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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 EMAIL logo and Ernst P. Stephan


We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for p(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.

MSC 2010: 65N30; 65N15; 74M15

Funding statement: 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.

A Basic Results

For the convenience of the reader, let us recall some, for the p-Laplace fundamental, inequalities. We refer to [1] for the proofs of these results.

Lemma 13 (cf. [23, Lemmas 5.1–5.3]).

For all p>1, δ0, and ξ, ηRn there holds


For all a, σ1, σ20, p>1, θ>0 there holds



γ={1if 1<p2,θ[1,) or 2<p<,θ(0,1),(p-1)-1if 1<p2,θ(0,1) or 2<p<,θ[1,).

For all a, σ1, σ20, p>1, and δ>0,


where β is such that δ-β=max{δ-1,δ-1p-1}.

Lemma 14.

Let a, bRn and sR. Then there holds


and for s0,


with C=1 if s[0,1] and C=2s-1 if s>1.

Lemma 15 ([15, Proposition 2.1]).

Let wW1,p(Ω).

  1. It holds |v|(1,w,p)0, and, when vW01,p(Ω), |v|(1,w,p)=0 if and only if v=0.

  2. There holds |v1+v2|(1,w,p)C(|v1|(1,w,p)+|v2|(1,w,p)) for any v1, v2W1,p(Ω).

  3. Furthermore, for 1<p2, there holds


  4. For 2p<, s[2,p], r=s(2-p)2-s, there holds


The constant in (A.1) and (A.2) can be stated explicitly and |v|W1,p(Ω) on the right-hand side of (A.1) can be eliminated.

Lemma 16.

For 1<p<2 there holds


and for p>2 there holds


for all v, wW1,p(Ω).

Corollary 17.

For 1<p<2 there exists a constant C>0 such that


for all v, wW1,p(Ω).

Lemma 18.

For p2 there exists a constant C(p)>0 such that


and for p2 there exists a constant C(p)>0 such that


for all u1, u2, v, wW1,p(Ω).


Ernst P. Stephan expresses his sincere thanks to Bishnu Lamichhane for his hospitality during the visit.


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Received: 2017-12-12
Revised: 2018-04-30
Accepted: 2018-06-07
Published Online: 2018-06-21
Published in Print: 2019-04-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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