A singularly perturbed elliptic problem is considered on the unit square. Its boundary data has a jump discontinuity at one corner of the square, so the solution of the problem exhibits a singularity there. To solve the problem numerically, the Galerkin finite element method is tested on various tensor-product meshes. It is demonstrated that the Shishkin mesh does not yield satisfactory results, but meshes with a sufficient degree of mesh grading will yield convergence in certain norms, uniformly in the singular perturbation parameter.
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