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A p-version finite element method for nonlinear elliptic variational inequalities in 2D

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Abstract

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u p from the p-version for the obstacle problem. We prove the convergence of u p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.

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Authors and Affiliations

  1. Institut für Mathematik, Brandenburgische Technische Universität Cottbus, 03046, Cottbus, Germany

    Andreas Krebs

  2. Institut für Angewandte Mathematik, Universität Hannover, 30167, Hannover, Germany

    Ernst P. Stephan

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  1. Andreas Krebs
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  2. Ernst P. Stephan
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Correspondence to Andreas Krebs.

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Krebs, A., Stephan, E.P. A p-version finite element method for nonlinear elliptic variational inequalities in 2D. Numer. Math. 105, 457–480 (2007). https://doi.org/10.1007/s00211-006-0035-0

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