is a bounded operator from H p (I ) to H p+1 (I ) for any p ∈ R (Sloan, 1992). The inverse operator Q −1 is a bounded operator from H p+1 (I ) to H p (I ) and can be written in terms of the differentiation operator D and the conjugation operator K as (Berrut, 1986) …”

A fast numerical solution for the first kind boundary integral equation for the Helmholtz equation

“…In [1,5,20,21], the quadrature method and the modified quadrature method are considered. In [22,23], the qualocation method has been developed as a new approximation, which tries to combine the advantage of the Galerkin method and the collocation method into a new scheme.…”