Maximum norm estimates in the finite element method on plane polygonal domains. I

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Abstract
The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case.

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