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

Lothar Banz; Bishnu P. Lamichhane; Ernst P. Stephan

doi:10.1515/cmam-2018-0015

Published by De Gruyter June 21, 2018

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

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2018-0015

Showing a limited preview of this publication:

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.

Keywords: A Priori Error Estimate; A Posteriori Error Estimate; Discrete Inf-Sup Constant

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, δ≥0, and ξ, η∈Rn there holds

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

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

For all a, σ1, σ2≥0, p>1, θ>0 there holds

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

where

γ={1if ⁢1<p≤2,θ∈[1,∞)⁢ or ⁢2<p<∞,θ∈(0,1),(p-1)-1if ⁢1<p≤2,θ∈(0,1)⁢ or ⁢2<p<∞,θ∈[1,∞).

For all a, σ1, σ2≥0, p>1, and δ>0,

σ1⁢σ2≤δ-β⁢(ap-1+σ1)p′-2⁢σ12+δ⁢(a+σ2)p-2⁢σ22,

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

Lemma 14.

Let a, b∈Rn and s∈R. Then there holds

2-|s|⁢(|a|+|b|)s≤(|a|+|a-b|)s≤2|s|⁢(|a|+|b|)s

and for s≥0,

(|a|+|b|)s≤C⁢(|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 w∈W1,p⁢(Ω).

It holds |v|(1,w,p)≥0, and, when v∈W01,p⁢(Ω), |v|(1,w,p)=0 if and only if v=0.
There holds |v1+v2|(1,w,p)≤C⁢(|v1|(1,w,p)+|v2|(1,w,p)) for any v1, v2∈W1,p⁢(Ω).
Furthermore, for 1<p≤2, 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.
For 2≤p<∞, s∈[2,p], r=s⁢(2-p)2-s, there holds
(A.2)|v|W1,p⁢(Ω)p≤|v|(1,w,p)2≤C⁢(|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)2⁢p⁢(|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)2⁢p+1⁢|w|W1,p⁢(Ω)2-p2⁢|v|(1,w,p)+p(2-p)p-2p⁢2(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)2≤max⁡{1,2p-3}⁢(|w|W1,p⁢(Ω)p-2⁢|v|W1,p⁢(Ω)2+|v|W1,p⁢(Ω)p)

for all v, w∈W1,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)2p 𝑎𝑛𝑑 C⁢|v-w|W1,p⁢(Ω)≤|w|W1,p⁢(Ω)+|v-w|(1,v,p)2p

for all v, w∈W1,p⁢(Ω).

Lemma 18.

For p≤2 there exists a constant C⁢(p)>0 such that

a⁢(u1;u1,v)-a⁢(u2;u2,v)≤C⁢|u1-u2|(1,u1,p)2p′⁢∥∇⁡v∥Lp⁢(Ω)

and for p≥2 there exists a constant C⁢(p)>0 such that

a⁢(u1;u1,v)-a⁢(u2;u2,v)≤C⁢max⁡{1,∥∇⁡u1∥Lp⁢(Ω)p-2}⁢(|u1-u2|(1,w,p)2p+|u1-u2|(1,w,p)2p′)⁢∥∇⁡v∥Lp⁢(Ω)

for all u1, u2, v, w∈W1,p⁢(Ω).

Acknowledgements

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

References

[1] L. Banz, B. P. Lamichhane and E. P. Stephan, Higher order FEM for the obstacle problem of the p-Laplacian – A variational inequality approach, preprint, (2017). 10.1016/j.camwa.2018.07.016Search in Google Scholar

[2] L. Banz and A. Schröder, Biorthogonal basis functions in hp-adaptive FEM for elliptic obstacle problems, Comput. Math. Appl. 70 (2015), no. 8, 1721–1742. 10.1016/j.camwa.2015.07.010Search in Google Scholar

[3] L. Banz and E. P. Stephan, A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle problems, Appl. Numer. Math. 76 (2014), 76–92. 10.1016/j.apnum.2013.10.004Search in Google Scholar

[4] L. Banz and E. P. Stephan, hp-adaptive IPDG/TDG-FEM for parabolic obstacle problems, Comput. Math. Appl. 67 (2014), 712–731. 10.1016/j.camwa.2013.03.003Search in Google Scholar

[5] J. W. Barrett and W. Liu, Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math. 68 (1994), no. 4, 437–456. 10.1007/s002110050071Search in Google Scholar

[6] S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math. 99 (2004), 225–249. 10.1007/s00211-004-0553-6Search in Google Scholar

[7] D. Braess, A posteriori error estimators for obstacle problems–another look, Numer. Math. 101 (2005), no. 3, 415–421. 10.1007/s00211-005-0634-1Search in Google Scholar

[8] D. Braess, C. Carstensen and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007), 455–471. 10.1007/s00211-007-0098-6Search in Google Scholar

[9] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3 ed., Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar

[10] C. Carstensen and J. Hu, An optimal adaptive finite element method for an obstacle problem, Comput. Methods Appl. Math. 15 (2015), 259–277. 10.1515/cmam-2015-0017Search in Google Scholar

[11] C. Carstensen and R. Klose, A posteriori finite element error control for the p-Laplace problem, SIAM J. Sci. Comput. 25 (2003), no. 3, 792–814. 10.1137/S1064827502416617Search in Google Scholar

[12] C. Carstensen, W. Liu and N. Yan, A posteriori FE error control for p-Laplacian by gradient recovery in quasi-norm, Math. Comp. 75 (2006), no. 256, 1599–1616. 10.1090/S0025-5718-06-01819-9Search in Google Scholar

[13] M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser, Basel, 2009. 10.1007/978-3-7643-9982-5Search in Google Scholar

[14] L. Diening and C. Kreuzer, Linear convergence of an adaptive finite element method for the p-Laplacian equation, SIAM J. Numer. Anal. 46 (2008), no. 2, 614–638. 10.1137/070681508Search in Google Scholar

[15] C. Ebmeyer and W. Liu, Quasi-norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems, Numer. Math. 100 (2005), no. 2, 233–258. 10.1007/s00211-005-0594-5Search in Google Scholar

[16] J. Gwinner, hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics, J. Comput. Appl. Math. 254 (2013), 175–184. 10.1016/j.cam.2013.03.013Search in Google Scholar

[17] S. Hüeber and B. Wohlmuth, A primal–dual active set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 27–29, 3147–3166. 10.1016/j.cma.2004.08.006Search in Google Scholar

[18] G. Jouvet and E. Bueler, Steady, shallow ice sheets as obstacle problems: Well-posedness and finite element approximation, SIAM J. Appl. Math. 72 (2012), no. 4, 1292–1314. 10.1137/110856654Search in Google Scholar

[19] R. Krause, B. Müller and G. Starke, An adaptive least-squares mixed finite element method for the signorini problem, Numer. Methods Partial Differential Equations 33 (2017), 276–289. 10.1002/num.22086Search in Google Scholar

[20] A. Krebs and E. Stephan, A p-version finite element method for nonlinear elliptic variational inequalities in 2D, Numer. Math. 105 (2007), no. 3, 457–480. 10.1007/s00211-006-0035-0Search in Google Scholar

[21] B. Lamichhane and B. Wohlmuth, Biorthogonal bases with local support and approximation properties, Math. Comp. 76 (2007), no. 257, 233–249. 10.1090/S0025-5718-06-01907-7Search in Google Scholar

[22] M. Lewicka and J. J. Manfredi, The obstacle problem for the p-Laplacian via optimal stopping of tug-of-war games, Probab. Theory Related Fields 167 (2017), no. 1–2, 349–378. 10.1007/s00440-015-0684-ySearch in Google Scholar

[23] W. Liu and N. Yan, Quasi-norm local error estimators for p-Laplacian, SIAM J. Numer. Anal. 39 (2001), no. 1, 100–127. 10.1137/S0036142999351613Search in Google Scholar

[24] W. Liu and N. Yan, On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian, SIAM J. Numer. Anal. 40 (2002), no. 5, 1870–1895. 10.1137/S0036142901393589Search in Google Scholar

[25] M. Maischak and E. P. Stephan, Adaptive hp-versions of BEM for Signorini problems, Appl. Numer. Math. 54 (2005), no. 3, 425–449. 10.1016/j.apnum.2004.09.012Search in Google Scholar

[26] D. Malkus, Eigenproblems associated with the discrete LBB condition for incompressible finite elements, Internat. J. Engrg. Sci. 19 (1981), no. 10, 1299–1310. 10.1016/0020-7225(81)90013-6Search in Google Scholar

[27] J. M. Melenk, hp-interpolation of nonsmooth functions and an application to hp-a posteriori error estimation, SIAM J. Numer. Anal. 43 (2005), no. 1, 127–155. 10.1137/S0036142903432930Search in Google Scholar

[28] N. Ovcharova and L. Banz, Coupling regularization and adaptive hp-BEM for the solution of a delamination problem, Numer. Math. 137 (2017), 303–337. 10.1007/s00211-017-0879-5Search in Google Scholar

[29] J. Qin, On the convergence of some low order mixed finite elements for incompressible fluids, Ph.D. thesis, The Pennsylvania State University, 1994. Search in Google Scholar

[30] A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal. 39 (2001), no. 1, 146–167. 10.1137/S0036142900370812Search in Google Scholar

[31] T. Wick, An error-oriented Newton/inexact augmented Lagrangian approach for fully monolithic phase-field fracture propagation, SIAM J. Sci. Comput. 39 (2017), no. 4, B589–B617. 10.1137/16M1063873Search in Google Scholar

Received: 2017-12-12

Revised: 2018-04-30

Accepted: 2018-06-07

Published Online: 2018-06-21

Published in Print: 2019-04-01

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

Abstract

A Basic Results

Lemma 13 (cf. [23, Lemmas 5.1–5.3]).

Lemma 14.

Lemma 15 ([15, Proposition 2.1]).

Lemma 16.

Corollary 17.

Lemma 18.

Acknowledgements

References

Journal and Issue

Articles in the same Issue