An Exact Boundary Technique for Improved Accuracy in the Finite Element Method

Abstract

Discretization procedures such as the finite difference and finite element methods for the solution of elliptic equations with Dirichlet boundary conditions suffer in general from the defect that for a given grid size, the solution is influenced only by a limited amount of the boundary data. Here blending function interpolants (Gordon, 1971) are used to construct an overall interpolant for a closed region which matches all the boundary information on the perimeter of the region for any value of the grid spacing. This overall interpolant is incorporated into the Ritz Galerkin version of the finite element method and error estimates obtained for this improved procedure. Two numerical examples are given which demonstrate the increased accuracy of the exact boundary scheme as compared with the discretized boundary scheme, and as expected, the improvement is particularly noticeable when the number of elements is small.