Finite element eigenvalues for the Laplacian over an L-shaped domain

Abstract The standard finite element schemes for computing eigenvalues of the Laplacian converge slowly with decreasing mesh size if the domain boundary contains a re-entrant corner. A superelement is constructed to cover the region surrounding a corner of angle 3π 2 . Inside the superelement, the mesh is refined and the trial function constrained to fit the known analytic form of the solution in the neighborhood of the corner singularity. Being compatible with linear and bilinear elements, the superelement is easily embodied into standard finite element programs. Calculations are made for an L-shaped domain with various boundary conditions. Comparison with other calculations show that the incorporation of the superelement has neutralised the deterioration in convergence which would otherwise have taken place due to the presence of the re-entrant corner.