A quadrature formula for integrals with nearby singularities

Abstract

SUMMARYThe purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities.In the boundary element method, the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point. Especially, field variables and their derivatives at a field point near a boundary cannot be computed accurately. In the present paper a quadrature formula for l-isolated singularities near the integration interval, based on Lagrange interpolatory polynomials, is obtained. The error estimation of the proposed formula is also given. Quadrature formulas for regular and singular integrals with conjugate poles are derived. Numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy.

On the evaluation of layer potentials close to their sources

“…The density µ(t) in (33,34) is assumed known at the points t nk on the panel l i , interpolated by P n [µ(t)](t), and evaluated at the points t mk . Polynomial interpolation at t nk cannot produce a better approximation to µ(t) than P n [µ(t)](t).…”

“…This section compares different implementations of the quadratures (33,34) for I i (z). The comparisons contain crude complexity estimates.…”

“…Combinations of singularity subtraction, smoothing, and various types of error correction, but without use of special quadrature points or overresolution, have also been suggested [3,26,27], as have schemes where variable transformations help to simplify the integrand [11,24]. Other interesting research directions involve generalized Gaussian quadrature [36] and interpolation of µ(τ ) only in combination with analytical integration [28,33] which is related to rational quadrature [9,12,29,35]. Here some implementations, involving the homotopy methods and various recurrence relations, could be costly and may require extended precision arithmetic for stability.…”

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

“…These include analytical evaluation [41,42], division of the interval about the singularity [43], subtraction of the singularity [42], special quadrature routines [44], and the use of non-linear coordinate transformations [45][46][47][48][49]. Each method has its own particular advantages and disadvantages depending on the type and order of the boundary elements used, and the form of the singular integral.…”

Inmost singularities and appropriate quadrature rules in the boundary element method

“…Many problems in the boundary element method can be reduced to a singular integral equation or to systems of singular integral equations [1]. For the solution of these equations where the known functions are Hölder continuous, approximate solutions through numerical techniques preserving the correct nature of singularities of the unknown function, have been developed [1][2][3][4][5][6][7][8][9][10].…”