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Supraconvergence and Supercloseness of a Discretisation for Elliptic Third-kind Boundary-value Problems on Polygonal Domains

Emmrich, Etienne

The third-kind boundary-value problem for a second-order elliptic equation on a polygonal domain with variable coefficients, mixed derivatives, and first-order terms is approximated by a linear finite element method with first-order accurate quadrature. The corresponding bilinear form does not need to be strongly positive. The discretisation is equivalent to a finite difference scheme. Although the discretisation is in general only first-order consistent, supraconvergence, i.e., convergence of higher order, is shown to take place even on nonuniform grids. If neither oblique boundary sections nor mixed derivatives occur, then the optimal order s is achieved. The supraconvergence result is equivalent to the supercloseness of the gradient.
Published in: Computational methods in applied mathematics, 10.2478/cmam-2007-0008, De Gruyter