TY - JOUR

T1 - Higher Order Mixed FEM for the Obstacle Problem of the p-Laplace Equation Using Biorthogonal Systems

AU - Banz, Lothar

AU - Lamichhane, Bishnu P.

AU - Stephan, Ernst P.

N1 - Funding information: 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.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p ϵ (1, ∞), 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.

AB - We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p ϵ (1, ∞), 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.

KW - A Posteriori Error Estimate

KW - A Priori Error Estimate

KW - Discrete Inf-Sup Constant

KW - hq-Adaptive Mixed FEM

KW - p-Laplace Obstacle Problem

UR - http://www.scopus.com/inward/record.url?scp=85049129510&partnerID=8YFLogxK

U2 - 10.1515/cmam-2018-0015

DO - 10.1515/cmam-2018-0015

M3 - Article

AN - SCOPUS:85049129510

VL - 19

SP - 169

EP - 188

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-4840

IS - 2

ER -