Selecting a Derivative Boundary Element Formulation for use with Hermitian Interpolation

Cubic Hermite interpolation ensures continuity of derivatives between elements, so as well as providing an excellent representation of a smooth function, it also accurately models its gradients. Another attractive feature of this interpolation is that because each node is shared by neighbouring elements, a mesh of cubic Hermite elements has no more unknowns than a mesh of quadratic Lagrange elements, even though the cubic elements provide the better modelling scheme. The difficulty with Hermite interpolation is that the conventional boundary element procedure does not provide enough linearly dependent equations even when solving Laplace’s equation on simple geometries. There are several other well-documented situations where the conventional boundary element method does not produce a well-conditioned system of equations. Amongst these are certain exterior acoustic problems[1], and elastic stress analysis in the neighbourhood of cracks[2][3]. The standard approach to overcome these difficulties is to use a derivative boundary integral equation. We investigate the use of this equation on its own and together with the conventional equation for problems governed by Laplace's equation with cubic Hermite interpolation.