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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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# Higher Order Mixed FEM for the Obstacle Problem of the p-Laplace Equation Using Biorthogonal Systems

Lothar Banz
/ Bishnu P. Lamichhane
• Corresponding author
• School of Mathematical & Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia
• Email
• Other articles by this author:
/ Ernst P. Stephan
Published Online: 2018-06-21 | DOI: https://doi.org/10.1515/cmam-2018-0015

## Abstract

We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for $p\in \left(1,\mathrm{\infty }\right)$, where the obstacle condition is imposed by using a Lagrange multiplier. In the discrete setting the Lagrange multiplier basis forms a biorthogonal system with the standard finite element basis so that the variational inequality can be realized in the point-wise form. We provide a general a posteriori error estimate for adaptivity and prove an a priori error estimate. We present numerical results for the adaptive scheme (mesh-size adaptivity with and without polynomial degree adaptation) for the singular case $p=1.5$ and the degenerated case $p=3$. We also present numerical results on the mesh independency and on the polynomial degree scaling of the discrete inf-sup constant when using biorthogonal basis functions for the dual variable defined on the same mesh with the same polynomial degree distribution.

MSC 2010: 65N30; 65N15; 74M15

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Revised: 2018-04-30

Accepted: 2018-06-07

Published Online: 2018-06-21

The visit of the third author to the University of Newcastle, Australia was partially supported by the priority research centre of the University of Newcastle for Computer-Assisted Research Mathematics and its Applications.

Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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