Anisotropic Adaptive Finite Elements for a p-Laplacian Problem

Paride Passelli; Marco Picasso

doi:10.1515/cmam-2022-0205

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Anisotropic Adaptive Finite Elements for a p-Laplacian Problem

Paride Passelli and Marco Picasso

Published/Copyright: June 26, 2024

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From the journal Computational Methods in Applied Mathematics Volume 25 Issue 2

Abstract

The p-Laplacian problem - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p = 3 . From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.

Keywords: A Posteriori Error Estimates; Adaptive Algorithm; Anisotropic Finite Elements; Nonlinear Equation; Aluminium Electrolysis

MSC 2020: 65N30; 65N50

1 Introduction

Adaptive meshes demonstrated their efficiency to solve PDEs at a reduced computational cost, for a given level of accuracy. When boundary or internal layers are involved, anisotropic meshes, that are meshes with large aspect ratio, should be considered [24, 3]. The adaptive criteria is, when possible, based on an a posteriori error estimates, see for instance [1, 4, 6, 18, 40, 47, 49] in the isotropic framework and [9, 10, 16, 21, 22, 23, 27, 35, 36, 39, 42, 43, 50] when anisotropic meshes are considered.

A posteriori error estimates for the p-Laplacian problem - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f were proposed in [14, 7, 32, 13, 8, 17] for isotropic finite elements. A quasi-norm was introduced to obtain efficient and reliable error estimates. In [8, 17] convergence of an adaptive finite elements method for the p-Laplace equation is proven, when isotropic meshes and recursive bisection refinement are considered.

In this paper the model problem is studied with anisotropic finite elements. Since the anisotropic meshes are not nested, the framework of [8, 17] cannot be used. A residual-based anisotropic error estimator is derived for the p-Laplacian problem and an anisotropic adaptive algorithm is presented. We show that the error estimator is equivalent to the error in a quasi-norm, up to higher order terms. The involved constants are independent of μ, the solution, the mesh size and aspect ratio. We derive an upper bound for the classical W 0 1 , p norm and a quasi-norm. A lower bound for the error in a quasi-norm is proved. Numerical experiments confirm that when μ is small, the error estimator is sharp with respect to the error in the quasi-norm only. When μ is large, the error estimator is sharp with respect to both the W 0 1 , 3 and the quasi-norm.

The outline is the following: in Section 2 the problem and the numerical method are presented. In Section 3, we introduce the anisotropic finite element setting and Clément interpolation estimates in L p norms. In Section 4, we introduce the new error estimator, we prove an upper bound for the quasi-norm and the W 0 1 , p norm and a lower bound for the quasi-norm only. In Section 5, we present numerical experiments with non-adapted meshes. In Section 6, the adaptive algorithm and numerical experiments with adapted anisotropic meshes are presented and confirm the theoretical findings. In Section 7, an application to the Navier–Stokes equation with the so-called Smagorinsky turbulence model [45] in the context of aluminium electrolysis is presented. This industrial problem has some multi-scale features (from meters to millimeters), thus anisotropic finite elements are particularly efficient.

2 Problem Statement and Numerical Method

Consider a polygonal domain Ω ⊂ ℝ d for d = 2 , 3 and a function u : Ω → ℝ solution of the following problem

(2.1) { - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω ,

where μ ≥ 0 , p ≥ d and | ⋅ | denotes the Euclidian norm in ℝ d . We assume throughout this paper that f ∈ L 2 ⁢ ( Ω ) . The weak formulation of problem (2.1) consists in finding u ∈ W 0 1 , p ⁢ ( Ω ) such that

(2.2) ∫ Ω ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ⋅ ∇ ⁡ v = ∫ Ω f ⁢ v for all ⁢ v ∈ W 0 1 , p ⁢ ( Ω ) .

When μ = 0 , existence and uniqueness of a solution u ∈ W 0 1 , p ⁢ ( Ω ) have been proved in [31, 25]. The case μ > 0 can be proved in a similar way.

To solve problem (2.2), we use classical continuous piecewise linear finite elements. Consider, for any 0 < h < 1 , a conforming triangulation 𝒯 h of Ω ¯ having triangles K of diameter h K ≤ h . Let V h ⊂ W 0 1 , p ⁢ ( Ω ) be the classical finite element space of continuous piecewise linear functions on triangles K ∈ 𝒯 h with zero value on the boundary ∂ ⁡ Ω . To approximate the solution of (2.2), we are looking for u h ∈ V h such that

(2.3) ∫ Ω ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ ∇ ⁡ v h = ∫ Ω f ⁢ v h for all ⁢ v h ∈ V h .

Existence and uniqueness of u h ∈ V h can be again proved similarly as in [31, 25]. To solve this nonlinear problem, Newton’s method is advocated.

3 Anisotropic Finite Elements and Clément’s Interpolation Estimates

Anisotropic finite elements, that is meshes with possibly large aspect ratio, are considered. The framework of [22, 23] is used. For any K ∈ 𝒯 h , we note T K : K ^ → K one of the affine transformation mapping the reference triangle K ^ into K defined by

(3.1) 𝐱 = T K ⁢ ( 𝐱 ^ ) = M K ⁢ 𝐱 ^ + 𝐭 K ,

where M K ∈ ℝ d × d and 𝐭 K ∈ ℝ d . Since M K is invertible, it admits a singular value decomposition M K = R K T ⁢ Λ K ⁢ P K , where R K and P K are orthogonal matrices and Λ K is a diagonal matrix

Λ K = ( λ 1 , K ⋱ λ d , K ) , λ 1 , K ≥ … ≥ λ d , K > 0 ,

and

R K = ( 𝐫 1 , K T ⋮ 𝐫 d , K T ) .

In the above notations, for a given K ∈ 𝒯 h , 𝐫 1 , K , … , 𝐫 d , K are the unit vectors corresponding to the stretching directions and λ 1 , K , … , λ d , K their respective amplitude. In the context of anisotropic meshes, the classical minimal angle condition is not necessary anymore. However some additional assumptions are necessary. From now on, we will assume the two following geometrical assumptions:

For each K, the cardinality of Δ ⁢ K , that is the union of triangles sharing a vertex with K, is uniformly bounded from above, independently of the mesh geometry.
For each K, the diameter of Δ ⁢ K ^ = T K - 1 ⁢ ( Δ ⁢ K ) is uniformly bounded from above, independently of the mesh geometry.

Both assumptions guarantee that the constants involved in the interpolation estimates are mesh aspect ratio independent, which is necessary when working with anisotropic meshes. In particular, the second assumption excludes too distorted reference patches, an example of acceptable and non-acceptable patch is presented in Figure 1. The two assumptions seem to be fulfilled when using the anisotropic mesh generators in our numerical experiment [22, 23, 38, 43]. The Clément’s interpolant [15] is an important tool for deriving residual based a posteriori error estimates in the H 0 1 norm. Note that when d = 2 and p > 2 , since W 1 , p ⁢ ( Ω ) ↪ C 0 ⁢ ( Ω ¯ ) , the Lagrangian interpolant could also be used. However, in Section 7, we use our results when d = 3 and p = 3 , thus we only consider Clément’s interpolant in this paper.

$Figure 1 Top: example of acceptable patch: the size of Δ ⁢ K ^ {\Delta\hat{K}} is independent of the aspect ratio H / h {H/h} . Bottom: example of non-acceptable patch where the size of Δ ⁢ K ^ {\Delta\hat{K}} depends of the aspect ratio H / h {H/h} .$

Figure 1

Top: example of acceptable patch: the size of Δ ⁢ K ^ is independent of the aspect ratio H / h . Bottom: example of non-acceptable patch where the size of Δ ⁢ K ^ depends of the aspect ratio H / h .

Let R h : W 1 , p ⁢ ( Ω ) → V h be Clément’s interpolant with p ≥ 2 . Proceeding as in [22, 23] for the case d = 2 and p = 2 , we obtain the following.

Proposition 1 (Anisotropic Clément Interpolation Error Estimate in the L p Norm).

For p ≥ 2 , there exists a constant C ^ > 0 depending only on the reference triangle K ^ and on p such that for all v ∈ W 1 , p ⁢ ( Ω ) , for all K ∈ T h , we have for i = 1 , … , d + 1 ,

(3.2) ( ∏ j = 1 d λ j , K | ∂ ⁡ K i | ) 1 p ⁢ ∥ v - R h ⁢ ( v ) ∥ L p ⁢ ( ∂ ⁡ K i ) + ∥ v - R h ⁢ ( v ) ∥ L p ⁢ ( K ) ≤ C ^ ⁢ ω p , K ⁢ ( v ) ,

where

(3.3) ( ω p , K ⁢ ( v ) ) p = ∑ i = 1 d λ i , K p ⁢ ∥ ∇ ⁡ v ⋅ 𝐫 i , K ∥ L p ⁢ ( Δ ⁢ K ) p ,

Δ ⁢ K is the union of triangles sharing a vertex with those of K and ∂ ⁡ K i is an edge (face if d = 3 ) of K.

Proof.

Let v ^ : K ^ ↦ ℝ be defined by

v ^ ⁢ ( 𝐱 ^ ) = v ⁢ ( M K ⁢ 𝐱 ^ + 𝐭 𝐤 ) .

Using the equivalence between the p-norm and the 2-norm in ℝ d and the orthogonality of P K there exists C ^ depending only on p such that

∥ ∇ ^ ⁢ v ^ ∥ L p ⁢ ( K ^ ) p = ∥ M K T ⁢ ∇ ⁡ v ∥ L p ⁢ ( K ^ ) p = ∫ K ^ ∥ P K T ⁢ Λ K ⁢ R K ⁢ ∇ ⁡ v ∥ p p ≤ C ^ ⁢ ∫ K ^ ∥ Λ K ⁢ R K ⁢ ∇ ⁡ v ∥ p p = C ^ ⁢ ∑ i = 1 d λ i , K p ⁢ ∫ K ^ ∥ ∇ ⁡ v ⋅ 𝐫 i , K ∥ p p .

After the change of variable (3.1), since det ⁡ M K = ∏ j = 1 d λ i , K , we obtain

(3.4) ∥ ∇ ^ ⁢ v ^ ∥ L p ⁢ ( K ^ ) p ≤ C ^ ⁢ ∑ i = 1 d ( λ i , K p ∏ j = 1 d λ j , K ⁢ ∥ ∇ ⁡ v ⋅ 𝐫 i , K ∥ L p ⁢ ( K ) p ) .

Moreover,

∥ v - R h ⁢ ( v ) ∥ L p ⁢ ( K ) p = ( ∏ j = 1 d λ j , K ) ⁢ ∥ v ^ - R ^ h ⁢ ( v ^ ) ∥ L p ⁢ ( K ^ ) p

≤ C ^ ⁢ h K ^ p ⁢ ( ∏ j = 1 d λ j , K ) ⁢ ∥ ∇ ^ ⁢ v ^ ∥ L p ⁢ ( Δ ⁢ K ^ ) p

= C ^ ⁢ ( ∏ j = 1 d λ j , K ) ⁢ ∑ T ∈ Δ ⁢ K ^ ∥ ∇ ^ ⁢ v ^ ∥ L p ⁢ ( T ) p .

Using (3.4), we therefore obtain

∥ v - R h ⁢ ( v ) ∥ L p ⁢ ( K ) p ≤ C ^ ⁢ ( ω p , K ⁢ ( v ) ) p .

Proceeding in a similar way we have for i = 1 , … , d + 1 ,

∥ v - R h ⁢ ( v ) ∥ L p ⁢ ( ∂ ⁡ K i ) p ≤ C ^ ⁢ | ∂ ⁡ K i | ⁢ ∥ v ^ - R ^ h ⁢ ( v ^ ) ∥ L p ⁢ ( ∂ ⁡ K ^ i ) p ≤ C ^ ⁢ | ∂ ⁡ K i | ⁢ ∥ ∇ ^ ⁢ v ^ ∥ L p ⁢ ( Δ ⁢ K ^ ) p .

Using again (3.4), we obtain

∥ v - R h ⁢ ( v ) ∥ L p ⁢ ( ∂ ⁡ K i ) p ≤ | ∂ ⁡ K i | ∏ j = 1 d λ j , K ⁢ ( ω p , K ⁢ ( v ) ) p ,

which concludes the proof. ∎

Remark 1.

When p = 2 , we have as in [22, 23]

( ω 2 , K ⁢ ( v ) ) 2 = ∑ i = 1 d λ i , K 2 ⁢ ( 𝐫 i , K T ⁢ G K ⁢ ( v ) ⁢ 𝐫 i , K ) ,

with G K ⁢ ( v ) ∈ ℝ d × d defined as ( G K ⁢ ( v ) ) i ⁢ j = ∫ Δ ⁢ K ∂ ⁡ v ∂ ⁡ x i ⁢ ∂ ⁡ v ∂ ⁡ x j for i , j = 1 , … , d .

4 The Anisotropic Error Estimator

Let u ∈ W 0 1 , p ⁢ ( Ω ) be the weak solution of (2.2) and u h ∈ V h the solution of the finite element approximation (2.3). We derive an estimator for the error in the norm

μ ⁢ ∥ ∇ ⁡ ( u - u h ) ∥ L 2 ⁢ ( Ω ) 2 + ∥ ∇ ⁡ ( u - u h ) ∥ L p ⁢ ( Ω ) p

and also in the quasi-norm

∫ Ω | ∇ ⁡ ( u - u h ) | 2 ⁢ ( μ + ( | ∇ ⁡ ( u - u h ) | + | ∇ ⁡ u | ) p - 2 ) .

We first focus on the quasi-norm.

Proposition 2.

Let μ ≥ 0 , for w ∈ W 0 1 , p ⁢ ( Ω ) and v ∈ W 0 1 , p ⁢ ( Ω ) let

∥ v ∥ ( w ) p = ∫ Ω | ∇ ⁡ v | 2 ⁢ ( μ + ( | ∇ ⁡ v | + | ∇ ⁡ w | ) p - 2 ) .

Then ∥ ⋅ ∥ ( w ) is a quasi-norm on W 0 1 , p ⁢ ( Ω ) .

The proof is as in [7]. We now state and prove a slight modification of [34, Lemma 2.1], where we set α 1 = α 2 = 0 , δ = 0 and μ has been added.

Lemma 1.

Let μ ≥ 0 , p ≥ 2 , d = 2 , 3 and let k ∈ C 1 ( 0 , ∞ ) ∩ C [ 0 , ∞ [ be defined by k ⁢ ( t ) = μ + t p - 2 . Then there exist two constants C , M > 0 such that for all μ ≥ 0 and for all x , y ∈ R d we have

(4.1) | k ⁢ ( | 𝐱 | ) ⁢ 𝐱 - k ⁢ ( | 𝐲 | ) ⁢ 𝐲 | ≤ C ⁢ | 𝐱 - 𝐲 | ⁢ ( μ + ( | 𝐱 | + | 𝐲 | ) p - 2 )

and

(4.2) ( k ⁢ ( | 𝐱 | ) ⁢ 𝐱 - k ⁢ ( | 𝐲 | ) ⁢ 𝐲 ) ⋅ ( 𝐱 - 𝐲 ) ≥ M ⁢ | 𝐱 - 𝐲 | 2 ⁢ ( μ + ( | 𝐱 | + | 𝐲 | ) p - 2 ) .

Proof.

We start by proving (4.1). First, observe that for all s , t > 0 ,

(4.3) | k ⁢ ( t ) ⁢ t - k ⁢ ( s ) ⁢ s | ≤ C ⁢ | t - s | ⁢ ( μ + ( t + s ) p - 2 ) .

Indeed, assume for instance 0 ≤ s ≤ t ; by the mean value theorem there exists s ≤ x ≤ t such that

k ⁢ ( t ) ⁢ t - k ⁢ ( s ) ⁢ s = ( t - s ) ⁢ ( ( p - 2 ) ⁢ x p - 3 ⁢ x + ( μ + x p - 2 ) )

≤ ( t - s ) ⁢ ( p - 1 ) ⁢ ( μ + x p - 2 )

and (4.3) follows directly. Proceeding as in [34, Lemma 2.1], we can obtain (4.1). To prove (4.2), we observe that there exists M > 0 for all t ≥ s ≥ 0 such that

k ⁢ ( t ) ⁢ t - k ⁢ ( s ) ⁢ s ≥ M ⁢ ( t - s ) ⁢ ( μ + ( t + s ) p - 2 ) .

Indeed, we have

( t - s ) ⁢ ( μ + ( t + s ) p - 2 ) = ( t - s ) ⁢ μ + t ⁢ exp ⁡ ( ( p - 2 ) ⁢ ln ⁡ ( t + s ) ) - s ⁢ exp ⁡ ( ( p - 2 ) ⁢ ln ⁡ ( t + s ) )

≤ ( t - s ) ⁢ μ + 2 p - 2 ⁢ ( t ⁢ exp ⁡ ( ( p - 2 ) ⁢ ln ⁡ ( t ) ) - s ⁢ exp ⁡ ( ( p - 2 ) ⁢ ln ⁡ ( s ) ) )

≤ 2 p - 2 ⁢ ( t ⁢ ( μ + t p - 2 ) - s ⁢ ( μ + s p - 2 ) ) .

Again following the idea of [34, Lemma 2.1], we prove (4.2). ∎

We now recall [32, Lemma 5.2] when θ = 1 .

Lemma 2.

For all σ 1 , σ 2 , σ 3 ≥ 0 we have

( σ 3 + σ 1 ) p - 2 ⁢ σ 1 ⁢ σ 2 ≤ ( σ 3 + σ 1 ) p - 2 ⁢ σ 1 2 + ( σ 3 + σ 2 ) p - 2 ⁢ σ 2 2 .

As in [33] we set

a ⁢ ( u , v ) = ∫ Ω ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ⋅ ∇ ⁡ v ,

where it must be noticed that a ⁢ ( ⋅ , ⋅ ) is not linear with respect to the first variable.

Following [32] and [7] we can prove the following result.

Proposition 3.

Let C and M be the constants of Lemma 1. Let u ∈ W 0 1 , p ⁢ ( Ω ) . Then for all v , w ∈ W 0 1 , p ⁢ ( Ω ) we have

(4.4) a ⁢ ( u , u - v ) - a ⁢ ( v , u - v ) ≥ M 2 ⁢ ∥ u - v ∥ ( u ) p

and

(4.5) | a ⁢ ( u , w ) - a ⁢ ( v , w ) | ≤ C ⁢ ( ∥ u - v ∥ ( u ) p + ∥ w ∥ ( u ) p ) .

Proof.

First we prove equation (4.4) using relation (4.2) of Lemma 1. We have

a ⁢ ( u , u - v ) - a ⁢ ( v , u - v ) = ∫ Ω ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u - ( μ + | ∇ ⁡ v | p - 2 ) ⁢ ∇ ⁡ v ) ⋅ ∇ ⁡ ( u - v )

≥ M ⁢ ∫ Ω | ∇ ⁡ ( u - v ) | 2 ⁢ ( μ + ( | ∇ ⁡ u | + | ∇ ⁡ v | ) p - 2 ) .

Using the fact that for all 𝐱 , 𝐲 ∈ ℝ d we have

2 ⁢ ( | 𝐱 | + | 𝐲 | ) ≥ | 𝐱 | + | 𝐱 - 𝐲 | ,

we obtain

a ⁢ ( u , u - v ) - a ⁢ ( v , u - v ) ≥ M 2 ⁢ ∫ Ω | ∇ ⁡ ( u - v ) | 2 ⁢ ( μ + ( | ∇ ⁡ u | + | ∇ ⁡ ( u - v ) | ) p - 2 )

= M 2 ⁢ ∥ u - v ∥ ( u ) p .

We now prove (4.5). We have

| a ( u , w ) - a ( v , w ) | ≤ ∫ Ω | ( ( μ + | ∇ u | p - 2 ) ∇ u - ( μ + | ∇ v | p - 2 ) ∇ v ) ∥ ∇ w | .

We apply (4.1) of Lemma 1 and the fact that for all 𝐱 , 𝐲 ∈ ℝ d we have

| 𝐱 | + | 𝐲 | 2 ≤ | 𝐱 | + | 𝐱 - 𝐲 |

to obtain

| a ⁢ ( u , w ) - a ⁢ ( v , w ) | ≤ C ⁢ ∫ Ω | ∇ ⁡ ( u - v ) | ⁢ ( μ + ( | ∇ ⁡ u | + | ∇ ⁡ ( u - v ) | ) p - 2 ) ⁢ | ∇ ⁡ w | .

Using Lemma 2, we obtain

| a ⁢ ( u , w ) - a ⁢ ( v , w ) | ≤ C ⁢ ( ∫ Ω | ∇ ⁡ ( u - v ) | 2 ⁢ ( μ + ( | ∇ ⁡ u | + | ∇ ⁡ ( u - v ) | ) p - 2 ) + ∫ Ω ( μ + ( | ∇ ⁡ u | + | ∇ ⁡ w | ) p - 2 ) ⁢ | ∇ ⁡ w | 2 ) ,

which concludes the proof. ∎

The next result follows directly taking w = u - v in (4.4)–(4.5).

Proposition 4.

Let u ∈ W 0 1 , p ⁢ ( Ω ) and v ∈ W 0 1 , p ⁢ ( Ω ) . Then there exist two constants M , C > 0 independent of μ, u, v such that

(4.6) 1 2 ⁢ C ⁢ ( a ⁢ ( u , u - v ) - a ⁢ ( v , u - v ) ) ≤ ∥ u - v ∥ ( u ) p ≤ 2 M ⁢ ( a ⁢ ( u , u - v ) - a ⁢ ( v , u - v ) ) .

We can also prove the following proposition.

Proposition 5.

There exists α > 0 such that for all μ ≥ 0 and u , v ∈ W 0 1 , p ⁢ ( Ω ) we have

(4.7) α ⁢ ( μ ⁢ ∥ ∇ ⁡ ( u - v ) ∥ L 2 ⁢ ( Ω ) 2 + ∥ ∇ ⁡ ( u - v ) ∥ L p ⁢ ( Ω ) p ) ≤ a ⁢ ( u , u - v ) - a ⁢ ( v , u - v ) .

Proof.

It suffices to prove that there exists α such that for all 𝐱 , 𝐲 ∈ ℝ d we have

(4.8) ( ( μ + | 𝐱 | p - 2 ) ⁢ 𝐱 - ( μ + | 𝐲 | p - 2 ) ⁢ 𝐲 ) ⋅ ( 𝐱 - 𝐲 ) ≥ α ⁢ ( μ ⁢ | 𝐱 - 𝐲 | 2 + | 𝐱 - 𝐲 | p ) .

Indeed, we have

( ( μ + | 𝐱 | p - 2 ) ⁢ 𝐱 - ( μ + | 𝐲 | p - 2 ) ⁢ 𝐲 ) ⋅ ( 𝐱 - 𝐲 ) = μ ⁢ | 𝐱 - 𝐲 | 2 + ( | 𝐱 | p - 2 ⁢ 𝐱 - | 𝐲 | p - 2 ⁢ 𝐲 ) ⋅ ( 𝐱 - 𝐲 )

so that (4.8) follows using [25, Lemma 5.1]. ∎

We can now state the main result of the paper.

Theorem 1.

Let u ∈ W 0 1 , p ⁢ ( Ω ) be a solution of (2.2), u h ∈ V h a solution of (2.3) and e = u - u h . Let p ≥ 2 . Then there exists C ^ 1 > 0 independent of μ, u, the mesh size and the aspect ratio such that

(4.9) μ ⁢ ∥ ∇ ⁡ ( u - u h ) ∥ L 2 ⁢ ( Ω ) 2 + ∥ ∇ ⁡ ( u - u h ) ∥ L p ⁢ ( Ω ) p + ∥ u - u h ∥ ( u ) p ≤ C ^ 1 ⁢ ( ∑ K ∈ 𝒯 h η 2 , K + ϵ 1 ) .

Moreover, if there exists a constant C ^ , dependent only on the reference triangle K ^ , such that for all K ∈ T h and for i = 1 , … , d - 1 ,

(4.10) λ i , K 2 ⁢ ( 𝐫 i , K T ⁢ G K ⁢ ( u - u h ) ⁢ 𝐫 i , K ) ≤ C ^ ⁢ λ d , K 2 ⁢ ( 𝐫 d , K ⁢ G K ⁢ ( u - u h ) ⁢ 𝐫 d , K )

and assuming λ i , K vary smoothly around K, then there exists C ^ 2 > 0 independent of μ , u , the mesh size and the aspect ratio such that

(4.11) ∑ K ∈ 𝒯 h η 2 , K ≤ C ^ 2 ⁢ ( ∥ u - u h ∥ ( u ) p + ∑ j = 1 4 ϵ j ) .

Here

(4.12)

η 2 , K = ( ∥ ∇ ⋅ ( ( μ + | ∇ u h | p - 2 ) ∇ u h ) + Π K f ∥ L 2 ⁢ ( K )

+ 1 2 ∑ i = 1 d + 1 ( | ∂ ⁡ K i | ∏ j = 1 d λ j , K ) 1 2 ∥ [ ( μ + | ∇ u h | p - 2 ) ∇ u h ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K i ) ) ω 2 , K ( u - u h )

and

ϵ 1 = ∑ K ∈ 𝒯 h ∥ f - Π K ⁢ f ∥ L 2 ⁢ ( K ) ⁢ ω 2 , K ⁢ ( e ) , ϵ 2 = ∑ K ∈ 𝒯 h λ d , K ⁢ ∥ f - Π K ⁢ f ∥ L 2 ⁢ ( K ) 2 ,

ϵ 3 = ∑ K ∈ 𝒯 h ∑ K ′ ∈ 𝒫 K ∫ K ′ | ( μ + | ∇ ⁡ u h | ) | K p - 2 - ( μ + | ∇ ⁡ u h | ) | K ′ p - 2 | ⁢ | ∇ ⁡ e | 2 , ϵ 4 = ∑ K ∈ 𝒯 h λ d , K ⁢ ∥ ∇ ⁡ e ∥ L 2 ⁢ ( Δ ⁢ K ) 2 .

Here ω 2 , K ⁢ ( ⋅ ) is defined in (3.3), n stands for the unit outer normal to triangle K, [ ⋅ ] denotes the jump across the edges (or faces if d = 3 ) ∂ ⁡ K i of K for i = 1 , … , d + 1 ( [ ⋅ ] = 0 if ∂ ⁡ K ⊂ ∂ ⁡ Ω ). For all K ∈ T h , Π K ⁢ f is the L 2 ⁢ ( K ) projection of f onto the set of piecewise constant functions defined by

Π K ⁢ f = 1 | K | ⁢ ∫ K f .

Moreover, we denote

𝒫 K = Δ ⁢ K ∪ ( ⋃ i = 1 d + 1 Δ ⁢ K i )

with K i the ith element sharing a facet ∂ ⁡ K i with K.

Remark 2.

Let 2 ≤ p ′ ≤ p and let q ′ be the Hölder conjugate of p ′ . Then the upper bound (4.9) can be generalized to

C ^ 1 ⁢ ( ∑ K ∈ 𝒯 h η p ′ , K + ∑ K ∈ 𝒯 h ∥ f - Π K ⁢ f ∥ L q ′ ⁢ ( K ) ⁢ ω p ′ , K ⁢ ( u - u h ) ) ,

where

η p ′ , K = ( ∥ ∇ ⋅ ( ( μ + | ∇ u h | p - 2 ) ∇ u h ) + Π K f ∥ L q ′ ⁢ ( K )

+ 1 2 ∑ i = 1 d + 1 ( | ∂ ⁡ K i | ∏ j = 1 d λ j , K ) 1 p ′ ∥ [ ( μ + | ∇ u h | p - 2 ) ∇ u h ⋅ 𝐧 ] ∥ L q ′ ⁢ ( ∂ ⁡ K i ) ) ω p ′ , K ( u - u h ) .

Remark 3.

Assume f ∈ H 1 ⁢ ( Ω ) . Then we have

∥ f - Π K ⁢ f ∥ L 2 ⁢ ( K ) 2 ≤ C ^ ⁢ ∑ i = 1 d λ i , K 2 ⁢ ∥ ∇ ⁡ f ⋅ 𝐫 i , K ∥ L 2 ⁢ ( K ) 2 .

In the isotropic case this yields ϵ 1 , ϵ 2 , ϵ 4 = O ⁢ ( h 3 ) , that are high order terms compared to ∥ u - u h ∥ ( u ) p which should be, according to [7], O ⁢ ( h 2 ) . In the anisotropic context if, for instance, f depends only on x 2 , d = 2 and for all K ∈ 𝒯 h , 𝐫 1 , K = ( 0 , 1 ) , then ϵ 1 , ϵ 2 , ϵ 4 = O ⁢ ( ( max K ∈ 𝒯 h ⁡ λ 2 , K ) 3 ) . Numerical experiments performed in Sections 5 and 6 confirm previous predictions and show that ϵ 3 is also of higher order with respect to ∥ u - u h ∥ ( u ) p , see Figures 2 and 3. Thus when max K ∈ 𝒯 h ⁡ λ d , K 2 → 0 , then ∥ u - u h ∥ ( u ) p and ∑ K ∈ 𝒯 h η 2 , K are equivalent up to higher order terms.

Remark 4.

Observe that local error estimator (4.12) is not standard since the exact solution is present in the term ω 2 , K ⁢ ( u - u h ) . In practice post-processing techniques can be applied in order to approximate the gradient of the exact solution. For instance Zienkiewicz–Zhu (ZZ) post-processing can be applied [2, 51, 52]. We will replace

∂ ⁡ ( u - u h ) ∂ ⁡ x i by Π h Z ⁢ Z ⁢ ∂ ⁡ u h ∂ ⁡ x i - ∂ ⁡ u h ∂ ⁡ x i , i = 1 , … , d ,

where for any v h ∈ V h , Π h Z ⁢ Z ⁢ ∂ ⁡ v h x i is an approximate L 2 ⁢ ( Ω ) projection of ∂ ⁡ v h ∂ ⁡ x i onto V h . The use of ZZ post-processing can be justified theoretically whenever superconvergence occurs. For elliptic equations and structured meshes, superconvergence of the ZZ recovery is presented in [12, 52]. Then, the post-processing is asymptotically exact, that is,

lim h → 0 ⁡ ∥ Π h Z ⁢ Z ⁢ ∇ ⁡ u h - ∇ ⁡ u h ∥ L 2 ⁢ ( Ω ) ∥ ∇ ⁡ u - ∇ ⁡ u h ∥ L 2 ⁢ ( Ω ) = 1 .

In [28] equivalence between ∥ Π h Z ⁢ Z ⁢ ∇ ⁡ u h - ∇ ⁡ u h ∥ L 2 ⁢ ( Ω ) and ∥ ∇ ⁡ u - ∇ ⁡ u h ∥ L 2 ⁢ ( Ω ) on general meshes has been proven, while the superconvergence of the ZZ gradient recovery has been shown for unstructured anisotropic meshes in [11]. Moreover, the efficiency of the ZZ post-processing has already been numerically demonstrated in [42, 43, 41, 26, 37, 44, 10] in the case of anisotropic meshes for elliptic, parabolic and hyperbolic equations. Notice that post-processing techniques for the p-Laplacian problem have already been studied in [14, 33].

Proof of the Upper Bound (4.9) of Theorem 1.

For all v ∈ W 0 1 , p ⁢ ( Ω ) and v h ∈ V h we have using (2.2) and (2.3),

a ⁢ ( u , v ) - a ⁢ ( u h , v ) = ∫ Ω f ⁢ ( v - v h ) - ∫ Ω ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ ∇ ⁡ ( v - v h )

= ∑ K ∈ 𝒯 h ( ∫ K ( f + ∇ ⋅ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ) ⁢ ( v - v h ) + 1 2 ⁢ ∫ ∂ ⁡ K [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ⁢ ( v - v h ) ) ,

where we integrated by parts. We set v = u - u h and v h = R h ⁢ ( u - u h ) , adding and subtracting Π K ⁢ f , using the Cauchy–Schwarz inequality and Proposition 1, we obtain

| a ⁢ ( u , v ) - a ⁢ ( u h , v ) |

≤ ∑ K ∈ 𝒯 h ( ∥ Π K ⁢ f + ∇ ⋅ ( ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ) ∥ L 2 ⁢ ( K ) + ∥ f - Π K ⁢ f ∥ L 2 ⁢ ( K ) ) ⁢ ∥ u - u h - R h ⁢ ( u - u h ) ∥ L 2 ⁢ ( K )

+ 1 2 ⁢ ∑ K ∈ 𝒯 h ∥ [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K ) ⁢ ∥ u - u h - R h ⁢ ( u - u h ) ∥ L 2 ⁢ ( ∂ ⁡ K )

≤ C ∑ K ∈ 𝒯 h ( ∥ Π K f + ∇ ⋅ ( ( μ + | ∇ u h | p - 2 ) ∇ u h ) ∥ L 2 ⁢ ( K ) + 1 2 ∑ i = 1 d + 1 ( | ∂ ⁡ K i | ∏ j = 1 d λ j , K ) 1 2 ∥ [ ( μ + | ∇ u h | p - 2 ) ∇ u h ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K i )

+ ∥ f - Π K f ∥ L 2 ⁢ ( K ) ) ω 2 , K ( u - u h ) ,

which, together with Propositions 4 and 5, yields (4.9). ∎

We now aim to prove the lower bound (4.11). The classical standard bubble functions [5, 48], adapted to the anisotropic case [43, 20] and modified to take into account the nonlinearity, are involved.

Proposition 6.

Let e = u - u h . Under the same assumptions of Theorem 1 there exist a function φ ∈ W 1 , p ⁢ ( Ω ) and a constant C ^ > 0 (that depends only on the reference triangle K ^ and on p) such that for any K ∈ T h and i = 1 , … , d + 1 ,

∫ ∂ ⁡ K i [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ⁢ φ = 1 2 ⁢ ( | ∂ ⁡ K i | 1 2 ( ∏ j = 1 d λ j , K ) 1 2 ⁢ ω 2 , K ⁢ ( e ) + | ∂ ⁡ K i | 1 2 ( ∏ j = 1 d λ j , K i ) 1 2 ⁢ ω 2 , K i ⁢ ( e ) )

(4.13) × ∥ [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K i ) ,

(4.14) ∫ K ( Π K ⁢ f + ∇ ⋅ ( ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ) ) ⁢ φ = ∥ Π k ⁢ f + ∇ ⋅ ( ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ) ∥ L 2 ⁢ ( K ) ⁢ ω 2 , K ⁢ ( e ) ,

(4.15) ( ∑ j = 1 d λ j , K 2 ⁢ ∥ ∇ ⁡ φ ⋅ 𝐫 j , K ∥ L 2 ⁢ ( K ) 2 ) 1 2 ≤ C ^ ⁢ ( ω 2 , K ⁢ ( e ) + ∑ i = 1 d + 1 ω 2 , K i ⁢ ( e ) ) ,

(4.16) ∫ K | ∇ ⁡ φ | 2 ≤ C ^ ⁢ ( ω 2 , K 2 ⁢ ( e ) λ d , K 2 + ∑ i = 1 d + 1 ω 2 , K i 2 ⁢ ( e ) λ d , K i 2 ) ,

(4.17) ∫ K | ∇ φ | p Z ≤ C ^ ∥ ∇ e ∥ L p ( 𝒫 K p ) .

We denote for i = 1 , … , d + 1 , K i the ith element sharing facet ∂ ⁡ K i with K.

Proof.

Following the proof of [43], we claim that

φ = ∑ K ∈ 𝒯 h C K ⁢ Ψ K + 1 2 ⁢ ∑ K ∈ 𝒯 h ∑ i = 1 d + 1 C ∂ ⁡ K i ⁢ Ψ ∂ ⁡ K i ,

where C K , C ∂ ⁡ K i are constants Ψ K , Ψ ∂ ⁡ K i are the usual bubble functions over K and its edges (faces if d = 3 ) ∂ ⁡ K i , i = 1 , … , d + 1 . We set C ∂ ⁡ K i = 0 if ∂ ⁡ K i ⊂ ∂ ⁡ Ω . First we compute C ∂ ⁡ K i for i = 1 , … , d + 1 . We require that the constants associated to the same facet shared by two elements are equal. Using the fact that Ψ K , Ψ ∂ ⁡ K j are zero over ∂ ⁡ K i for all i ≠ j , the fact that ( Π K f + ∇ ⋅ ( ( μ + | ∇ u h | p - 2 ) ∇ u h ) and [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] are constants over K and ∂ ⁡ K i respectively, | K | = C ^ ⁢ ∏ j = 1 d λ j , K and (4.14) and (6), we obtain

C ∂ ⁡ K i = ± 1 2 ⁢ ( ( | ∂ ⁡ K i | 2 ∏ j = 1 d λ j , K ) 1 2 ⁢ ω 2 , K ⁢ ( e ) + ( | ∂ ⁡ K i | 2 ∏ j = 1 d λ j , K i ) 1 2 ⁢ ω 2 , K i ⁢ ( e ) ) ⁢ 1 ∫ ∂ ⁡ K i Ψ ∂ ⁡ K i

and

C K = ± 1 ∫ K Ψ K ⁢ ( C ^ ⁢ ( ∏ j = 1 d λ j , K ) 1 2 ⁢ ω 2 , K ⁢ ( e ) ∓ ∑ i = 1 d + 1 1 2 ⁢ ( ( | ∂ ⁡ K i | 2 ∏ j = 1 d λ j , K ) 1 2 ⁢ ω 2 , K ⁢ ( e ) + ( | ∂ ⁡ K i | 2 ∏ j = 1 d λ j , K i ) 1 2 ⁢ ω 2 , K i ⁢ ( e ) ) ⁢ ∫ K Ψ ∂ ⁡ K i ∫ ∂ ⁡ K i Ψ ∂ ⁡ K i ) ,

where the signs are chosen again accordingly to the signs of [ Π ∂ ⁡ K i ⁢ μ ⁢ ∇ ⁡ u h ⋅ 𝐧 ] and Π K f + ∇ ⋅ ( ( μ + | ∇ u h | p - 2 ) ∇ u h ) ) . By construction φ satisfies thus both equations (6)–(4.14). We now prove (4.17). We have

∫ K | ∇ ⁡ φ | p ≤ C ^ ⁢ ∫ K ( | C K | p ⁢ | ∇ ⁡ Ψ K | p + ∑ i = 1 d + 1 | C ∂ ⁡ K i | p ⁢ | ∇ ⁡ Ψ ∂ ⁡ K i | p ) .

Using the change of variables (3.1), we have

∫ K Ψ K = | K | ⁢ ∫ K ^ Ψ ^ K ^ = C ^ ⁢ ∏ j = 1 d λ j , K ,

∫ K Ψ ∂ ⁡ K i = | K | ⁢ ∫ K ^ Ψ ^ ∂ ⁡ K ^ i = C ^ ⁢ ∏ j = 1 d λ j , K , i = 1 , … , d + 1 ,

∫ ∂ ⁡ K i Ψ ∂ ⁡ K i = | ∂ ⁡ K i | ⁢ ∫ ∂ ⁡ K ^ i Ψ ^ ∂ ⁡ K ^ i = C ^ ⁢ | ∂ ⁡ K i | , i = 1 , … , d + 1 .

Thus we obtain the following bounds:

| C ∂ ⁡ K i | p ≤ C ^ ⁢ ( ω 2 , K p ⁢ ( e ) ( ∏ j = 1 d λ j , K ) p 2 + ω 2 , K i p ⁢ ( e ) ( ∏ j = 1 d λ j , K i ) p 2 ) ,

| C K | p ≤ C ^ ⁢ ( ω 2 , K p ⁢ ( e ) ( ∏ j = 1 d λ j , K ) p 2 + ∑ i = 1 d + 1 ω 2 , K i p ⁢ ( e ) ( ∏ j = 1 d λ j , K i ) p 2 ) ,

hence

∫ K | ∇ ⁡ φ | p ≤ C ^ ⁢ ( ω 2 , K p ⁢ ( e ) ( ∏ j = 1 d λ j , K ) p 2 + ∑ i = 1 d + 1 ω 2 , K i p ⁢ ( e ) ( ∏ j = 1 d λ j , K i ) p 2 ) ⁢ ( ∫ K | ∇ ⁡ Ψ K | p + ∑ j = 1 d + 1 ∫ K | ∇ ⁡ Ψ ∂ ⁡ K j | p ) .

Using assumption (4.10), we have

ω 2 , K p ⁢ ( e ) ≤ C ^ ⁢ ( λ d , K 2 ⁢ ∥ ∇ ⁡ e ∥ L 2 ⁢ ( Δ ⁢ K ) 2 ) p 2 .

Proceeding in the same way as we did for (3.4), we can show that there exists a constant C ^ > 0 such that for all v ∈ W 0 1 , p ⁢ ( Ω ) ,

(4.18) ∥ ∇ ⁡ v ∥ L p ⁢ ( K ) p ≤ C ^ ⁢ ∑ i = 1 d ( ∏ j = 1 d λ j , K λ i , K p ) ⁢ ∥ ∇ ^ ⁢ v ^ ⋅ 𝐩 i , K ∥ L p ⁢ ( K ^ ) p .

Since λ 1 , K ≥ ⋯ ≥ λ d , K > 0 , we obtain from (4.18) that

∥ ∇ ⁡ Ψ K ∥ L p ⁢ ( K ) p ≤ C ^ ⁢ ∏ j = 1 d λ j , K λ d , K p ⁢ ∥ ∇ ^ ⁢ Ψ ^ K ^ ∥ L p ⁢ ( K ^ ) p

and the same occurs for ∇ ⁡ Ψ ∂ ⁡ K i . Altogether we obtain using Hölder’s inequality

∫ K | ∇ ⁡ φ | p ≤ C ^ ⁢ ( ∏ j = 1 d λ j , K ) 2 - p 2 ⁢ ( ( ∫ Δ ⁢ K | ∇ ⁡ e | 2 ) p 2 + ∑ i = 1 d + 1 ( ∫ Δ ⁢ K i | ∇ ⁡ e | 2 ) p 2 )

≤ C ^ ⁢ ( ∏ j = 1 d λ j , K ) 2 - p 2 ⁢ ( | Δ ⁢ K | p - 2 2 ⁢ ∫ Δ ⁢ K | ∇ ⁡ e | p + ∑ i = 1 d + 1 | Δ ⁢ K i | p - 2 2 ⁢ ∫ Δ ⁢ K i | ∇ ⁡ e | p ) ,

and using assumption (1) of Section 3 and the fact that λ i , K vary smoothly around K, we obtain (4.17). To prove (4.15), we proceed in the same way to get

∫ K | ∇ ⁡ φ ⋅ 𝐫 i , K | 2 ≤ C ^ ⁢ ( ω 2 , K 2 ⁢ ( e ) ∏ j = 1 d λ j , K + ∑ l = 1 d + 1 ω 2 , K l 2 ⁢ ( e ) ∏ j = 1 d λ j , K l ) ⁢ ( ∫ K | ∇ ⁡ Ψ K ⋅ 𝐫 i , K | 2 + ∑ j = 1 d + 1 ∫ K | ∇ ⁡ Ψ ∂ ⁡ K j ⋅ 𝐫 i , K | 2 ) .

Using again (3.4) (which is an equality when p = 2 ), we obtain the desired result. Finally, we proceed in a similar way to prove (4.16) ∎

We are now ready to prove (4.11).

Proof of the Lower Bound (4.11) of Theorem 1.

Using the definition of η 2 , K and identities (6)–(4.14), one can write

∑ K ∈ 𝒯 h η 2 , K = ∑ K ∈ 𝒯 h ∫ K ( Π K ⁢ f + ∇ ⋅ ( ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ) ⁢ φ ) + 1 2 ⁢ ∫ ∂ ⁡ K [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ⁢ φ ,

where φ is the bubble function introduced in proposition 6. Using (2.2) and integration by parts we obtain

∑ K ∈ 𝒯 h η 2 , K = ∑ K ∈ 𝒯 h ∫ K ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u - ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ) ⋅ ∇ ⁡ φ - ∑ K ∈ 𝒯 h ∫ K ( f - Π K ⁢ f ) ⁢ ( φ - Π K ⁢ φ )

≤ | a ⁢ ( u , φ ) - a ⁢ ( u h , φ ) | + ∑ K ∈ 𝒯 h ∫ K | f - Π K ⁢ f | ⁢ | φ - Π K ⁢ φ | .

Using Proposition 3 together with the Cauchy–Schwarz inequality, we get

(4.19) ∑ K ∈ 𝒯 h η 2 , K ≤ C ^ ⁢ ( ∥ e ∥ ( u h ) p + ∥ φ ∥ ( u h ) p ) + ∑ K ∈ 𝒯 h ∥ f - Π K ⁢ f ∥ L 2 ⁢ ( K ) ⁢ ∥ φ - Π K ⁢ φ ∥ L 2 ⁢ ( K ) .

Using the change of variable (3.1) and the Poincaré–Wirtinger inequality, we can show that

∥ φ - Π K ⁢ φ ∥ L 2 ⁢ ( K ) 2 = ∏ j = 1 d λ j , K ⁢ ∫ K ^ ( φ ^ - Π ^ K ^ ⁢ φ ^ ) 2 ≤ C ^ ⁢ ∏ j = 1 d λ j , K ⁢ ∫ K ^ | ∇ ^ ⁢ φ ^ | 2 ,

which together with (3.4) and (4.15) gives

∥ φ - Π K ⁢ φ ∥ L 2 ⁢ ( K ) ≤ C ^ ⁢ ( ω 2 , K ⁢ ( e ) + ∑ i = 1 d + 1 ω 2 , K i ⁢ ( e ) ) .

We obtain thus, using Young’s inequality,

∑ K ∈ 𝒯 h ∥ f - Π K ⁢ f ∥ L 2 ⁢ ( K ) ⁢ ∥ φ - Π K ⁢ φ ∥ L 2 ⁢ ( K ) ≤ C ^ ⁢ ( ϵ 1 + ϵ 2 + ϵ 4 ) .

We have

∥ φ ∥ ( u h ) p = ∑ K ∈ 𝒯 h ∫ K ( μ + | ∇ ⁡ u h | + | ∇ ⁡ φ | ) p - 2 ⁢ | ∇ ⁡ φ | 2 ≤ ∑ K ∈ 𝒯 h C ^ ⁢ ∫ K ( μ + | ∇ ⁡ u h | ) p - 2 ⁢ | ∇ ⁡ φ | 2 + C ^ ⁢ ∑ K ∈ 𝒯 h ∫ K | ∇ ⁡ φ | p

≤ C ^ ⁢ ∑ K ∈ 𝒯 h ∫ 𝒫 K ( μ + | ∇ ⁡ u h | p - 2 ) | K ⁢ | ∇ ⁡ e | 2 + | ∇ ⁡ e | p ,

where we used (4.16)–(4.17) and (4.10). We have

C ^ ⁢ ∑ K ∈ 𝒯 h ∫ 𝒫 K ( μ + | ∇ ⁡ u h | ) | K p - 2 ⁢ | ∇ ⁡ e | 2 = C ^ ⁢ ∑ K ∈ 𝒯 h ∫ 𝒫 K ( μ + | ∇ ⁡ u h | ) p - 2 ⁢ | ∇ ⁡ e | 2

+ C ^ ⁢ ∑ K ∈ 𝒯 h ∑ K ′ ∈ 𝒫 K ∫ K ′ ( ( μ + | ∇ ⁡ u h | ) | K p - 2 - ( μ + | ∇ ⁡ u h | ) | K ′ p - 2 ) ⁢ | ∇ ⁡ e | 2 .

Altogether, we have

∑ K ∈ 𝒯 h η 2 , K ≤ C ^ ⁢ ( ∥ e ∥ ( u h ) p + ϵ 1 + ϵ 2 + ϵ 4 ) + C ^ ⁢ ∑ K ∈ 𝒯 h ∑ K ′ ∈ 𝒫 K ∫ K ′ ( ( μ + | ∇ ⁡ u h | ) | K p - 2 - ( μ + | ∇ ⁡ u h | ) | K ′ p - 2 ) ⁢ | ∇ ⁡ e | 2 .

Therefore, applying the triangle inequality ∥ e ∥ ( u h ) p ≤ C ^ ⁢ ∥ e ∥ ( u ) p , we get the desired result. ∎

5 Numerical Experiments with Non-Adapted Meshes

The goal of this section is to check numerically the sharpness of the error estimator (4.12) derived in the previous section. We are interested by the case p = 3 , since it corresponds to the so called Smagorinsky model [46, 45] discussed in Section 7. We define

e Q ⁢ N = ∥ u - u h ∥ ( u ) 3 , e 3 = ∥ ∇ ⁡ ( u - u h ) ∥ L 3 ⁢ ( Ω ) 3 and e 2 = μ ⁢ ∥ ∇ ⁡ ( u - u h ) ∥ L 2 ⁢ ( Ω ) 2 .

We also define the following effectivity indices:

(5.1) ei N = ∑ K ∈ 𝒯 h η 2 , K e 2 + e 3 ,

(5.2) ei Q ⁢ N = ∑ K ∈ 𝒯 h η 2 , K e Q ⁢ N ,

ei Z ⁢ Z = ∥ ∇ ⁡ u h - Π h Z ⁢ Z ⁢ ∇ ⁡ u h ∥ L 2 ⁢ ( Ω ) ∥ ∇ ⁡ ( u - u h ) ∥ L 2 ⁢ ( Ω ) .

Let Ω = ( 0 , 1 ) 2 and consider problem (2.1). Results with various mesh sizes ( h 1 and h 2 being the mesh sizes in directions x 1 and x 2 ) and various values of μ are reported in Table 1 for the exact solution

(5.3) u ⁢ ( x , y ) = 4 ⁢ ( 1 - e - α ⁢ x - ( 1 - e - α ) ⁢ x ) ⁢ y ⁢ ( 1 - y )

as in [23], with α = 50 . Similarly, in Tables 2 and 3 results are reported for the exact solution

(5.4) u ⁢ ( x , y ) = tanh ⁡ ( x - 0.5 ϵ )

with ϵ = 0.1 and ϵ = 0.05 respectively. Note that u defined by (5.4) is not zero on the boundary, thus the theory does not apply as is anymore. Nevertheless, numerical results indicate that the error estimator is still sharp.

We observe that the error estimator is accurate for both the W 0 1 , 3 norm and the quasi-norm when μ is large. When μ is large and h is small enough the effectivity indices are around 9 which corresponds to the linear Laplace problem [20, 43]. When μ is small, only the quasi-norm should be considered. Indeed, when h is small enough, the corresponding effectivity indices remain between 11 and 18 uniformly with respect to μ and h. We can therefore conclude that the upper bound (4.9) seems to be sharp with p = 3 . We now discuss the higher order terms of the upper and lower bounds, ϵ 1 , ϵ 2 , ϵ 3 and ϵ 4 (see Theorem 1). Consider again the first numerical experiment when μ = 0 and u is given by (5.3) with α = 50 , let h 1 = 1 N x and h 2 = 10 ⁢ h 1 . In Figure 2 left we can observe the order of convergence of the error estimator ∑ K ∈ 𝒯 h η 2 , K = O ⁢ ( h 2 ) , the quasi-norm error e Q ⁢ N = O ⁢ ( h 2 ) and the W 0 1 , 3 error e 3 = O ⁢ ( h 3 ) . Moreover, in Figure 2 right, we observe ϵ 1 , ϵ 2 , ϵ 4 = O ⁢ ( h 3 ) and ϵ 3 = O ⁢ ( h 5 ) , which are higher order terms with respect to e Q ⁢ N as discussed in Remark 3.

$Figure 2 Left: error estimator, quasi-norm error, W 0 1 , 3 {W^{1,3}_{0}} norm error with corresponding slopes, when u is given by (5.3), μ = 0 {\mu=0} and α = 50 {\alpha=50} . Right: Higher order terms ϵ 1 {\epsilon_{1}} , ϵ 2 {\epsilon_{2}} , ϵ 3 {\epsilon_{3}} , ϵ 4 {\epsilon_{4}} with corresponding slopes, when u is given by (5.3), μ = 0 {\mu=0} and α = 50 {\alpha=50} .$

Figure 2

Left: error estimator, quasi-norm error, W 0 1 , 3 norm error with corresponding slopes, when u is given by (5.3), μ = 0 and α = 50 . Right: Higher order terms ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 with corresponding slopes, when u is given by (5.3), μ = 0 and α = 50 .

Table 1

True errors and effectivity indices for various non-adapted meshes and various choices of μ when solving problem (2.1) with f such that the exact solution is given by (5.3), with α = 50 .

	h 1	h 2	ei N	ei Q ⁢ N	e Q ⁢ N	e 3	e 2	e ⁢ i Z ⁢ Z
μ = 0	0.025	0.025	8.43	2.71	81.50	26.21	0	0.59
	0.0125	0.0125	20.08	6.14	15.32	3.11	0	0.82
	0.00625	0.00625	85.17	9.66	3.12	3.53e - 01	0	0.93
	0.003125	0.003125	208.47	11.99	7.11e - 01	4.07e - 02	0	0.98
	0.01	0.1	55.12	10.07	7.75	1.40	0	0.96
	0.005	0.05	111.62	12.10	1.86	2.02e - 01	0	0.98
	0.0025	0.025	235.82	13.89	4.60e - 01	2.70e - 02	0	0.99
	0.00125	0.0125	510.447	15.33	1.02e - 01	3.05e - 03	0	0.99
μ = 1	0.025	0.025	8.13	2.72	82.71	26.06	1.64	0.59
	0.0125	0.0125	27.28	6.13	15.70	3.12	4.02e - 01	0.82
	0.00625	0.00625	68.39	9.57	3.21	3.54e - 01	9.60e - 02	0.93
	0.003125	0.003125	134.81	11.84	7.34e - 01	4.09e - 02	2.36e - 02	0.98
	0.01	0.1	44.00	9.94	8.17	1.40	4.42e - 01	0.97
	0.005	0.05	73.06	11.83	1.98	2.01e - 01	1.20e - 01	0.98
	0.0025	0.025	111.91	13.48	4.92e - 01	2.70e - 02	3.22e - 02	0.99
	0.00125	0.0125	148.42	14.78	1.10e - 01	3.07e - 03	7.91e - 03	0.99
μ = 100	0.025	0.025	4.16	3.23	222.81	23.33	149.50	0.63
	0.0125	0.0125	7.22	5.65	53.55	3.00	38.89	0.84
	0.00625	0.00625	9.40	7.36	12.54	3.51e - 01	9.47	0.94
	0.003125	0.003125	10.69	8.36	3.05	4.09e - 02	2.35	0.98
	0.01	0.1	8.66	7.60	51.04	1.37	43.42	0.97
	0.005	0.05	9.08	7.99	13.72	1.99e - 01	11.87	0.98
	0.0025	0.025	9.45	8.34	3.67	2.69e - 02	3.21	0.99
	0.00125	0.0125	9.66	8.58	8.92e - 01	3.06e - 03	7.90e - 01	0.99

Table 2

True errors and effectivity indices for various non-adapted meshes and various choices of μ when solving problem (2.1) with f such that the exact solution is given by (5.4), with ϵ = 0.1 .

	h 1	h 2	ei N	ei Q ⁢ N	e Q ⁢ N	e 3	e 2	e ⁢ i Z ⁢ Z
μ = 0	0.1	0.1	36.63	8.08	8.13	1.79	0	1.15
	0.05	0.05	85.12	11.59	1.30	1.77e - 01	0	1.04
	0.025	0.025	159.87	13.17	2.73e - 01	2.25e - 02	0	1.02
	0.0125	0.0125	309.63	13.82	6.11e - 02	2.73e - 03	0	1.00
	0.00625	0.00625	643.61	14.22	1.51e - 02	3.33e - 04	0	1.00
	0.003125	0.003125	1280.79	14.34	3.74e - 03	4.19e - 05	0	1.00
	0.01	0.1	378.17	15.30	3.61e - 02	1.46e - 03	0	1.00
	0.005	0.05	772.78	15.81	8.26e - 03	1.69e - 04	0	0.99
	0.0025	0.025	1547.62	15.57	1.92e - 03	1.94e - 05	0	1.00
	0.00125	0.0125	3098.95	15.62	4.79e.04	2.41e - 06	0	1.00
	0.002	0.2	2574.62	18.56	1.25e - 03	9.02e - 06	0	1.00
	0.001	0.1	4704.70	17.58	3.16e.04	1.18e - 06	0	0.99
	0.0005	0.05	8910.96	17.36	7.57e - 05	1.47e - 07	0	0.99
μ = 1	0.1	0.1	27.07	8.15	8.76	1.70	9.37e - 01	1.16
	0.05	0.05	45.73	11.27	1.47	1.73e - 01	1.88e - 01	1.06
	0.025	0.025	59.82	12.46	3.15e - 01	2.24e - 02	4.33e - 02	1.02
	0.0125	0.0125	71.31	12.95	7.12e - 02	2.73e - 03	1.02e - 02	1.01
	0.00625	0.00625	80.49	13.29	1.76e - 02	3.33e - 04	2.57e - 03	1.00
	0.003125	0.003125	85.29	13.39	4.39e - 03	4.10e - 05	6.47e - 04	1.00
	0.01	0.1	79.92	14.25	4.20e - 02	1.46e - 03	6.04e - 03	1.00
	0.005	0.05	88.81	14.67	9.68e - 02	1.69e - 04	1.43e - 03	1.00
	0.0025	0.025	91.69	14.42	2.26e - 03	1.94e - 05	3.36e - 04	1.00
	0.00125	0.0125	95.00	14.46	5.61e - 04	2.41e - 06	8.30e - 05	1.00
	0.002	0.2	111.81	17.23	1.47e - 03	9.03e - 06	2.17e - 04	1.00
	0.001	0.1	108.85	16.32	3.70e - 04	1.18e - 06	5.44e - 05	0.99
	0.0005	0.05	107.82	16.09	8.88e - 05	1.47e - 07	1.31e - 05	0.99
μ = 100	0.1	0.1	8.28	7.80	88.35	1.36	81.91	1.17
	0.05	0.05	8.82	8.33	18.90	1.63e - 01	17.70	1.08
	0.025	0.025	8.66	8.19	4.56	2.22e - 02	4.29	1.03
	0.0125	0.0125	8.55	8.09	1.08	2.73e - 03	1.02	1.01
	0.00625	0.00625	8.65	8.19	2.73e - 01	3.34e - 04	2.58e - 01	1.00
	0.003125	0.003125	8.64	8.17	6.83e - 02	4.21e - 05	6.46e - 02	1.00
	0.01	0.1	8.88	8.40	6.37e - 01	1.46e - 03	6.00e - 01	1.01
	0.005	0.05	8.92	8.44	1.51e - 01	1.79e - 04	1.42e - 01	1.00
	0.0025	0.025	8.69	8.22	3.55e - 02	1.94e - 05	3.36e - 02	1.00
	0.00125	0.0125	8.68	8.21	8.78e - 03	2.42e - 06	8,30e - 03	1.00
	0.001	0.1	10.07	9.52	5.74e - 03	1.17e.06	5.42e - 03	0.99
	0.0005	0.05	9.75	9.22	1.38e - 03	1.46e - 07	1.31e - 03	0.99

Table 3

True errors and effectivity indices for various non-adapted meshes and various choices of μ when solving problem (2.1) with f such that the exact solution is given by (5.4), with ϵ = 0.05 .

	h 1	h 2	ei N	ei Q ⁢ N	e Q ⁢ N	e L 3	e L 2	e ⁢ i Z ⁢ Z
μ = 0	0.05	0.05	14.42	4.66	53.93	17.41	0	0.70
	0.025	0.025	74.61	11.67	4.93	7.70e - 01	0	1.06
	0.0125	0.0125	153.99	13.08	1.05	8.89e - 02	0	1.00
	0.00625	0.00625	324.70	13.90	2.50e - 01	1.07e - 02	0	1.01
	0.003125	0.003125	650.11	14.17	6.12e - 02	1.33e - 03	0	1.00
	0.01	0.1	198.78	15.46	6.62e - 01	5.15e - 02	0	0.90
	0.005	0.05	384.54	14.91	1.38e - 01	5.36e - 03	0	0.89
	0.0025	0.025	766.47	15.20	3.13e - 02	6.22e - 04	0	0.99
	0.00125	0.0125	1544.29	15.53	7.71e - 03	7.76e - 05	0	1.00
	0.002	0.2	1184.92	17.28	2.01e - 02	2.93e - 04	0	1.00
	0.001	0.1	2189.84	17.27	5.26e - 03	4.14e - 05	0	1.00
	0.0005	0.05	4364.82	17.12	2.21e - 03	4.77e - 06	0	1.00
μ = 1	0.05	0.05	14.14	5.03	51.76	14.25	4.19	0.76
	0.025	0.025	53.56	11.46	5.25	7.64e - 01	3.58e - 01	1.08
	0.0125	0.0125	83.35	12.73	1.13	8.85e - 02	8.33e - 02	1.02
	0.00625	0.00625	115.70	13.44	2.70e - 01	1.07e - 02	2.07e - 02	1.01
	0.003125	0.003125	138.78	13.68	6.64e - 02	1.33e - 03	5.21e - 03	1.00
	0.01	0.1	97.92	14.94	7.01e - 01	4.96e - 02	5.74e - 02	0.95
	0.005	0.05	121.83	14.53	1.48e - 01	5.23e - 03	1.24e - 02	0.95
	0.0025	0.025	150.69	14.62	3.40e - 02	6,22e - 04	2.68e - 03	1.00
	0.00125	0.0125	167.63	14.92	8.39e - 03	7.76e - 05	6.68e - 04	1.00
	0.002	0.2	179.26	16.61	2.18e - 02	2.94e - 04	1.73e - 03	1.00
	0.001	0.1	191.54	16.58	5.71e - 03	4.14e - 05	4.53e - 05	1.00
	0.0005	0.05	196.31	16.44	1.32e - 03	4.77e - 06	1.06e - 05	1.00
μ = 100	0.05	0.05	8.22	7.36	185.30	6.02	159.75	1.16
	0.025	0.025	9.29	8.34	38.88	7.36e - 01	34.19	1.09
	0.0125	0.0125	9.35	8.39	9.16	8.78e - 02	8.14	1.03
	0.00625	0.00625	9.47	8.50	2.31	1.07e - 02	2.06	1.01
	0.003125	0.003125	9.47	8.50	5.80e - 01	1.34e - 03	5.19e - 01	1.00
	0.01	0.1	9.83	8.83	5.49	4.72e - 02	4.88	1.02
	0.005	0.05	9.76	8.77	1.24	5.16e - 03	1.11	1.00
	0.0025	0.025	9.51	8.53	2.98e - 01	6.22e - 04	2.67e - 01	1.00
	0.00125	0.0125	9.58	8.60	7.45e - 02	7.77e - 05	6.68e - 02	1.00
	0.002	0.2	10.91	9.79	1.93e - 01	2.93e - 04	1.73e - 01	1.00
	0.001	0.1	10.72	9.61	5.05e - 02	4.12e - 06	4.52e - 02	1.00
	0.0005	0.05	10.58	9.49	1.18e - 02	4.75e - 05	1.09e - 01	0.99

In the next section, we introduce an adaptive algorithm aiming to control the relative error in the quasi norm ∥ ⋅ ∥ ( u ) considering error estimator presented in Theorem 1 when p = 3 .

6 Adaptive Algorithm and Numerical Experiments with Adapted Meshes

The adaptive algorithm is similar to the one presented in [19, 43, 41, 20]. The goal of the adaptive algorithm is to build a sequence of meshes with possibley large aspect ratio, such that the relative estimated error is closed to a prescribed tolerance TOL, i.e.

(6.1) 0.75 ⁢ TOL ≤ ( ∑ K ∈ 𝒯 h η 2 , K ∥ u h ∥ ( u ) 3 ) 1 2 ≤ 1.25 ⁢ TOL .

In the term

∥ u h ∥ ( u ) 3 = ∫ Ω | ∇ ⁡ u h | 2 ⁢ ( μ + | ∇ ⁡ u | + | ∇ ⁡ u h | ) ,

∇ ⁡ u is in practice replaced by its approximate L 2 ⁢ ( Ω ) projection onto V h , as discussed in Remark 4. Condition (6.1) holds whenever, for all K ∈ 𝒯 h ,

(6.2) 1 N K ⁢ 0.75 2 ⁢ TOL 2 ⁢ ∥ u h ∥ ( u ) 3 ≤ η 2 , K ≤ 1 N K ⁢ 1.25 2 ⁢ TOL 2 ⁢ ∥ u h ∥ ( u ) 3 ,

where N K is the total number of elements of 𝒯 h .

Assumption (4.10) suggests that the error estimator should be equidistributed in all stretching directions. We recall that λ i , K is the length of the stretching direction 𝐫 i , K of the triangle K and ω 2 , K ⁢ ( v ) and G K ⁢ ( v ) are defined in Remark 1. We define for all v ∈ W 0 1 , 2 ⁢ ( Ω ) and for i = 1 , … , d ,

(6.3) ( ω 2 , K , i ⁢ ( v ) ) 2 = λ i , K 2 ⁢ 𝐫 i , K T ⁢ G K ⁢ ( v ) ⁢ 𝐫 i , K

and the local error estimator in direction x i

(6.4) η 2 , K , i = ( ∥ ∇ ⋅ ( ( μ + | ∇ ⁡ u h | ) ⁢ ∇ ⁡ u h ) + Π K ⁢ f ∥ L 2 ⁢ ( K ) + 1 2 ⁢ ∑ l = 1 d + 1 ( | ∂ ⁡ K l | ∏ j = 1 d λ j , K ) 1 2 ⁢ ∥ [ ( μ + | ∇ ⁡ u h | ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K l ) ) ⁢ ω 2 , K , i ⁢ ( u - u h ) ,

so that

ω 2 , K ⁢ ( v ) = ( ∑ i = 1 d ( ω 2 , K , i ⁢ ( v ) ) 2 ) 1 2

and

η 2 , K = ( ∑ i = 1 d η 2 , K , i 2 ) 1 2 .

Thus in order to satisfies (6.2), we require that for all K ∈ 𝒯 h and for i = 1 , … , d ,

(6.5) 1 d ⁢ N K 2 ⁢ 0.75 4 ⁢ TOL 4 ⁢ ∥ u h ∥ ( u ) 6 ≤ η 2 , K , i 2 ≤ 1 d ⁢ N K 2 ⁢ 1.25 4 ⁢ TOL 4 ⁢ ∥ u h ∥ ( u ) 6 .

In this way we will equidistribute the local error estimator in all directions 𝐫 1 , K , … , 𝐫 d , K . The adaptive strategy consists in aligning each triangle K (i.e. its directions of anisotropy 𝐫 1 , K , … , 𝐫 d , K ) with the eigenvectors of the matrix G K and to define a new mesh size based on (6.5), details can be found in [19, 43, 41, 20] (we use the mesh generator BL2D [30] in two dimensions and the 3D Precise Mesh software [53] in three dimensions).

Let Ω = ( 0 , 1 ) 2 and consider again problem (2.1) with exact solution (5.3) with α = 100 . We apply the adaptive algorithm with a starting mesh of size h = 0.1 . Starting with a tolerance TOL = 0.5 , 40 iterations of the adaptive algorithm are performed. The tolerance is then halved and the process is repeated. We report in Table 4 the results obtained for different values of μ. We observe that, for any choice of μ, the value of ei Q ⁢ N does not increase together with the average aspect ratio av ar and the maximum aspect ratio max ar . For μ = 100 , we report in Figure 3 the graph of convergence with respect to TOL = 0.5 × 2 - N of the errors (left) and the higher order terms ϵ 1 , ϵ 2 , ϵ 3 and ϵ 4 (right). Theoretical predictions of Remark 3 are again numerically verified.

Table 4

True errors, effectivity indices, number of vertices and aspect ratio for different values of tolerance TOL and μ, when solving problem (2.1) with f such that the exact solution is given by (5.3) with α = 100 .

	TOL	ei N	ei Q ⁢ N	e Q ⁢ N	ei Z ⁢ Z	N v	av ar	max ar
μ = 0	0.5	33.52	10.25	20.50	0.97	89	25	116
	0.25	70.84	12.97	3.48	0.98	249	32	272
	0.125	98.70	13.67	8.17e - 01	0.97	838	33	239
	0.0625	338.71	17.13	1.61e - 01	1.01	2975	36	421
	0.03125	756.82	18.82	3.63e - 02	1.01	11582	37	825
	0.015625	1539.65	18.33	9.43e - 03	1.00	44454	39	962
μ = 1	0.5	38.02	11.01	14.23	0.99	95	26	131
	0.25	61.19	13.65	3.38	0.99	254	29	305
	0.125	83.56	15.21	7.28e - 01	0.99	863	33	257
	0.0625	112.37	16.89	1.58e - 01	1.01	3182	36	607
	0.03125	135.67	17.64	3.93e - 02	1.00	11936	37	695
	0.015625	150.25	17.90	9.84e - 03	1.00	46032	41	670
μ = 100	0.5	9.94	8.37	40.89	0.93	171	18	202
	0.25	10.39	8.91	10.16	0.95	502	23	225
	0.125	10.65	9.18	2.42	0.95	1713	26	253
	0.0625	11.85	10.15	5.23e - 01	0.99	6649	26	436
	0.03125	12.44	10.63	1.26e - 02	1.00	25536	27	635
	0.015625	13.07	11.13	3.02e - 02	1.00	98424	29	1064

$Figure 3 Left: error estimator, quasi-norm error, W 0 1 , 3 {W^{1,3}_{0}} and W 0 1 , 2 {W^{1,2}_{0}} norm error with corresponding slope, when u is given by (5.3), μ = 100 {\mu=100} and α = 100 {\alpha=100} . Right: Higher order terms ϵ 1 {\epsilon_{1}} , ϵ 2 {\epsilon_{2}} , ϵ 3 {\epsilon_{3}} , ϵ 4 {\epsilon_{4}} with corresponding slopes, when u is given by (5.3), μ = 100 {\mu=100} and α = 100 {\alpha=100} .$

Figure 3

Left: error estimator, quasi-norm error, W 0 1 , 3 and W 0 1 , 2 norm error with corresponding slope, when u is given by (5.3), μ = 100 and α = 100 . Right: Higher order terms ϵ 1 , ϵ 2 , ϵ 3 , ϵ 4 with corresponding slopes, when u is given by (5.3), μ = 100 and α = 100 .

Consider again problem (2.1) with exact solution (5.4) with ϵ = 0.05 and Ω = ( 0 , 1 ) 2 . In Table 5 we can observe results obtained applying the adaptive algorithm with a starting mesh of size h = 0.1 . Starting with a tolerance TOL = 0.5 , 40 iterations of the adaptive algorithm are performed. The tolerance is then halved and the process is repeated. Results are again reported for different choices of μ.

The obtained results confirm that the effectivity index ei Q ⁢ N is independent from the aspect ratio, but slightly depends on μ. In fact, when μ varies between 0 and 100 and the average aspect ratio varies between 1 and 15000, then the effectivity index ei Q ⁢ N is between 11 and 24 for TOL small enough. In Figure 4 left, an example of obtained mesh is reported.

Table 5

True errors, effectivity indices, number of vertices and aspect ratio for different values of tolerance TOL and μ, when solving problem (2.1) with f given such that the exact solution is given by (5.4) with ϵ = 0.05 .

	TOL	ei N	ei Q ⁢ N	e Q ⁢ N	ei Z ⁢ Z	N v	av ar	max ar
μ = 0	0.5	58.15	11.68	3.34	0.91	52	20	61
	0.25	139.26	18.36	6.68e - 01	0.97	70	47	142
	0.125	225.39	17.53	1.64e - 01	0.96	150	88	325
	0.0625	412.32	20.95	4.00e - 02	0.99	246	189	865
	0.03125	558.11	20.95	9.93e - 03	0.97	492	351	1365
	0.015625	1282.96	22.44	2.34e - 03	0.98	863	845	4025
	0.0078125	2426.23	23.76	5.73e - 04	0.99	1703	1661	6348
	0.00390625	3384.70	22.91	1.45e - 04	0.98	3434	3286	16889
	0.00195312	7852.33	23.20	3.61e - 05	0.99	6821	6640	38352
	0.000976562	14905.10	23.23	9.12e - 06	0.99	13510	13250	61063
μ = 1	0.5	51.02	12.56	2.88	0.93	80	14	46
	0.25	72.59	15.53	6.46e - 01	0.96	76	51	171
	0.125	104.87	19.87	1.50-01	0.99	137	104	359
	0.0625	95.51	19.04	4.37e - 02	0.98	272	166	575
	0.03125	108.03	20.36	1.11e - 02	0.99	488	367	1444
	0.015625	107.27	20.01	2.63e - 03	0.99	967	788	3325
	0.0078125	104.77	20.58	6.76e - 04	0.97	1828	1644	7717
	0.00390625	120.28	21.25	1.66e - 04	1.00	3520	3315	15048
	0.00195312	116.67	20.94	4.10e - 05	0.99	7169	6587	40360
	0.000976562	119.16	21.27	1.03e - 05	1.00	14026	13530	66011
μ = 100	0.5	13.03	12.03	17.13	0.97	59	23	90
	0.25	10.08	9.42	6.18	0.91	105	45	131
	0.125	11.39	10.53	1.18	0.97	210	92	364
	0.0625	13.48	12.51	2.91e - 01	0.97	401	180	971
	0.03125	12.51	11.61	6.89e - 02	0.99	732	424	2046
	0.015625	13.04	12.07	1.79e - 02	1.00	1354	872	4121
	0.0078125	13.14	12.15	4.50e - 03	0.99	2622	1735	7951
	0.00390625	13.48	12.45	1.10e - 03	1.00	5080	3770	17808
	0.00195312	13.54	12.48	6.82e - 05	1.00	9976	15286	77916
	0.000976562	13.60	12.54	7.01e - 05	1.00	19640	14987	71481

In [29] it is proven, up to higher order terms, that edge residuals dominate for the Laplace problem and linear finite elements method on anisotropic meshes. We numerically compare now, error estimator (4.12) with the simplified error indicator, where only edge residuals are considered

(6.6) η ~ 2 , K = ( 1 2 ⁢ ∑ i = 1 d + 1 ( | ∂ ⁡ K i | ∏ j = 1 d λ j , K ) 1 2 ⁢ ∥ [ ( μ + | ∇ ⁡ u h | p - 2 ) ⁢ ∇ ⁡ u h ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K i ) ) ⁢ ω 2 , K ⁢ ( u - u h ) .

Let again Ω = ( 0 , 1 ) 2 and consider again problem (2.1) with exact solution (5.4) with ϵ = 0.05 and μ = 0 . As before, we apply the adaptive algorithm to a starting mesh of size h = 0.1 . The starting tolerance is TOL = 0.5 , 40 iterations of the adaptive algorithm are performed. The tolerance is then halved and the process is repeated. We compare results obtained in Table 5 when adapting with the local error estimator (4.12) and results obtained when considering (6.6). We define as in (5.1)–(5.2) the effectivity index of the modified error indicator

ei ~ Q ⁢ N = ∑ K ∈ 𝒯 h η ~ 2 , K e Q ⁢ N .

In Table 6 results obtained for both error indicators η 2 , K and η ~ 2 , K are reported. Both error indicators give similar results, in particular the effectivity index ei Q ⁢ N is around 23 while ei ~ Q ⁢ N is around 9. In Figure 4 two adapted meshes with their respective solution can be observed. Error indicators η 2 , K and η ~ 2 , K seem to be equivalent with different magnitudes.

Table 6

True errors, effectivity indices for different values of tolerance TOL, when solving problem (2.1) with f such that the exact solution is given by (5.4) with ϵ = 0.05 and μ = 0 . Left: Adapting with respect to the local error estimator (4.12). Right: Adapting with respect to the local error indicator (6.6).

TOL	ei Q ⁢ N	e Q ⁢ N	N v	av ar	ei ~ Q ⁢ N	e Q ⁢ N	N v	av ar
0.03125	20.95	9.93e - 03	492	350	8.24	2.47e - 02	306	241
0.015625	22.45	2.34e - 03	863	844	8.80	5.90e - 03	587	491
0.0078125	23.76	5.73e - 04	1703	1661	9.03	1.49e - 03	1133	1008
0.00390625	22.91	1.46e - 04	3425	3286	9.37	3.68e - 04	2220	2040

$Figure 4 Left: Adapted mesh obtained when TOL = 0.25 {\mathrm{TOL}=0.25} using the local error estimator (4.12), when u is given by (5.4) with ϵ = 0.05 {\epsilon=0.05} and μ = 0 {\mu=0} . Right: Adapted mesh obtained when TOL = 0.125 {\mathrm{TOL}=0.125} using the local error indicator (6.6), when u is given by (5.4) with ϵ = 0.05 {\epsilon=0.05} and μ = 0 {\mu=0} .$

Figure 4

Left: Adapted mesh obtained when TOL = 0.25 using the local error estimator (4.12), when u is given by (5.4) with ϵ = 0.05 and μ = 0 . Right: Adapted mesh obtained when TOL = 0.125 using the local error indicator (6.6), when u is given by (5.4) with ϵ = 0.05 and μ = 0 .

7 An Industrial Application: Aluminium Electrolysis

Our objective is to apply our adaptive strategy to a fluid flow simulation arising from aluminium electrolysis [46]. Let Ω ⊂ ℝ 3 be the domain occupied by two fluids, namely the liquid electrolyte and the liquid aluminium, which are immiscible and have different densities. Let 𝐮 : Ω → ℝ 3 and p : Ω → ℝ be the velocity and the pressure of the fluids in the cell. Let 𝐅 denote the gravity and electromagnetic forces. For a given interface between the two fluids, we are looking for 𝐮 , p such that

(7.1) { ρ ⁢ ( 𝐮 ⋅ ∇ ) ⁢ 𝐮 - ∇ ⋅ ( 2 ⁢ ( μ ⁢ ( | ϵ ⁢ ( 𝐮 ) | ) ) ⁢ ϵ ⁢ ( 𝐮 ) ) + ∇ ⁡ p = 𝐅 in ⁢ Ω , ∇ ⋅ 𝐮 = 0 in ⁢ Ω .

On ∂ ⁡ Ω no slip boundary conditions are enforced. Here

ϵ i ⁢ j ⁢ ( 𝐮 ) = 1 2 ⁢ ( ∂ ⁡ u i ∂ ⁡ x j + ∂ ⁡ u j ∂ ⁡ x i )

and ρ is the density, which is piecewise constant in the liquid electrolyte and the liquid aluminium. Smagorinsky’s model [46, 45] is considered

μ ⁢ ( | ϵ ⁢ ( 𝐮 ) | ) = μ L + ξ turb ⁢ ρ ⁢ l 2 ⁢ | ϵ ⁢ ( 𝐮 ) | ,

where 𝝁 L , ξ turb and l are positive numbers and

| ϵ ⁢ ( 𝐮 ) | = ( ∑ i , j 3 ( ϵ i ⁢ j ⁢ ( 𝐮 ) ) 2 ) 1 2 .

At the interface between the two fluids a non-penetration condition is imposed: 𝐮 ⋅ 𝐧 = 0 so as zero tangential forces [ ( 2 ⁢ μ ⁢ ϵ ⁢ ( 𝐮 ) - p ⁢ I d ) ⁢ 𝐧 ⋅ 𝐭 i ] = 0 for i = 1 , 2 . Our goal is to reduce the CPU time using an anisotropic adapted mesh. The numerical results obtained using this adapted mesh will be compared to those obtained with a reference industrial mesh. The classical mini element is used, leading to an approximation 𝐮 h and p h of (7.1) as in [46]. Starting from a given mesh, we apply the adaptive strategy of Section 6. We denote 𝒯 h a mesh of Ω in tetrahedron K. From discussion of the previous section, the following simplified error indicator with d = 3 is considered:

∑ K ∈ 𝒯 h 1 λ 3 , K 1 2 ⁢ ∥ [ ( μ L + ξ turb ⁢ ρ ⁢ l 2 ⁢ | ϵ ⁢ ( 𝐮 h ) | ) ⁢ ϵ ⁢ ( 𝐮 h ) ⋅ 𝐧 ] ∥ L 2 ⁢ ( ∂ ⁡ K ) ⁢ ω 2 , K ⁢ ( 𝐮 - 𝐮 h ) .

Here u h is the linear part of the finite element velocity, for 𝐯 ∈ ( H 0 1 ⁢ ( Ω ) ) 3 we have defined

( ω 2 , K ⁢ ( 𝐯 ) ) 2 = ∑ i = 1 3 ( ω 2 , K ⁢ ( v i ) ) 2 ,

where v i are the components of 𝐯 and ω 2 , K ⁢ ( v i ) is defined in (3.3). We choose a starting mesh of 10693 vertices, we run ten iterations of the adaptive algorithm with TOL = 1 , ten additional iterations with TOL = 0.5 and 20 more iterations with TOL = 0.375 . The final mesh has 24255 vertices and the CPU time of the whole iterative process is 4.08 hours.

Once the anisotropic adapted mesh is obtained, it is used to solve a free interface problem [46] to equilibrate the normal forces at the electrolyte-aluminium surface [ ( 2 ⁢ μ ⁢ ϵ ⁢ ( 𝐮 ) - p ⁢ I d ) ⁢ 𝐧 ⋅ 𝐧 ] = 0 . This requires to compute ten more Navier–Stokes approximated solutions, the CPU time needed being 0.96 hours. On the other hand, the reference (non-adapted) industrial mesh has 326099 vertices and the computational time to solve the free-surface problem is 5.78 hours. Therefore, once the adapted mesh is available, solving the interface Navier–Stokes problem requires five times less CPU than when using the standard mesh. It should be mentioned that the same adapted mesh can be used for several computations. In Figures 5, 6 and 7, results obtained with the reference industrial mesh and the adapted one are shown. In Figures 8 and 9, we report a plot over two different lines of the velocity magnitude. Similar accuracy between the adapted mesh and the reference one can be observed.

Figure 5

View of the computational domain with corresponding mesh.

(a)

Reference industrial mesh (326099 vertices).

(b)

Adapted mesh (24255 vertices).

Figure 6

Cut at x = - 6 of the fluid domain.

(a)

Reference industrial mesh (326099 vertices) and solution after solving the free-surface problem.

(b)

Adapted mesh (24255 vertices) and solution after solving the free-surface problem.

Figure 7

Cut at z = 0.3 of the fluid domain, view from below.

(a)

Reference industrial mesh (326099 vertices) and solution after solving the free-surface problem.

(b)

Adapted mesh (24255 vertices) and solution after solving the free-surface problem.

Figure 8

Plot along y at x = - 6.0 and z = 0.3 . The velocity magnitude, computed solving the free interface problem, for the adapted mesh and the reference one is reported.

Figure 9

Plot along z at x = - 6.0 and y = - 1.0 . The velocity magnitude, computed solving the free interface problem, for the adapted mesh and the reference one is reported.

8 Conclusion and Perspectives

An anisotropic error estimator for the approximation of a p-Laplacian problem is introduced. The constants involved in the upper and lower bounds are shown to be independent from the aspect ratio, the solution and μ, up to higher order terms. An adaptive algorithm based on the derived error estimator (4.12) is presented. Numerical experiments for p = 3 show that the proposed error estimator is accurate for both the W 0 1 , 3 ⁢ ( Ω ) norm and the quasi-norm, when μ is large and only for the quasi-norm when μ is small, as already shown for isotropic finite elements. Effectivity indices for the quasi-norm remain in [ 11 , 24 ] interval, when μ ranges from 0 to 100 and when adapted anisotropic meshes are used when TOL is small enough. Finally, a simplified error estimator, which is used to obtain an adapted mesh in the framework of aluminium electrolysis is presented. The results show a reduction of the computational time, while keeping a comparable accuracy with respect to a reference industrial mesh. An extension to higher order finite elements should be considered first when p = 2 , then when p ≥ 2 .

Funding statement: Paride Passelli is financed by Rio Tinto Aluminium LRF Research Center at Saint Jean de Maurienne (EPFL industrial grant).

Acknowledgements

Alexei Lozinski is acknowledged for advice. Maude Girardin, Samuel Dubuis and Jacques Rappaz are acknowledged for discussions. The referees are acknowledged for fruitful suggestions.

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Received: 2022-10-12

Revised: 2024-06-04

Accepted: 2024-06-06

Published Online: 2024-06-26

Published in Print: 2025-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cmam-2022-0205

Keywords for this article

A Posteriori Error Estimates; Adaptive Algorithm; Anisotropic Finite Elements; Nonlinear Equation; Aluminium Electrolysis

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