An efficient finite element method for treating singularities in Laplace's equation

We present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape. Terms from the local solution supplement the ordinary basis functions in the finite element solution. All singular contributions reduce to boundary integrals after a double application of the divergence theorem to the Galerkin integrals, and the essential boundary conditions are weakly enforced using Lagrange multipliers. The proposed method eliminates the need for high-order integration, improves the overall accuracy, and yields very accurate estimates for the singular coefftcients. It also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions. Although here we solve the Laplace equation in two dimensions, the method is applicable to a more general class of problems.

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