A spectral collocation method with triangular boundary elements

A spectral collocation method is proposed for solving integral equations arising from boundary integral formulations over surfaces discretized into flat or curved triangular elements. In the numerical approximation, a function of interest defined over a triangular element is approximated with an arbitrary-degree complete polynomial of two local triangle barycentric coordinates. Collocation points are then deployed at the nodes of a triangular Lobatto grid constructed on the basis of the zeros of the Lobatto polynomials, so that the number of collocation points over each element is equal to the number of terms in a complete polynomial expansion. The node interpolation functions are computed from the Proriol polynomial base using the generalized Vandermonde matrix approach. The spectral element method is applied to solve integral equations of the second kind arising from the double-layer representation of a harmonic function in the interior or exterior of a sphere. The numerical results confirm a rapid convergence with respect to the order of the polynomial expansion.