Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains

Error estimates in the numerical evaluation of some BEM singular integrals

“…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).…”

“…We recall that if it is too large, then the recurrence relationships used to compute the weights of the internal quadrature rule are unstable (see [7]); if, on the contrary, it is too small, then the Gauss-Legendre rule that we use to compute the inner integral whenever −1 < y ≤ y 0 would require too many points to produce the required accuracy. A good choice of y 0 should be suggested by a criterion analogous to that used in [7].…”

“…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…”

Numerical Integration of Functions with Endpoint Singularities and/or Complex Poles in 3D Galerkin Boundary Element Methods

“…Indeed, in the case of weak singularities a proper change of variable can make the integrand function arbitrarily smooth. This, combined with the use of a standard rule like the Gauss-Legendre one or the composite trapezoidal formula, allows to achieve high accuracy using a low number of abscissas (see [2], [8], [9], [11], [14], [16]). Here we recall some of the known changes of variable, all having as co-domain the interval (0, 1), which have been proposed in the literature.…”

New Numerical Integration Schemes for Applications of Galerkin Bem to 2-D Problems

“…u(x) = u(x) on 1 (Dirichlet condition) (12) t(x) = @u @n = t(x) on 2 (Neumann condition) (13) where u(x) is the potential, t(x) is the ux and n is the unit outward normal vector of , u and t are given functions, then we obtain the equivalent BIE formulation of type (4), (5) with the G hk kernels replaced by the g hk ones. These are deÿned as follows:…”