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

• Lothar Banz , Bishnu P. Lamichhane and Ernst P. Stephan

## Abstract

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

||ξ|p-2ξ-|η|p-2η|C|ξ-η|1-δ(|ξ|+|η|)p-2+δ,
(|ξ|p-2ξ-|η|p-2η,ξ-η)C|ξ-η|2+δ(|ξ|+|η|)p-2-δ.

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

(a+σ1)p-2σ1σ2θ-γ(a+σ1)p-2σ12+θ(a+σ2)p-2σ22,

where

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,

σ1σ2δ-β(ap-1+σ1)p-2σ12+δ(a+σ2)p-2σ22,

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

## Lemma 14.

Let a, bRn and sR. Then there holds

2-|s|(|a|+|b|)s(|a|+|a-b|)s2|s|(|a|+|b|)s

and for s0,

(|a|+|b|)sC(|a|s+|b|s)

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

(A.1)|v|W1,p(Ω)C(|w|W1,p(Ω),|v|W1,p(Ω))|v|(1,w,p)𝑎𝑛𝑑|v|(1,w,p)2|v|W1,p(Ω)p.

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

(A.2)|v|W1,p(Ω)p|v|(1,w,p)2C(|w|W1,r(Ω),|v|W1,r(Ω))|v|W1,s(Ω)2.

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

|v|W1,p(Ω)2(p-1)(2-p)2p(|w|W1,p(Ω)+|v|W1,p(Ω))2-p2|v|(1,w,p)
(A.3)1.062(|w|W1,p(Ω)+|v|W1,p(Ω))2-p2|v|(1,w,p),
|v|W1,p(Ω)2(p-1)(2-p)2p+1|w|W1,p(Ω)2-p2|v|(1,w,p)+p(2-p)p-2p2(p-1)(2-p)p2|v|(1,w,p)2p
(A.4)2.124|w|W1,p(Ω)2-p2|v|(1,w,p)+2|v|(1,w,p)2p

and for p>2 there holds

|v|(1,w,p)2max{1,2p-3}(|w|W1,p(Ω)p-2|v|W1,p(Ω)2+|v|W1,p(Ω)p)

for all v, wW1,p(Ω).

## Corollary 17.

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

C|v-w|W1,p(Ω)|w|W1,p(Ω)2-p2|v-w|(1,v,p)+|v-w|(1,v,p)2p𝑎𝑛𝑑C|v-w|W1,p(Ω)|w|W1,p(Ω)+|v-w|(1,v,p)2p

for all v, wW1,p(Ω).

## Lemma 18.

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

a(u1;u1,v)-a(u2;u2,v)C|u1-u2|(1,u1,p)2pvLp(Ω)

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

a(u1;u1,v)-a(u2;u2,v)Cmax{1,u1Lp(Ω)p-2}(|u1-u2|(1,w,p)2p+|u1-u2|(1,w,p)2p)vLp(Ω)

for all u1, u2, v, wW1,p(Ω).

## Acknowledgements

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

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