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

Article

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

Published/Copyright: June 21, 2018

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From the journal Computational Methods in Applied Mathematics Volume 19 Issue 2

Abstract

We consider a mixed finite element method for an obstacle problem with the p-Laplace differential operator for $p \in (1, \infty)$ $p∈(1,∞)$ , 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 $p = 1.5$ $p=1.5$ and the degenerated case $p = 3$ $p=3$ . 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 $p > 1$ $p>1$ , $δ \geq 0$ $δ≥0$ , and ξ, $η \in R^{n}$ $η∈Rn$ there holds

| {| ξ |}^{p - 2} ξ - {| η |}^{p - 2} η | \leq C {| ξ - η |}^{1 - δ} {(| ξ | + | η |)}^{p - 2 + δ},

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

({| ξ |}^{p - 2} ξ - {| η |}^{p - 2} η, ξ - η) \geq C {| ξ - η |}^{2 + 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, $σ_{1}$ $σ1$ , $σ_{2} \geq 0$ $σ2≥0$ , $p > 1$ $p>1$ , $θ > 0$ $θ>0$ there holds

{(a + σ_{1})}^{p - 2} σ_{1} σ_{2} \leq θ^{- γ} {(a + σ_{1})}^{p - 2} σ_{1}^{2} + θ {(a + σ_{2})}^{p - 2} σ_{2}^{2},

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

where

γ = {\begin{matrix} 1 & if 1 < p \leq 2, θ \in [1, \infty) or 2 < p < \infty, θ \in (0, 1), \\ {(p - 1)}^{- 1} & if 1 < p \leq 2, θ \in (0, 1) or 2 < p < \infty, θ \in [1, \infty) . \end{matrix}

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

For all a, $σ_{1}$ $σ1$ , $σ_{2} \geq 0$ $σ2≥0$ , $p > 1$ $p>1$ , and $δ > 0$ $δ>0$ ,

σ_{1} σ_{2} \leq δ^{- β} {(a^{p - 1} + σ_{1})}^{p^{'} - 2} σ_{1}^{2} + δ {(a + σ_{2})}^{p - 2} σ_{2}^{2},

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

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

Lemma 14.

Let a, $b \in R^{n}$ $b∈Rn$ and $s \in R$ $s∈R$ . Then there holds

2^{- | s |} {(| a | + | b |)}^{s} \leq {(| a | + | a - b |)}^{s} \leq 2^{| s |} {(| a | + | b |)}^{s}

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

and for $s \geq 0$ $s≥0$ ,

{(| a | + | b |)}^{s} \leq C ({| a |}^{s} + {| b |}^{s})

$(|a|+|b|)s≤C⁢(|a|s+|b|s)$

with $C = 1$ $C=1$ if $s \in [0, 1]$ $s∈[0,1]$ and $C = 2^{s - 1}$ $C=2s-1$ if $s > 1$ $s>1$ .

Lemma 15 ([15, Proposition 2.1]).

Let $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

It holds ${| v |}_{(1, w, p)} \geq 0$ $|v|(1,w,p)≥0$ , and, when $v \in W_{0}^{1, p} (Ω)$ $v∈W01,p⁢(Ω)$ , ${| v |}_{(1, w, p)} = 0$ $|v|(1,w,p)=0$ if and only if $v = 0$ $v=0$ .
There holds ${| v_{1} + v_{2} |}_{(1, w, p)} \leq C ({| v_{1} |}_{(1, w, p)} + {| v_{2} |}_{(1, w, p)})$ $|v1+v2|(1,w,p)≤C⁢(|v1|(1,w,p)+|v2|(1,w,p))$ for any $v_{1}$ $v1$ , $v_{2} \in W^{1, p} (Ω)$ $v2∈W1,p⁢(Ω)$ .
Furthermore, for $1 < p \leq 2$ $1<p≤2$ , there holds
(A.1) ${| v |}_{W^{1, p} (Ω)} \leq C ({| w |}_{W^{1, p} (Ω)}, {| v |}_{W^{1, p} (Ω)}) {| v |}_{(1, w, p)} 𝑎𝑛𝑑 {| v |}_{(1, w, p)}^{2} \leq {| v |}_{W^{1, p} (Ω)}^{p} .$ $|v|W1,p⁢(Ω)≤C⁢(|w|W1,p⁢(Ω),|v|W1,p⁢(Ω))⁢|v|(1,w,p) 𝑎𝑛𝑑 |v|(1,w,p)2≤|v|W1,p⁢(Ω)p.$
For $2 \leq p < \infty$ $2≤p<∞$ , $s \in [2, p]$ $s∈[2,p]$ , $r = \frac{s (2 - p)}{2 - s}$ $r=s⁢(2-p)2-s$ , there holds
(A.2) ${| v |}_{W^{1, p} (Ω)}^{p} \leq {| v |}_{(1, w, p)}^{2} \leq C ({| w |}_{W^{1, r} (Ω)}, {| v |}_{W^{1, r} (Ω)}) {| v |}_{W^{1, s} (Ω)}^{2} .$ $|v|W1,p⁢(Ω)p≤|v|(1,w,p)2≤C⁢(|w|W1,r⁢(Ω),|v|W1,r⁢(Ω))⁢|v|W1,s⁢(Ω)2.$

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

Lemma 16.

For $1 < p < 2$ $1<p<2$ there holds

{| v |}_{W^{1, p} (Ω)} \leq 2^{\frac{(p - 1) (2 - p)}{2 p}} {({| w |}_{W^{1, p} (Ω)} + {| v |}_{W^{1, p} (Ω)})}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)}

$|v|W1,p⁢(Ω)≤2(p-1)⁢(2-p)2⁢p⁢(|w|W1,p⁢(Ω)+|v|W1,p⁢(Ω))2-p2⁢|v|(1,w,p)$

(A.3)

\leq 1.062 {({| w |}_{W^{1, p} (Ω)} + {| v |}_{W^{1, p} (Ω)})}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)},

$≤1.062⁢(|w|W1,p⁢(Ω)+|v|W1,p⁢(Ω))2-p2⁢|v|(1,w,p),$

{| v |}_{W^{1, p} (Ω)} \leq 2^{\frac{(p - 1) (2 - p)}{2 p} + 1} {| w |}_{W^{1, p} (Ω)}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)} + \frac{p}{{(2 - p)}^{\frac{p - 2}{p}}} 2^{\frac{(p - 1) (2 - p)}{p^{2}}} {| v |}_{(1, w, p)}^{\frac{2}{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)

\leq 2.124 {| w |}_{W^{1, p} (Ω)}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)} + 2 {| v |}_{(1, w, p)}^{\frac{2}{p}}

$≤2.124⁢|w|W1,p⁢(Ω)2-p2⁢|v|(1,w,p)+2⁢|v|(1,w,p)2p$

and for $p > 2$ $p>2$ there holds

{| v |}_{(1, w, p)}^{2} \leq \max {1, 2^{p - 3}} ({| w |}_{W^{1, p} (Ω)}^{p - 2} {| v |}_{W^{1, p} (Ω)}^{2} + {| v |}_{W^{1, p} (Ω)}^{p})

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

for all v, $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

Corollary 17.

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

C {| v - w |}_{W^{1, p} (Ω)} \leq {| w |}_{W^{1, p} (Ω)}^{\frac{2 - p}{2}} {| v - w |}_{(1, v, p)} + {| v - w |}_{(1, v, p)}^{\frac{2}{p}} 𝑎𝑛𝑑 C {| v - w |}_{W^{1, p} (Ω)} \leq {| w |}_{W^{1, p} (Ω)} + {| v - w |}_{(1, v, p)}^{\frac{2}{p}}

$C⁢|v-w|W1,p⁢(Ω)≤|w|W1,p⁢(Ω)2-p2⁢|v-w|(1,v,p)+|v-w|(1,v,p)2p 𝑎𝑛𝑑 C⁢|v-w|W1,p⁢(Ω)≤|w|W1,p⁢(Ω)+|v-w|(1,v,p)2p$

for all v, $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

Lemma 18.

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

a (u_{1}; u_{1}, v) - a (u_{2}; u_{2}, v) \leq C {| u_{1} - u_{2} |}_{(1, u_{1}, p)}^{\frac{2}{p^{'}}} {∥ \nabla v ∥}_{L^{p} (Ω)}

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

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

a (u_{1}; u_{1}, v) - a (u_{2}; u_{2}, v) \leq C \max {1, {∥ \nabla u_{1} ∥}_{L^{p} (Ω)}^{p - 2}} ({| u_{1} - u_{2} |}_{(1, w, p)}^{\frac{2}{p}} + {| u_{1} - u_{2} |}_{(1, w, p)}^{\frac{2}{p^{'}}}) {∥ \nabla v ∥}_{L^{p} (Ω)}

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

for all $u_{1}$ $u1$ , $u_{2}$ $u2$ , v, $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

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

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Abstract

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

MSC 2010: 65N30; 65N15; 74M15

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 $p > 1$ $p>1$ , $δ \geq 0$ $δ≥0$ , and ξ, $η \in R^{n}$ $η∈Rn$ there holds

| {| ξ |}^{p - 2} ξ - {| η |}^{p - 2} η | \leq C {| ξ - η |}^{1 - δ} {(| ξ | + | η |)}^{p - 2 + δ},

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

({| ξ |}^{p - 2} ξ - {| η |}^{p - 2} η, ξ - η) \geq C {| ξ - η |}^{2 + 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, $σ_{1}$ $σ1$ , $σ_{2} \geq 0$ $σ2≥0$ , $p > 1$ $p>1$ , $θ > 0$ $θ>0$ there holds

{(a + σ_{1})}^{p - 2} σ_{1} σ_{2} \leq θ^{- γ} {(a + σ_{1})}^{p - 2} σ_{1}^{2} + θ {(a + σ_{2})}^{p - 2} σ_{2}^{2},

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

where

γ = {\begin{matrix} 1 & if 1 < p \leq 2, θ \in [1, \infty) or 2 < p < \infty, θ \in (0, 1), \\ {(p - 1)}^{- 1} & if 1 < p \leq 2, θ \in (0, 1) or 2 < p < \infty, θ \in [1, \infty) . \end{matrix}

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

For all a, $σ_{1}$ $σ1$ , $σ_{2} \geq 0$ $σ2≥0$ , $p > 1$ $p>1$ , and $δ > 0$ $δ>0$ ,

σ_{1} σ_{2} \leq δ^{- β} {(a^{p - 1} + σ_{1})}^{p^{'} - 2} σ_{1}^{2} + δ {(a + σ_{2})}^{p - 2} σ_{2}^{2},

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

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

Lemma 14.

Let a, $b \in R^{n}$ $b∈Rn$ and $s \in R$ $s∈R$ . Then there holds

2^{- | s |} {(| a | + | b |)}^{s} \leq {(| a | + | a - b |)}^{s} \leq 2^{| s |} {(| a | + | b |)}^{s}

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

and for $s \geq 0$ $s≥0$ ,

{(| a | + | b |)}^{s} \leq C ({| a |}^{s} + {| b |}^{s})

$(|a|+|b|)s≤C⁢(|a|s+|b|s)$

with $C = 1$ $C=1$ if $s \in [0, 1]$ $s∈[0,1]$ and $C = 2^{s - 1}$ $C=2s-1$ if $s > 1$ $s>1$ .

Lemma 15 ([15, Proposition 2.1]).

Let $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

It holds ${| v |}_{(1, w, p)} \geq 0$ $|v|(1,w,p)≥0$ , and, when $v \in W_{0}^{1, p} (Ω)$ $v∈W01,p⁢(Ω)$ , ${| v |}_{(1, w, p)} = 0$ $|v|(1,w,p)=0$ if and only if $v = 0$ $v=0$ .
There holds ${| v_{1} + v_{2} |}_{(1, w, p)} \leq C ({| v_{1} |}_{(1, w, p)} + {| v_{2} |}_{(1, w, p)})$ $|v1+v2|(1,w,p)≤C⁢(|v1|(1,w,p)+|v2|(1,w,p))$ for any $v_{1}$ $v1$ , $v_{2} \in W^{1, p} (Ω)$ $v2∈W1,p⁢(Ω)$ .
Furthermore, for $1 < p \leq 2$ $1<p≤2$ , there holds
(A.1) ${| v |}_{W^{1, p} (Ω)} \leq C ({| w |}_{W^{1, p} (Ω)}, {| v |}_{W^{1, p} (Ω)}) {| v |}_{(1, w, p)} 𝑎𝑛𝑑 {| v |}_{(1, w, p)}^{2} \leq {| v |}_{W^{1, p} (Ω)}^{p} .$ $|v|W1,p⁢(Ω)≤C⁢(|w|W1,p⁢(Ω),|v|W1,p⁢(Ω))⁢|v|(1,w,p) 𝑎𝑛𝑑 |v|(1,w,p)2≤|v|W1,p⁢(Ω)p.$
For $2 \leq p < \infty$ $2≤p<∞$ , $s \in [2, p]$ $s∈[2,p]$ , $r = \frac{s (2 - p)}{2 - s}$ $r=s⁢(2-p)2-s$ , there holds
(A.2) ${| v |}_{W^{1, p} (Ω)}^{p} \leq {| v |}_{(1, w, p)}^{2} \leq C ({| w |}_{W^{1, r} (Ω)}, {| v |}_{W^{1, r} (Ω)}) {| v |}_{W^{1, s} (Ω)}^{2} .$ $|v|W1,p⁢(Ω)p≤|v|(1,w,p)2≤C⁢(|w|W1,r⁢(Ω),|v|W1,r⁢(Ω))⁢|v|W1,s⁢(Ω)2.$

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

Lemma 16.

For $1 < p < 2$ $1<p<2$ there holds

{| v |}_{W^{1, p} (Ω)} \leq 2^{\frac{(p - 1) (2 - p)}{2 p}} {({| w |}_{W^{1, p} (Ω)} + {| v |}_{W^{1, p} (Ω)})}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)}

$|v|W1,p⁢(Ω)≤2(p-1)⁢(2-p)2⁢p⁢(|w|W1,p⁢(Ω)+|v|W1,p⁢(Ω))2-p2⁢|v|(1,w,p)$

(A.3)

\leq 1.062 {({| w |}_{W^{1, p} (Ω)} + {| v |}_{W^{1, p} (Ω)})}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)},

$≤1.062⁢(|w|W1,p⁢(Ω)+|v|W1,p⁢(Ω))2-p2⁢|v|(1,w,p),$

{| v |}_{W^{1, p} (Ω)} \leq 2^{\frac{(p - 1) (2 - p)}{2 p} + 1} {| w |}_{W^{1, p} (Ω)}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)} + \frac{p}{{(2 - p)}^{\frac{p - 2}{p}}} 2^{\frac{(p - 1) (2 - p)}{p^{2}}} {| v |}_{(1, w, p)}^{\frac{2}{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)

\leq 2.124 {| w |}_{W^{1, p} (Ω)}^{\frac{2 - p}{2}} {| v |}_{(1, w, p)} + 2 {| v |}_{(1, w, p)}^{\frac{2}{p}}

$≤2.124⁢|w|W1,p⁢(Ω)2-p2⁢|v|(1,w,p)+2⁢|v|(1,w,p)2p$

and for $p > 2$ $p>2$ there holds

{| v |}_{(1, w, p)}^{2} \leq \max {1, 2^{p - 3}} ({| w |}_{W^{1, p} (Ω)}^{p - 2} {| v |}_{W^{1, p} (Ω)}^{2} + {| v |}_{W^{1, p} (Ω)}^{p})

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

for all v, $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

Corollary 17.

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

C {| v - w |}_{W^{1, p} (Ω)} \leq {| w |}_{W^{1, p} (Ω)}^{\frac{2 - p}{2}} {| v - w |}_{(1, v, p)} + {| v - w |}_{(1, v, p)}^{\frac{2}{p}} 𝑎𝑛𝑑 C {| v - w |}_{W^{1, p} (Ω)} \leq {| w |}_{W^{1, p} (Ω)} + {| v - w |}_{(1, v, p)}^{\frac{2}{p}}

$C⁢|v-w|W1,p⁢(Ω)≤|w|W1,p⁢(Ω)2-p2⁢|v-w|(1,v,p)+|v-w|(1,v,p)2p 𝑎𝑛𝑑 C⁢|v-w|W1,p⁢(Ω)≤|w|W1,p⁢(Ω)+|v-w|(1,v,p)2p$

for all v, $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

Lemma 18.

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

a (u_{1}; u_{1}, v) - a (u_{2}; u_{2}, v) \leq C {| u_{1} - u_{2} |}_{(1, u_{1}, p)}^{\frac{2}{p^{'}}} {∥ \nabla v ∥}_{L^{p} (Ω)}

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

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

a (u_{1}; u_{1}, v) - a (u_{2}; u_{2}, v) \leq C \max {1, {∥ \nabla u_{1} ∥}_{L^{p} (Ω)}^{p - 2}} ({| u_{1} - u_{2} |}_{(1, w, p)}^{\frac{2}{p}} + {| u_{1} - u_{2} |}_{(1, w, p)}^{\frac{2}{p^{'}}}) {∥ \nabla v ∥}_{L^{p} (Ω)}

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

for all $u_{1}$ $u1$ , $u_{2}$ $u2$ , v, $w \in W^{1, p} (Ω)$ $w∈W1,p⁢(Ω)$ .

Acknowledgements

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

References

[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

[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

[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

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

[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

[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

Received: 2017-12-12

Revised: 2018-04-30

Accepted: 2018-06-07

Published Online: 2018-06-21

Published in Print: 2019-04-01

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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

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

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