On the Choice of a Derivative Boundary Element Formulation Using Hermite Interpolation

Abstract

SUMMARYThis paper reports on some problems that can arise with the use of regularized derivative boundary integral equations. It concentrates on developing a formulation for the simple Laplace equation using a cubic Hermite interpolation and shows how certain combinations of derivative and conventional boundary integral equations can result in a solution scheme severely lacking in stability. With some simple two-and three-dimensional geometries, the derivative equations on their own do not provide enough information to solve a Dirichlet problem. Even combinations of the conventional and derivative equations fail for some simple geometries. We conclude that the only consistently successful combination is that of the conventional equation with the tangential derivative equation, which showed cubic convergence of results with mesh refinement. Numerical results are presented for this scheme in both two and three dimensions.KEY WORDS derivative boundary integral equations; Hermite interpolation

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“…We expand the normal boundary B d (U (s, x) = ln(r) + c) in Equation (23) to the 'new degenerate scale', B d * , by using the modified fundamental solution as shown in Figure 2(a). The homogeneous Equation (23) reduces to…”

Degenerate scale for multiply connected Laplace problems

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Symmetric Galerkin boundary integral formulation for interface and multi-zone problems

“…At the risk of being repetitive, these rows do not include equations for the interface, i.e. equations (9) and (10) with the weighting functions l centred on an interface node. The minus sign multiplying S BFI is due to the change in sign for the interface ux, as mentioned above.…”

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