The adaptive finite element method for the P-Laplace problem

Journal Pre-proof The adaptive finite element method for the P-Laplace problem

D.J. Liu, Z.R. Chen

PII:

S0168-9274(19)30327-7

DOI:

https://doi.org/10.1016/j.apnum.2019.11.018

Reference:

APNUM 3709

To appear in:

Applied Numerical Mathematics

Received date:

29 April 2019

Revised date:

1 October 2019

Accepted date:

25 November 2019

Please cite this article as: D.J. Liu, Z.R. Chen, The adaptive finite element method for the P-Laplace problem, Appl. Numer. Math. (2019), doi: https://doi.org/10.1016/j.apnum.2019.11.018.

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The adaptive finite element method for the P-Laplace problem D. J. Liu · Z. R. Chen

Received: date / Accepted: date

Abstract We consider the adaptive finite element method (AFEM) for the P-Laplace problem, −div(|∇u| p−2 ∇u) = f . A posteriori and priori error analysis of conforming and nonconforming finite element method are measured in the new framework. A number of experiments confirm the effective decay rates of the AFEM. Keywords AFEM · conforming · nonconforming · p-Laplacian · quasi-norm. Mathematics Subject Classification (2000) 65N12 · 65N30 · 65Y20 1 Introduction Let Ω be a bounded polyhedral domain in Rd , d ∈ N. We consider the following nonlinear p-Laplacian with homogeneous Dirichlet data  −div(|∇u| p−2 ∇u) = f in Ω (1) u=0 on ∂ Ω . Here, 2 < p < ∞ and the given f ∈ Lq (Ω ) (q conjugate of p). The P-Laplace problem (1) admits a unique weak solution satisfying u = arg min E(v) for v ∈ V := W01,p (Ω ) = {v ∈ W 1,p (Ω ) : v|∂ Ω = 0}. where



E(v) :=

Ω

 W (∇v)dx − F(v).

(2)

(3)

This author’s research was supported by National Natural Science Foundation of China (No.11571226). D. J. Liu Department of Mathematics, Shanghai University, Shanghai 200444, PR China. E-mail: [email protected] Z. R. Chen Department of Mathematics, Shanghai University, Shanghai 200444, PR China.

2

D. J. Liu, Z. R. Chen

The energy density function W : Rd → R reads W (a) := ϕ (|a|) = |a| p /p with the  := derivative DW (a) = ϕ  (|a|)a/|a| for all a ∈ Rd \{0} and F(v) Ω f v dx. Denote the p−2 flow σ := DW (∇u) = |∇u| ∇u, then the Euler-Lagrange equation of (2) consists in finding u ∈ V with  Ω

 =0 σ · ∇vdx − F(v)

f or all v ∈ V.

(4)

The imbedding of V into W01,2 (Ω ) is continuous when 2 < p < ∞ and Ω is bounded domain (see [2]). Finite element analysis for (1) has been studied extensively in the literature, and one can find some previous work ,for example in [16,15], and some resent work in [1,21,17]. The purpose of this paper is to present a new result of an adaptive finite element method (AFEM) for the nonlinear p-Laplace equation (1). As a common practice the AFEM consists of a loop SOLV E → EST IMAT E → MARK → REFINE. starting from a initial triangulation of Ω . To be more precise description of AFEM, the problem is solved on the current mesh, then a posteriori error estimator is computed, with its help some element are marked for refinement. The first convergence analysis of nonlinear Laplacian is due to [27], the residual based estimators for the W 1,p norm are used. Since a gap appears in the power between the upper and lower estimates, therefore this approach does not seem to work well. Sharper a priori error estimates were derived in [24,19,5] by developing the so-called quasi-norm techniques, and these techniques were extended to establish improved a posteriori error estimators of residual type for the P1 conforming finite element methods (CFEM) and nonconforming finite element methods (NCFEM) [25,26,14]. Effective numerical algorithms for (2) can refer to [28,22,18,7]. This paper aims at the conforming and nonconforming adaptive finite element method with different mathematical ideas. We derive upper bounds for the estimators in the new defined quasi-norm, and the error of the energy can be presented at the same time in an easy way. The remaining parts of this paper are organized as follows. In Section 2, we introduce the error notation [5] F(∇u) − F(∇v)2 to quantify the quality of approximations via F(a) := ϕ  (|a|)|a|−p/2 a, a ∈ Rd \{0}. The resulting error quantity is equivalent to the quasi-norm, which was introduced to finite element analysis by Barrett and Liu [3,4]. The following Lemma 1 shows that it also equivalent to the quasi-norm used in [19,23]. Section 3 states the P1 CFEM and Crouzeix-Raviart NCFEM for the P-Laplacian. In Section 4 and Section 5, we construct a posteriori error estimators based on new quasi-norm of P1 CFEM and CR-NCFEM, and we prove upper error bounds for these estimators. Some numerical experiments conclude the paper in Section 6 with empirical evidence of the expected optimal rates, provided the error measure is the quasi-norm. Standard notation applies throughout this paper to Lebesgue and Sobolev spaces L p (Ω ), H s (Ω ), and H(div, Ω), as well as to the associated norms · p,Ω := ·L p (Ω ) , |||·||| p,Ω := ∇ · L p (Ω ) , and |||·|||NC,p,Ω := ∇NC · L p (Ω ) with the piecewise gradient

The adaptive finite element method for the P-Laplace problem

3

∇NC · |T := ∇(·|T ) for all T in a regular triangulation T of the polygonal Lipschitz domain Ω . Here and throughout, ”:” denotes the scalar product in Rm×n and the expression ”” abbreviates an inequality up to some multiplicative generic constant, i.e. A  B means A ≤ C B with some generic constant 0 ≤ C < ∞, which depends on the interior angles of the triangles but not their sizes. We write A ∼ B if A  B and B  A. 2 The concept of distance We first recall suitable convex function called N-function [20,5] before we analyze the problem (2). A real convex function ϕ : R≥0 → R≥0 is said to be an N-function if it satisfies the following condition: There exists the derivative ϕ  of ϕ . This derivative is right continuous, non-decreasing and satisfies ϕ  (0) = 0, ϕ  (t) > 0 for t > 0. Assumption 1 Let ϕ be an N-function such that ϕ is C1 on [0, ∞) and C2 on (0, ∞). Further assume that ϕ  (t) ∼ t ϕ  (t) uniformly in t > 0. The constants hidden in ∼ are called the characteristics of ϕ . Define (5) F(a) := ϕ  (|a|)|a|−p/2 a, a ∈ Rd \{0}. Then we have the following result. a , β := DW (b) = Lemma 1 Let ϕ be as in Assumption 1, then α := DW (a) = ϕ  (|a|) |a|

b ϕ  (|b|) |b| for a, b ∈ Rd \ {0} satify

|α − β | ∼ ϕ  (|a| + |b|)|a − b|, |F(a) − F(b)|2 ∼

(6)

|α − β |2 |α − β |2 . ∼ p−2 p−2 (|a| + |b|) (|a| + |b| p−2 )

(7) 

(t) Proof The proof of (6) can be found in reference [19]. Define ψ  (t) := t ϕp/2−1 for t >

0, it is not difficult to prove that ψ  (t) ∼

ϕ  (t) , t p/2−1

|F(a) − F(b)|2 ∼ [ψ  (|a| + |b|)|a − b|]2 ∼

this and the technique in [19] lead to

| α − β |2 ϕ  (|a| + |b|)2 2 |a − b| ∼ (|a| + |b|) p−2 (|a| + |b|) p−2

This completes the proof. In proving the estimates of FEM, an important role is played by the convexity control of the energy density function W , the convexity control property is formulated more precisely in the following lemma. Lemma 2 Given 2 ≤ p < ∞ and the conjugate q, there exist positive constants c1 (p), c2 (p) such that for any a, b ∈ Rd \ {0}, α := DW (a), β := DW (b) satisfy 1 |α − β |2 (|a| + |b|) p−2 ≤ c2 (p)(W (b) −W (a) − α · (b − a)).

c1 (p)(W (b) −W (a) − α · (b − a)) ≤

(8)

4

D. J. Liu, Z. R. Chen

Proof Given a, b ∈ Rd \ {0} with a  b, set t := |b|/|a| and g := a · b/(|a| · |b|) (−1 ≤ g ≤ 1), it is obvious that 1 |α − β |2 (|a| + |b|) p−2 (W (b) −W (a) − α · (b − a)) =

1 + t 2(p−1) − 2gt p−1 := f (t, g). (1 + t) p−2 (t p /p + 1/q − gt)

(9)

A direct calculation verifies that ∂ f /∂ g as a function of g has one sign (which depends on t and p), hence it is monotone increasing or decreasing. Therefore for all 0 < t < ∞, there exist constants c1 (p), c2 (p) satisfy that min{ f (t, 1), f (t, −1)} := c1 (p) ≤ f (t, p) ≤ c2 (p) := max{ f (t, 1), f (t, −1)} < ∞. The case g = 1 is the crucial one because t p /p + 1/q − t vanishes for t=1, f (t, 1) =

(1 − t p−1 )2 (1 + t) p−2 (t p /p + 1/q − t)

L’Hospital rule yields f (1, 1) = p − 1 > 0. The monotone decreasing and monotone increasing of t p /p + 1/q − t on (0, 1) and (1, ∞) show that t p /p + 1/q − t > 0, that is f (t, 1) > 0. The analysis of f (t, −1) > 0 is simpler and hence omitted, hence c1 (p) > 0, c2 (p) > 0, this concludes the proof. Define the distance  Ω

|α − β |2 dx := F(a) − F(b)22,Ω , for any a, b ∈ Rd \ {0}. (|a| + |b|) p−2

(10)

It is not difficult to find that the defined distance (10) is an quasi-norm [23], and equivalent to the norm in Belenki and Diening[5] and the quasi-norm in Barrett and Liu [24–26]. The need for such an error concept comes from the nonlinear behaviour of the residual, which is not reflected by standard Sobolev norms. Hence, in this case there is no equivalence between error in the energy norm and the residual. In contrast, the quasi-norm concept permits us to prove a best approximation property for Galerkin solutions as well as optimal interpolation estimates, which are crucial ingredients in a posteriori finite element analysis.

3 FEMs for p-Laplacian 3.1 Triangulations Let T be a regular triangulation of the simply-connected bounded Lipschitz domain Ω ⊆ Rd with polygonal boundary ∂ Ω into closed triangles. Let E denote the set of all edges and E (Ω ) (resp. E (∂ Ω )) denote the set of all interior (resp. boundary) edges, N denote the set of vertices and N (Ω ) (resp. N (∂ Ω )) denote the interior (resp.

The adaptive finite element method for the P-Laplace problem

5

boundary) nodes. For any triangle T ∈ T , set hT := diam(T ) and let E (T ) denote the set of three edges of T , write hE := diam(E) for an edge E ∈ E (T ). Let Pk (T ) = {vk : Ω → R | ∀T ∈ T , vk |T is a polynomial of total degree ≤ k} denote the set of piecewise polynomials and let hT ∈ P0 (T ) denote the T -piecewise constant mesh size function with hT |T = hT for all T ∈ T and the maximum hmax := hT ∞ . Assume that T is shape-regular so that hT ≈ hE ≈ |T |1/d for all E ∈ E (T ) and T ∈ T . Let [•]E := •|T+ − •|T− denote the jump across the common edge E = ∂ T+ ∩ ∂ T− with T+ , T− ∈ T and unit normal νE pointing into T− . Let Π0 : Lq (Ω) → P0 (T ) denote the Lq projection onto T piecewise constant, i.e., (Π0 f )|T = T f dx for all T ∈ T (the same notation Π0 is also used for vectors and understood componentwise), and let osc( f , T ) := hT ( f − Π0 f )q,Ω . 3.2 P1 conforming FEM The P1 conforming finite element approximation uC to (2) minimizes the energy E in VC (T ) := P1 (T ) ∩V , written uC ∈ arg min E(vC ), vC ∈ VC (T ).

(11)

The discrete minimizers uC from (11) exist and unique. Denote σC := DW (∇uC ), then the Euler-Lagrange equations of (11) consists in finding uC ∈ VC (T ) with  Ω

 C) = 0 σC · ∇vC dx − F(v

f or all vC ∈ VC (T ).

(12)

A priori error and a posteriori error estimates will be analyzed in Section 4.

3.3 Crouzeix-Raviart nonconforming FEM The Crouzeix-Raviart finite element space is defined as CR10 (T ) := {vh ∈ P1 (T ) | vh is continuous at midpoints of interior edges and vanishes at midpoints of boundary edges}. The nonconforming FEM is based on CR10 (T ) and the nonconforming energy ENC  with Fh (•) := F ◦ Π0 (•) = Ω (Π0 f ) • dx and ENC (vCR ) :=



Ω

W (∇NC vCR )dx − Fh (vCR )

for vCR ∈ CR10 (T ).

(13)

The Crouzeix-Raviart finite element approximation uCR to (2) minimizes the energy ENC in CR10 (T ), written uCR ∈ arg min ENC (vCR ), vCR ∈ CR10 (T ).

(14)

6

D. J. Liu, Z. R. Chen

The discrete minimizers uCR exist and unique. Denote σCR := DW (∇NC uCR ), then the Euler-Lagrange equations of (14) consists in finding uCR ∈ CR10 (T ) with 

Ω

σCR · ∇NC vCR dx − Fh (vCR ) = 0

f or all vCR ∈ CR10 (T ).

(15)

A priori and a posteriori error analysis will be analyzed in Section 5. 4 Error analysis of P1 conforming FEM The trace inequality [6] reads: For any T ∈ T with F ∈ T and for all v ∈ W 1,p (T ) −1/p

v p,F  hF

1/q

v p,T + hF ∇v p,T .

Definition Given the nodal basis function ϕz of z in VC (T ), set Ωz := {x ∈ Ω : 0 < ϕz (x)} with patch-size hz := diam(Ωz ), ψz := ϕz /ψ , ψ = ∑z∈N (Ω ) ϕz . Then, for all v ∈ V , define the quasi-interpolant operator JC : V → VC (T ) [8,9] by 

Ω vz ϕz , vz :=  z

∑

JC v :=

vψz dx

Ωz ϕz dx

z∈N (Ω )

.

The following Lemma 3 shows the property of quasi-interpolation operator JC . Lemma 3 (Property of quasi-interpolation operator JC ). For all g ∈ V and f ∈ Lq (Ω ), there holds 

f · (g − JC g)dx ≤ |||g||| p,Ω · osc( f , T ).

Ω

−1

−1

hT p (g − JC g) p,Ω  hT q ∇g p,Ω . 1 q

1 p

hT ∇(g − JC g) p,Ω  hT ∇g p,Ω . 1 q

∑ g − JC g pp,E ≤ hT ∇g pp,Ω .

(16) (17) (18) (19)

E∈E

Proof The key estimate of this proof is the following inequality gz ϕz − gψz  p,Ωz  hz ∇g p,Ωz

(20)

Let gz denote the integral mean of g on Ωz . The Triangle inequality and H¨older inequality shows that 1

ϕzp (gz − gz ) pp,Ωz ≤ =



Ωz

 Ωz

ϕz |gz − gz | p−1 |g − gz |dx +

ϕz |gz − gz | p−1 |g − gz |dx −

1

1

 Ωz

 Ωz

ϕz |gz − gz | p−1 |gz − g|dx

(ψz − ϕz )|gz − gz | p−1 |gz − g|dx

≤ ϕzq (gz − gz ) p−1 q,Ωz ϕzp (g − gz ) p,Ωz + (ψz − ϕz )g p,Ωz (gz − gz ) p−1 q,Ωz

The adaptive finite element method for the P-Laplace problem

7

The equality (p − 1)q = p and the Young inequality lead to 1

1

ϕzq (gz − gz ) p−1 q,Ωz ϕzp (g − gz ) p,Ωz + (ψz − ϕz )g p,Ωz (gz − gz ) p−1 q,Ωz p

1

p

1

= ϕzp (gz − gz ) p,q Ωz ϕzp (g − gz ) p,Ωz + (ψz − ϕz )g p,Ωz gz − gz  p,q Ωz p

p

p

1 1 2 q C0 2q 1 ϕzp (g − gz ) pp,Ωz + ϕzp (gz − gz ) pp,Ωz + (ψz − ϕz )g pp,Ωz p 2q p 1 + gz − gz  pp,Ωz . 2qC0q

≤

with C0 =

1q

p 1 q gz − gz  p,Ω z C0

, absorbing

1 q ϕz q

1

≤ ϕzp (gz − gz ) pp,Ωz ,we deduce that

1 1 1p ϕz (gz − gz ) pp,Ωz  ϕzp (g − gz ) pp,Ωz + (ψz − ϕz )g pp,Ωz p

A Poincar´e inequality shows 1

ϕzp (g − gz ) p,Ωz  hz ∇g p,Ωz 

Note that (ψz − ϕz )g is nonzero only if ΓD (∂ Ωz ) has positive surface measure, where g = 0, the Friedrichs inequality yields g p,Ωz  hz ∇g p,Ωz . Combing all the analysis results in 1

ϕzp (gz − gz ) p,Ωz  hz ∇g p,Ωz . The triangle inequality, (21) and Friedrichs inequality lead to gz ϕz − gψz  p,Ωz  (gz − gz )ϕz  p,Ωz + (g − gz )ϕz  p,Ωz + (ψz − ϕz )g p,Ωz  hz ∇g p,Ωz That is (20). For any fz ∈ R, The (20) and H¨older inequality lead to 

Ω

f · (g − JC g)dx =

∑

z∈NΩ



∑

z∈NΩ



Ω

( f − fz ) · (gψz − gz ϕz )dx

hz  f − fz q,Ωz ∇g p,Ωz

 |||g||| p,Ω · osc( f , T ). p−1 in (16), it holds Take f := h−1 T (g − JC g) −1

hT p (g − JC g) pp,Ω = 

∑

z∈NΩ

∑

z∈NΩ



−1

Ω

−1

hT q (g − JC g) p−1 hT p (gψz − gz ϕz )dx

−1

−1

hT q (g − JC g) p−1 q,Ω hT p (gψz − gz ϕz ) p,Ω

(21)

8 p q

D. J. Liu, Z. R. Chen

= p − 1 and (20) lead to

∑

−1

z∈NΩ

−1

hT q (g − JC g) p−1 q,Ω hT p (gψz − gz ϕz ) p,Ω 1

−1

p−1 q  hT p (g − JC g) p, Ω hT ∇g p,Ω

That is (17). To prove (18), let g denote the mean of g on Ωz . The triangle inequality shows that 1

hTq ∇(g − JC g) pp,Ω 

1

∑

z∈NΩ

hzq ∇(gψz − gz ϕz ) pp,Ωz

1

1

 hzq ∇g · (ψz − ϕz ) pp,Ωz + hzq g · ∇(ψz − ϕz ) pp,Ωz 1

(22)

1

+ hzq ∇(ϕz (gz − gz )) pp,Ωz + hzq ∇(ϕz (g − gz )) pp,Ωz . The Friedrichs inequality and Poincar´e in equality yield (18). let v = g − JC g in the trace inequality, we get the (19), which conclude the proof of this lemma. 4.1 A priori error analysis Theorem 1 (A priori error estimate). The discrete minimizers uC and σC := DW (∇uC ) satisfy c1 (p)



Ω

1 (σ − σC ) · ∇(u − uC )dx ≤ F(∇u) − F(∇uC )22 + c1 (p)|E(u) − E(uC )| 2 ≤ c2 (p)



Ω

(σ − σC ) · ∇(u − uC )dx.

Proof The choice a := ∇uC , b := ∇u, and α := σC in right inequality of Lemma 2 and Euler-Lagrange equations (4) lead to F(∇u) − F(∇uC )22 + c2 (p)(E(uC ) − E(u))  − uC ) + ≤ c2 (p)(F(u = c2 (p)



Ω



Ω

(σ − σC ) · ∇(u − uC )dx −

 Ω

σ · ∇(u − uC )dx)

(23)

(σ − σC ) · ∇(u − uC )dx.

The choice a := ∇u, b := ∇uC , and α := σ in the right inequality of Lemma 2 and (12) lead to F(∇u) − F(∇uC )22 + c2 (p)(E(u) − E(uC )) ≤ c2 (p)



Ω

(σ − σC ) · ∇(u − uC )dx. (24)

The (23)-(24) and the Cauchy inequality show that F(∇u) − F(∇uC )22 + c2 (p)|E(u) − E(uC )| ≤ c2 (p)



Ω

(σ − σC ) · ∇(u − uC )dx

(25)

The adaptive finite element method for the P-Laplace problem

9

The same technique in left inequality of Lemma 2 leads to F(∇u) − F(∇uC )22 + c1 (p)|E(u) − E(uC )| ≥ c1 (p)



Ω

(26)

(σ − σC ) · ∇(u − uC )dx

The combination of preceding displayed inequalities concludes the proof. 4.2 A posteriori error analysis This subsection is devoted to an a posteriori error analysis of the P1 CFEM. Define ηE := hE [σC ]E · νE q,E for the jump [σC ]E · νE of the discrete stress σC in the normal direction νE across an interior edge E. Recall that any v ∈ V satisfies the Friedrichs inequality v p,Ω ≤ CF |||v||| p,Ω with CF ≤ width(Ω )/π . The energy density W (a) = ity shows that

|a| p and the Friedrichs inequalp

1 |||u||| pp,Ω −CF  f q,Ω |||u||| p,Ω ≤ E(u). p Since E(u) ≤ E(0) = 0, this implies 1

|||u||| p,Ω ≤ (pCF  f q,Ω ) p−1 .

(27)

Theorem 2 (A posteriori error estimate). The discrete minimizers uC and the constants M1 := 2c2 (p)CP satisfy F(∇u) − F(∇uC )22 + c2 (p)|E(u) − E(uC )|   ≤ M1 osc( f , T ) + ( ∑ ηE q )1/q .

(28)

E∈E

1

where CP := (pCF  f q,Ω ) p−1 . Proof Denote eC := u−uC . The choice a := ∇uC , b := ∇u, α := σC in Lemma 2 leads to (29) F(∇u) − F(∇uC )22 + c2 (p)(E(uC ) − E(u)) ≤ c2 (p)res(eC ) for the residual  − uC ) − res(eC ) := F(u

 Ω

σC · ∇(u − uC )dx.

 C − JC eC ) − res(eC − JC eC ) = F(e

 Ω

σC · ∇(eC − JC eC )dx

 C eC ) + = res(eC ) − F(J



Ω

 C eC − eC ). = res(eC ) − F(J

σC · ∇eC dx

10

D. J. Liu, Z. R. Chen

That is

 C eC − eC ). res(eC ) = res(eC − JC eC ) + F(J

The Lemma 3 and H¨older inequality together with σ ∈ P0 (T ; R2 ) imply that res(eC − JC eC ) =



Ω

f · (eC − JC eC )dx −



Ω

≤ ∇eC  p,Ω · osc( f , T ) +

σC · ∇(eC − JC eC )dx

∑ [σC ]E · νF q,E eC − JC eC  p,E .

E∈E

The Young inequality and Lemma 3 show that

∑ [σC ]E · νE q,E eC − JC eC  p,E

E∈E

≤ ( ∑ [σC ]E · νE qq,E )1/q · ( ∑ eC − JC eC  pp,E )1/p E∈E

E∈E

≤ ( ∑ ηE q )1/q · ∇eC  p,Ω E∈E

The combination of the above estimates results in F(∇u) − F(∇uC )22 + c2 (p)(E(uC ) − E(u))   ≤ ∇eC q,Ω · osc( f , T ) + ( ∑ ηE q )1/q .

(30)

E∈E

The choice a := ∇u, b := ∇uC , and α := σ in right inequality of Lemma 2 leads to F(∇u) − F(∇uC )22 + c2 (p)(E(u) − E(uC )) ≤ 0.

(31)

The sum of (30)-(31) and the Young inequality show that F(∇u) − F(∇uC )22 + c2 (p)|E(u) − E(uC )|   ≤ c2 (p)∇eC  p,Ω · osc( f , T ) + ( ∑ ηE q )1/q .

(32)

E∈E

This and estimate (27) concludes the proof.

5 Error analysis of Crouzeix-Raviart NCFEM This section analyzes the error estimates of the Crouzeix-Raviart NCFEM which is based on the following lemmas and the Crouzeix-Raviart interpolation operator INC : W01,p (Ω ) → CR10 (T ) (2 ≤ p ≤ ∞),  vds for all E ∈ E . (INC v)(mid(E)) := E

Lemma 4 (Property of the Crouzeix-Raviart interpolant)[16,12,11] Any v ∈ W 1,p (Ω ) with its interpolation INC v and the constant κ satisfy ∇NC (INC v) = Π0 ∇v and v − INC v p,Ω ≤ κ hT (I − Π0 )∇v p,Ω ≤ κ hT ∇v p,Ω .

The adaptive finite element method for the P-Laplace problem

11

Lemma 5 (Conforming P3 companion) Given any vCR ∈ CR10 (T ), there exists some v3 ∈ P3 (T ) ∩W01,p (Ω ) with vCR = INC v3 , Π0 vCR = Π0 v3 , and h−1 T (vCR − v3 ) p,Ω + vCR − v3

NC,p,Ω

 min v − vCR

NC,p,Ω

v∈V

.

This lemma is a generalization of [13], where p = 2.

5.1 A priori error analysis The following Theorem 3 guarantees the convergence estimates of the CrouzeixRaviart NCFEM. Theorem 3 (A priori error estimate). The discrete minimizers uCR satisfies  F(∇u) − F(∇NC uCR )22 + c2 (p)|E(u) − ENC (uCR )| + κ osc( f , T ) u − uCR

+|

NC,p,Ω

 Ω

≤ c2 (p) osc( f , T ) u p,Ω

(I − Π0 )σ · (I − Π0 )∇u3 dx| .

Proof The choice a := ∇NC uCR , b := ∇u, and α := σCR in Lemma 2 leads to    )− σ ·∇ (u−u )dx . F(∇u)−F(∇NC uCR )22 ≤ c2 (p) E(u)−ENC (uCR )+ F(u−u CR CR NC CR Ω

Since σCR ∈ P0 (T ; R2 ) and  Ω

σCR · ∇udx =

 Ω

σCR · ∇NC INC udx = Fh (INC u).

The combination with the previous estimate verifies   − Fh (I u)). F(∇u) − F(∇NC uCR )22 + c2 (p) ENC (uCR ) − E(u) ≤ c2 (p)(F(u) NC (33) The choice a := ∇u, b := ∇NC uCR , and α := σ in Lemma 2 leads to     )− σ ·∇ u dx . F(∇u)−F(∇NC uCR )22 +c2 (p) E(u)−ENC (uCR ) ≤ c2 (p) F(u CR NC CR Ω

The conforming P3 companion u3 ∈ P3 (T ) ∩V with uCR = INC u3 shows −

 Ω

σ · ∇NC uCR dx = −

 Ω

σ · ∇u3 dx +

 3) + = −F(u



Ω

 Ω

σ · ∇NC (u3 − INC u3 )dx

(I − Π0 )σ · (I − Π0 )∇u3 dx.

12

D. J. Liu, Z. R. Chen

The combination of the preceding estimates results in  F(∇u) − F(∇NC uCR )22 + c2 (p) E(u) − ENC (uCR )    ≤ c2 (p) F(INC u3 − u3 ) + (I − Π0 )σ · (I − Π0 )∇u3 dx .

(34)

Ω

The (33)-(34) imply that (35) F(∇u) − F(∇NC uCR )22 + c2 (p)|E(u) − ENC (uCR )|

   − Fh (I u)| + |F(I  ≤ c2 (p) |F(u) (I − Π0 )σ · (I − Π0 )∇u3 dx| . NC NC u3 − u3 ) + Ω

The H¨older inequality and Lemma 4 prove  F(I NC u3 − u3 ) =  − Fh (I u) = F(u) NC =

 Ω

 Ω Ω

( f − Π0 f )(INC u3 − u3 )dx ≤ κ osc( f , T ) u3 − INC u3

NC,p,Ω

f (u − INC u)dx +

 Ω

,

( f − Π0 f )INC udx

( f − Π0 f )(u − INC u)dx +



Ω

( f − Π0 f )INC udx

≤ osc( f , T ) u p,Ω This and Lemma 5 prove the assertion.

5.2 A posteriori error analysis This subsection is devoted to an a posteriori error analysis of the CR-NCFEM. Recall that any vCR ∈ CR10 (T ) satisfies the discrete Friedrichs inequality ([6], P. 301) with some constant CdF ≈ 1. vCR  p,Ω ≤ CdF vCR

NC,p,Ω

.

Theorem 4 (A posteriori error estimate). The discrete minimizers uCR and the conp−2 and M3 := κ CP satisfy stants M2 := 2c2 (p)CP 1 F(∇u) − F(∇NC uCR )22 + c2 (p)|E(u) − ENC (uCR )| 2   2  ≤ c2 (p) F(u + M3 · osc( f , T ) . CR − u3 ) + M2 uCR − u3 NC,2,Ω

1

where CP := (pCF  f q,Ω ) p−1 .

(36)

The adaptive finite element method for the P-Laplace problem

13

Proof The H¨older inequality implies that 

(I − Π0 )σ · (I − Π0 )∇u3 dx  1/2 (|∇u| p−2 + |Π0 ∇u| p−2 )|(I − Π0 )∇u3 |2 dx . ≤ F(∇u) − F(Π0 ∇u)2 · Ω

Ω

The estimate (35), the Friedrichs inequality, and the H¨older inequality imply  F(∇u) − F(∇NC uCR )22 + c2 (p)|E(u) − ENC (uCR )| ≤ c2 (p) κ osc( f , T ) |||u||| p,Ω  + F(u CR − u3 ) + 2 sup |∇u|

p/2−1

x∈Ω

F(∇u) − F(Π0 ∇u)2 uCR − u3

NC,2,Ω

.

The Young inequality shows 2c(p) sup |∇u| p/2−1 F(∇u) − F(Π0 ∇u)2 uCR − u3

NC,2,Ω

x∈Ω

2 1 ≤ F(∇u) − F(Π0 ∇u)22 + 2c22 (p) sup |∇u| p−2 uCR − u3 . NC,2,Ω 2 x∈Ω The combination of preceding displayed inequalities concludes the proof.

6 Numerical experiments The numerical experiments concern the practical application of the a posteriori error estimates (28) and (36) and their efficiency. Denote the left-hand side (LHS) of the two estimates by LHS(28) and LHS(36). The guaranteed upper bounds (GUB) read   GUB(28) = M1 osc( f , T ) + ( ∑ ηE q )1/q ; E∈E



2  GUB(36) = c2 (p) F(u CR − u3 ) + M2 uCR − u3

NC,2,Ω



+ M3 · osc( f , T )

The triangulations are either uniform with successive red-refinement or with an adaptive mesh-refinement algorithm with initial mesh T0 and then, for any triangle T of a triangulation T at level  = 0, 1, 2, 3 . . . set 2 η (T ) = INC u3 − u3 + oscq ( f , T ). NC,2,T

Given all those contributions, mark some set M of triangles in T of minimal cardinality with the bulk criterion 1/2

∑

T ∈T

η 2 (T ) ≤

∑

T ∈M

η 2 (T ).

The refinement of all triangles in M plus minimal further refinements to avoid hanging nodes lead to the triangulation T+1 within the newest-vertex bisection. The

14

D. J. Liu, Z. R. Chen

choice of the refinement-indicator η (T ) is motivated by the convergence theory of adaptive mesh-refining algorithms e.g. in the review article [10] with further details on the mesh-refinement. The convergence history plots display the left-hand sides LHS(28), LHS(36) and the upper bounds GUB(28), GUB(36) as function of the number of degrees of freedom (ndof) in a log-log scale.

6.1 Example 1 Consider the P-Laplace Problem on the square domain Ω := [− 12 , 12 ]2 with the exact solution.

⎧ r2 1 1 ⎪ ⎨( − r2 )2 e− ε for r ≤ 4 2 u(r) = 1 ⎪ ⎩ 0 for r > 2

and right-hand side

r2 1 f (r) = − 2 p−1 g p−2 (2 + g) p−2 e−(p−1) ε r p−2 ε 1 2 1 r2 1 × [2(p − 1)(2 + g)r + 2(p − 1)g(3 + g) − pg(2 + g)] ε ε ε ε

where g = 14 − r2 , we test this example for p = 4,ε = 0.005. The reference value for the minimal energy E = −0.00374332 stems from Aitken extrapolation. Figure 1 and Figure 2 display the global upper bounds (GUB) and the corresponding error terms (LHS) of the estimates from (28), (36) for uniform and adaptive mesh-refinement. Figure 3 and Figure 4 displays the corresponding sequences of triangulations generated by adaptive FEM for (28), (36).

The adaptive finite element method for the P-Laplace problem

15

1

10

GUB(28)(uniform) LHS(28)(uniform) GUB(28)(adaptive) LHS(28)(adaptive)

0

10

−1

10

−2

10

−3

10

0.69 −4

10

1

−5

10

1

2

10

3

10

4

10

5

10

10

Fig. 1 Convergence history of P1 method on square domain.

GUB(36)(uniform) LHS(36)(uniform) GUB(36)(adaptive) LHS(36)(adaptive)

0

10

−1

10

−2

10

−3

10

0.78 1

−4

10

−5

10

2

10

3

10

4

10

Fig. 2 Convergence history of CR method on square domain.

5

10

16

D. J. Liu, Z. R. Chen

0.5

0.5

0

0

−0.5 −0.5

0 T , nrdof=114

0.5

−0.5 −0.5

2

0.5

0

0

0 T4, nrdof=832

0.5

0 T5, nrdof=2129

0.5

3

0.5

−0.5 −0.5

0 T , nrdof=319

0.5

−0.5 −0.5

Fig. 3 Adaptively generated triangulation T for  = 2, 3, 4, 5 on square domain.

0.5

0.5

0

0

−0.5 −0.5

0 T2, nrdof=190

0.5

−0.5 −0.5

0.5

0.5

0

0

−0.5 −0.5

0 T6, nrdof=4249

0.5

−0.5 −0.5

0 T4, nrdof=733

0.5

0 T8, nrdof=22501

0.5

Fig. 4 Adaptively generated triangulation T for  = 2, 4, 6, 8 on square domain.

The adaptive finite element method for the P-Laplace problem

17

6.2 Example 2 Consider the P-Laplace Problem on the square domain Ω := (0, 1)2 with the exact solution.  0 for r ≤ a u(r) = (r − a)4 for r > a and right-hand side

u(r) =

⎧ ⎨

0 for r ≤ a a ⎩4 p−1 (r − a)3p−4 (2 + − 3p) for r > a r

for p = 4, a = 0.8 The reference value for the minimal energy E = 0.00824393 stems from Aitken extrapolation. Figure 5 and Figure 6 display the global upper bounds (GUB) and the corresponding error terms (LHS) of the estimates from (28), (36) for uniform and adaptive mesh-refinement. Figure 7 and Figure 8 displays the corresponding sequences of triangulations generated by adaptive FEM for (28), (36).

GUB(28)(uniform) LHS(28)(uniform) GUB(28)(adaptive) LHS(28)(adaptive)

0

10

−1

10

−2

10

−3

10

0.91

−4

10

1

−5

10

−6

10

1

10

2

10

3

10

Fig. 5 Convergence history of P1 method on square domain.

4

10

5

10

18

D. J. Liu, Z. R. Chen

1

10

GUB(36)(uniform) LHS(36)(uniform) GUB(36)(adaptive) LHS(36)(adaptive)

0

10

−1

10

−2

10

−3

10

−4

10

1

−5

10

1

−6

10

0

10

1

10

2

10

3

10

4

10

Fig. 6 Convergence history of CR method on square domain.

Fig. 7 Adaptively generated triangulation T for  = 2, 4, 6, 8 on square domain.

5

10

The adaptive finite element method for the P-Laplace problem

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5 T , nrdof=43

1

0

0

2

1

1 0.8

0.6

0.6

0.4

0.4

0.2

0.2 0

0.5 T6, nrdof=1297

0.5 T , nrdof=229

1

0.5 T8, nrdof=6679

1

4

0.8

0

19

1

0

0

Fig. 8 Adaptively generated triangulation T for  = 2, 4, 6, 8 on square domain.

6.3 Conclusions We consider the conforming and nonconforming adaptive finite element method based on the new defined quasi-norm. The main features of our technique is that the a posteriori error estimates provide reliability upper bounds with discretization error, and the error of the energy can be presented at the same time. The numerical examples shows that the convergence results are consistent with the theoretical analysis.

References 1. M. Ainsworth, D. Kay, The approximation theory for the p-version finite element method and application to non-linear PDEs, Numer. Math., 83, 351-388. 2. H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential, Springer, New York (2010). 3. J. W. Barrett, W. B. Liu, Finite element approximation of the p-Laplacian, Math. Comput., 61 (1993), 523-537. 4. J. W. Barrett, W. B. Liu, Quasi-norm error bounds for the finite element approximation of a nonNewtonian flow, Numer. Math., 68 (1994), 437-456. 5. L. Belenki, L. Diening, Optimality of an adaptive finite element method for the p-Laplacian equation, IMA J Numer Anal., 32 (2012), 484-510. 6. S. C. Brenner, L. Scott, The mathematical theory of finite element methods, Third ed., Texts in Applied Mathematics, 15, Springer, 2008. 7. M. Caliari, S. Zuccher, Quasi-Newton minimization for the p(x)-Laplacian problem, J. Comput. Appl. Math., 309 (2017), 122-131. 8. C.Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, Mathematical Modelling and Numerical Analysis, 33, 1999, 1187-1202.

20

D. J. Liu, Z. R. Chen

9. C. Carstensen, S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Math. Comput., 71 (2002), 945-969. 10. C. Carstensen, M. Feischl, M. Page, D. Praetorius, Axioms of adaptivity, Comput. Methods Appl. Math., 67 (2014), 1195-1253. 11. C. Carstensen, D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math., 126 (2014), 33-51. 12. C. Carstensen, J. Gedicke, D. Rim, Explicit error estimates for Courant, Crouzeix-Raviart and RaviartThomas finite element methods, J. Comput. Math., 30 (2012), 337-353. 13. C. Carstensen, D. J. Liu, Nonconforming FEMs for an optimal design problem, SIAM J. Numer. Anal., 53 (2015), 874-894. 14. C.Carstensen, W. Liu, N.N. Yan, A posteriori error control for P-Laplacian by gradient recovery in quasi-norm, Math. Comp., 75 (2006), 1599-1616. 15. S. S. Chow, Finite element error estimaytors foro non-linear elliptic equations of monitone type, Numer. Math., 54, 373-393. 16. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978. 17. E. Creuse, M. Farhloul, L. Paquet, A posteriori error estimation for the dual mixed finite element method for the p-Laplacian in a polygonal domain, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2570-2582. 18. L.M. Del Pezzo, S. Mart´ınez, Order of convergence of the finite element method for the p(x)Laplacian, IMA J. Numer. Anal. 35 (4) (2015) 1864-1887. 19. L. Diening, F. Ettwein, Fraction Estimates for Non-Differentiable Elliptic Systems with general Growth,Forum Mathematicum, 20 (2008), no.3, 523-556. 20. L. Diening, M. Rˇuzˇ iˇcka, Non-Newtonian fluids and function spaces, Nonlinear Anal. Funct. Spaces Appl., 8 (2007), 95-143. 21. M. Farhloul, H. Manouzi, On a mixed finite element method for the p-Laplacian, Canad. Appl. Math., 8(2000), 67-78. 22. Y. Q. Huang, R. Li, W. B. Liu, Preconditioned Descent Algorithms for p-Laplacian, J. Sci. Comput., 32 (2007), no. 2, 343-371. 23. D. J. Liu, A.Q. Li, Z.R. Chen, Nonconforming FEMs for the P-Laplace Problem, Adv. Appl. Math. Mech., 10 (2018), no. 6, 1365-1383. 24. W. B. Liu, J. W. Barrett, Finite element approximaton of some degenerate monotone quasilinear elliptic system, SIAM J. Numer. Anal., 33 (1996), 88-106. 25. W. B. Liu, N. N. Yan, Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian, Numer. Math., 89 (2001), 341-378. 26. W. B. Liu, N. N. Yan, On quasi-norm interpolation error estimation and a posteriori error estimates for p-Laplacian, SIAM J. Numer. Anal., 40 (2002), 1870-1895. 27. A. Veeser, Convergent adaptive finite elements for nonlinear Laplacian, Numer. Math., 92, 743-770. 28. G. Zhou, Y. Huang, C. Feng, Preconditioned hybrid conjugate gradient algorithm for p-Laplacian, Int. J. Numer. Anal. Model. 2 (Suppl.) (2005) 123-130.