Numerical integration schemes for the BEM solution of hypersingular integral equations

Abstract

SUMMARYIn this paper we consider singular and hypersingular integral equations associated with 2D boundary value problems deÿned on domains whose boundaries have piecewise smooth parametric representations. In particular, given any (polynomial) local basis, we show how to compute e ciently all integrals required by the Galerkin method. The proposed numerical schemes require the user to specify only the local polynomial degrees; therefore they are quite suitable for the construction of p-and h-p versions of Galerkin BEM.

Error estimates in the numerical evaluation of some BEM singular integrals

“…and apply the procedure suggested in [1]. Since in this case the associated function F (y) (see (1.3)) has a log singularity only at the origin, it is sufficient to introduce which leads to the bound R n (f 1 ) = O(n −2q log n).…”

“…Since in this case the associated function F (y) (see (1.3)) has a log singularity only at the origin, it is sufficient to introduce which leads to the bound R n (f 1 ) = O(n −2q log n). For the evaluation of the inner integral we proceed as suggested in [1] (see also [7]): we use (1.2) when, for example, −0.05 = y 0 < y < 0, and the n-point Gauss-Legendre formula otherwise. We recall that in this particular case the remainder term of the latter rule is O(n −l ) with l arbitrarily large; therefore it can also be bounded by (2.34).…”

“…In the applications of Galerkin boundary element methods, for the numerical solution of one-dimensional singular and hypersingular integral equations, one has to deal (see [1]) with integrals which after proper normalization are of the form (1, b), and f (x, y) is a smooth function of both variables. When y ∈ (−1, 1), the inner integral is defined in the Cauchy principal value sense and will be denoted by the symbol .…”

“…A recent new numerical approach (see [1], [7]) suggests approximating the inner integral by a quadrature formula of interpolatory type, based on the zeros of Legendre polynomials, which exactly integrates the Cauchy kernel times an arbitrary polynomial of degree n − 1 and assumes the form…”

“…The composition of (1.2) and (1.3) then leads to the final formula Following [1], here we are referring to a p-formulation of the Galerkin BEM and are interested in the accurate calculation of the corresponding integrals of form (1.1). Therefore, we are not concerned with the behaviour of the quadrature rule, having fixed the number of its nodes, as the size of the domain of integration tends to zero.…”