Abstract
Applications of three-dimensional Galerkin boundary element methods requirethe numerical evaluation of many four-dimensional integrals.In this paper we explore the possibility of using extrapolation quadrature.To do so, one needs appropriate error functional expansions.The treatment here is limited tointegration over a region $\mathcal T_1 \times \mathcal T_2$,where $\mathcal T_1$ and $\mathcal T_2$ are planar triangularelements in a hanging-chadconfiguration; that is, they have one vertex in common but are otherwise disjoint.We derive error expansions for product trapezoidal rulesvalid for integrands having an $|r_{12}|^{-1}$ factor. This factor gives riseto a weak singularity at the common vertex.
Highlights
Two-dimensional boundary value problems are set in IR3
In a conventional application of the Galerkin method, a surface is discretized into a set of plane triangular elements Ti, with 1 ≤ i ≤ n
This integral is evaluated for all pairs Ti × Tj, with 1 ≤ i ≤ j ≤ n
Summary
Two-dimensional boundary value problems are set in IR3. In a conventional application of the Galerkin method, a (two-dimensional) surface is discretized into a set of (twodimensional) plane triangular elements Ti, with 1 ≤ i ≤ n. We remark that, when the integrand function is regular and the domain is R1 × R2 the error expansion is the four-dimensional product of a simple modification of the classical Euler Maclaurin summation formula. This is: Theorem 1.1 When Φ, together with all partial derivatives of order p − 1 or less are integrable and those of order p are absolutely integrable over R1 × R2, .
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