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Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps

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Let u and u V ∈ V be the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem Δu=f on ℙ, u|∂ℙ=0. For any m ∈ ℕ and any bounded polygonal domain ℙ, we provide a construction of a new sequence of finite dimensional subspaces V n such that where f ∈ Hm−1(ℙ) is arbitrary and C is a constant that depends only on ℙ and not on n (we do not assume u ∈ Hm+1(ℙ)). The existence of such a sequence of subspaces was first proved in a ground–breaking paper by Babuška [8]. Our method is different; it is based on the homogeneity properties of Sobolev spaces with weights and the well–posedness of non-homogeneous Dirichlet problem in suitable Sobolev spaces with weights, for which we provide a new proof, and which is a substitute of the usual “shift theorems” for boundary value problems in domains with smooth boundary. Our results extended right away to domains whose boundaries have conical points. We also indicate some of the changes necessary to deal with domains with cusps. Our numerical computation are in agreement with our theoretical results.

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Authors and Affiliations

  1. University of Delaware, Mathematical Sciences, 19716, Newark, DE, USA

    Constantin Băcuţă

  2. Mathematics Department, Pennsylvania State University, 16802, University Park, PA, USA

    Victor Nistor & Ludmil T. Zikatanov

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  1. Constantin Băcuţă
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  2. Victor Nistor
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  3. Ludmil T. Zikatanov
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Correspondence to Constantin Băcuţă.

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The authors were supported in part by the NSF grant DMS 02-09497. Victor Nistor was also partially supported by NSF grant DMS 02-00808.

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Băcuţă, C., Nistor, V. & Zikatanov, L. Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. Numer. Math. 100, 165–184 (2005). https://doi.org/10.1007/s00211-005-0588-3

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