Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem

Ruma Rani Maity; Apala Majumdar; Neela Nataraj

doi:10.1515/cmam-2020-0185

Published by De Gruyter December 16, 2020

Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem

Ruma Rani Maity , Apala Majumdar and Neela Nataraj

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2020-0185

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Abstract

We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau–de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche’s and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient a posteriori analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.

Keywords: Non-linear Elliptic PDE; Non-homogeneous Dirichlet Boundary Data; Lower Regularity; Nitsche’s Method; dG Method; A Priori and A Posteriori Error Estimates; Adaptive Finite Element Methods

MSC 2010: 65N30; 35J65

Funding statement: Ruma Rani Maity gratefully acknowledges support from institute Ph.D. fellowship and Neela Nataraj gratefully acknowledges the support by DST SERB MATRICS grant MTR/2017/000 199. Apala Majumdar acknowledges support from the DST-UKIERI and British Council funded project on “Theoretical and Experimental Studies of Suspensions of Magnetic Nanoparticles, Their Applications and Generalizations” and support from IIT Bombay, and a Visiting Professorship from the University of Bath.

A Appendix

This section discusses the proofs of the local efficiency results in Lemmas 3.8–3.10. The local cut off functions play an important role to establish the local efficiency results. Consider the interior bubble function (see [1, 34]) b ^ T = 27 ⁢ λ ^ 1 ⁢ λ ^ 2 ⁢ λ ^ 3 supported on a reference triangle T ^ with the barycentric coordinate functions λ ^ 1 , λ ^ 2 , λ ^ 3 . For T ∈ 𝒯 , let ℱ T : T ^ → T be a continuous, affine and invertible transformation. Define the bubble function on the element T by b T = b ^ T ∘ ℱ T - 1 . Three edge bubble functions on the reference triangle T ^ are given by b ^ 1 = 4 ⁢ λ ^ 2 ⁢ λ ^ 3 , b ^ 2 = 4 ⁢ λ ^ 1 ⁢ λ ^ 3 and b ^ 3 = 4 ⁢ λ ^ 1 ⁢ λ ^ 2 . On the edge E of any triangle T ∈ 𝒯 , define the edge bubble function to be b E := b ^ E ∘ ℱ T - 1 , where b ^ E is the corresponding edge bubble function on T ^ . Here, b E is supported on the pair of triangles sharing the edge E .

Lemma A.1 ([1, 34]).

Let P ^ ⊂ H 1 ⁢ ( T ^ ) be a finite-dimensional subspace on the reference triangle T ^ and consider P = { v ^ ∘ F T - 1 : v ^ ∈ P ^ } to be the finite-dimensional space of functions defined on T. Then the following inverse estimates hold for all v ∈ P ;

(A.1) ∥ v ∥ L 2 ⁢ ( T ) 2 ≲ ∫ T b T ⁢ v 2 ⁢ dx ≲ ∥ v ∥ L 2 ⁢ ( T ) 2 , ∥ v ∥ L 2 ⁢ ( T ) ≲ ∥ b T ⁢ v ∥ L 2 ⁢ ( T ) + h T ⁢ ∥ ∇ ⁡ ( b T ⁢ v ) ∥ L 2 ⁢ ( T ) ≲ ∥ v ∥ L 2 ⁢ ( T ) .

Let E ⊂ ∂ ⁡ T be an edge and let b E be the corresponding edge bubble function supported on the patch of triangles ω E sharing the edge E. Let P ⁢ ( E ) be the finite-dimensional space of functions defined on E obtained by mapping P ^ ⁢ ( E ^ ) ⊂ H 1 ⁢ ( E ^ ) . Then for all v ∈ P ⁢ ( E ) ,

(A.2) ∥ v ∥ L 2 ⁢ ( E ) 2 ≲ ∫ E b E ⁢ v 2 ⁢ dx ≲ ∥ v ∥ L 2 ⁢ ( E ) 2 , h E - 1 2 ⁢ ∥ b E ⁢ v ∥ L 2 ⁢ ( ω E ) + h E 1 2 ⁢ ∥ ∇ ⁡ ( b E ⁢ v ) ∥ L 2 ⁢ ( ω E ) ≲ ∥ v ∥ L 2 ⁢ ( E ) ,

where the hidden constants in “ ≲ ” are independent of h T and h E .

Proof of Lemma 3.8.

(i) Let T ∈ 𝒯 be arbitrary and b T be the interior bubble function supported on the triangle T. Choose

𝝆 T := { ( - Δ ⁢ Φ h + 2 ⁢ ϵ - 2 ⁢ ( | Φ h | 2 - 1 ) ⁢ Φ h ) ⁢ b T in ⁢ T , 0 in ⁢ Ω ∖ T ,

utilize (A.1), (2.1) with Φ := 𝝆 T and apply an integration by parts for the first term (which is a zero term) on the right-hand side below to obtain

|| | 𝜼 T | || 0 , T 2 ≲ ∫ T ( - Δ ⁢ Φ h + 2 ⁢ ϵ - 2 ⁢ ( | Φ h | 2 - 1 ) ⁢ Φ h ) ⋅ 𝝆 T ⁢ dx

(A.3) = A T ⁢ ( Φ h - Ψ , 𝝆 T ) + ( B T ⁢ ( Φ h , Φ h , Φ h , 𝝆 T ) - B T ⁢ ( Ψ , Ψ , Ψ , 𝝆 T ) ) + C T ⁢ ( Φ h - Ψ , 𝝆 T ) .

Together with Hölder’s inequality, Lemma 3.6 and (A.1), the terms on the right-hand side of (A) are estimated as

(A.4) A T ⁢ ( Φ h - Ψ , 𝝆 T ) ≲ || | ∇ ⁡ ( Φ h - Ψ ) | || 0 , T ⁢ || | ∇ ⁡ 𝝆 T | || 0 , T ≲ || | Ψ - Φ h | || 1 , T ⁢ h T - 1 ⁢ || | 𝜼 T | || 0 , T .

(A.5) C T ⁢ ( Φ h - Ψ , 𝝆 T ) ≲ ϵ - 2 ⁢ || | Φ h - Ψ | || 0 , T ⁢ || | 𝝆 T | || 0 , T ≲ ϵ - 2 ⁢ || | Ψ - Φ h | || 0 , T ⁢ || | 𝜼 T | || 0 , T ,

B T ⁢ ( Φ h , Φ h , Φ h , 𝝆 T ) - B T ⁢ ( Ψ , Ψ , Ψ , 𝝆 T )

(A.6) ≲ ϵ - 2 ⁢ || | Ψ - Φ h | || 1 , T ⁢ ( || | Ψ - Φ h | || 1 , T ⁢ ( || | Φ h | || 1 , T + || | Ψ | || 1 , T ) + || | Ψ | || 1 , T 2 ) ⁢ h T - 1 ⁢ || | 𝜼 T | || 0 , T .

A combination of the above three displayed estimates in (A) plus Lemma A.1 establishes

(A.7) h T ⁢ || | 𝜼 T | || 0 , T ≲ || | Ψ - Φ h | || h , T ⁢ ( 1 + ϵ - 2 ⁢ ( 1 + || | Ψ | || 1 , T 2 + || | Ψ - Φ h | || h , T ⁢ ( || | Φ h | || 1 , T + || | Ψ | || 1 , T ) ) ) .

To find the estimate corresponding to 𝜼 E , consider the edge bubble function b E supported on the patch of triangles ω E sharing the edge E. Define

𝝆 E := { [ ∇ ⁡ Φ h ⁢ ν ] ⁢ b E in ⁢ ω E , 0 in ⁢ Ω ∖ ω E ,

and use (A.2), [ 𝝆 E ] = 0 for E ∈ ℰ h i and an integration by parts to obtain

|| | 𝜼 E | || 0 , E 2 ≲ ∫ E [ ∇ ⁡ Φ h ⁢ ν ] ⋅ 𝝆 E ⁢ ds = ∫ E [ ∇ ⁡ Φ h ⁢ ν ] ⋅ { 𝝆 E } ⁢ ds + ∫ E { ∇ ⁡ Φ h ⁢ ν } ⋅ [ 𝝆 E ] ⁢ ds

(A.8) = ∑ T ∈ ω E ∫ T ( Δ ⁢ Φ h ⋅ 𝝆 E + ∇ ⁡ Φ h ⋅ ∇ ⁡ 𝝆 E ) ⁢ dx .

Add and subtract ∑ T ∈ ω E ∫ T 2 ⁢ ϵ - 2 ⁢ ( | Φ h | 2 - 1 ) ⁢ Φ h ⋅ 𝝆 E ⁢ dx on the right-hand side of (A) to rewrite the expression with the help of 𝜼 T = Δ ⁢ Φ h - 2 ⁢ ϵ - 2 ⁢ ( | Φ h | 2 - 1 ) ⁢ Φ h (with a - Δ ⁢ Φ h = 0 added). The expression (2.1) with Φ = 𝝆 E , a re-grouping of terms and Hölder’s inequality lead to

| | | 𝜼 E | | | 0 , E 2 ≲ ( ∑ T ∈ ω E | | | 𝜼 T | | | 0 , T 2 ) 1 2 ( ∑ T ∈ ω E | | | 𝝆 E | | | 0 , T 2 ) 1 2 + ∑ T ∈ ω E ( A T ( Φ h - Ψ , 𝝆 E ) + C T ( Φ h - Ψ , 𝝆 E )

(A.9) + ( B T ( Φ h , Φ h , Φ h , 𝝆 E ) - B T ( Ψ , Ψ , Ψ , 𝝆 E ) ) ) .

A combination of Hölder’s inequality, Lemma 3.6 and (A.2) yields

(A.10) ∑ T ∈ ω E A T ⁢ ( Φ h - Ψ , 𝝆 E ) ≲ ∑ T ∈ ω E || | ∇ ⁡ ( Ψ - Φ h ) | || 0 , T ⁢ || | ∇ ⁡ 𝝆 E | || 0 , T ≲ h E - 1 2 ⁢ || | 𝜼 E | || 0 , E ⁢ || | ∇ ⁡ ( Ψ - Φ h ) | || 0 , ω E ,

(A.11) ∑ T ∈ ω E C T ⁢ ( Φ h - Ψ , 𝝆 E ) ≲ ϵ - 2 ⁢ ∑ T ∈ ω E || | Ψ - Φ h | || 0 , T ⁢ || | 𝝆 E | || 0 , T ≲ ϵ - 2 ⁢ h E 1 2 ⁢ || | 𝜼 E | || 0 , E ⁢ || | Ψ - Φ h | || 0 , ω E ,

and

∑ T ∈ ω E ( B T ⁢ ( Φ h , Φ h , Φ h , 𝝆 E ) - B T ⁢ ( Ψ , Ψ , Ψ , 𝝆 E ) )

(A.12) ≲ ϵ - 2 ⁢ h E - 1 2 ⁢ || | 𝜼 E | || 0 , E ⁢ ∑ T ∈ ω E || | Ψ - Φ h | || 1 , T ⁢ ( || | Ψ - Φ h | || 1 , T ⁢ ( || | Φ h | || 1 , T + || | Ψ | || 1 , T ) + || | Ψ | || 1 , T 2 ) .

The estimate of || | 𝜼 T | || 0 , T in (A.7) and (A.2) together with the above three displayed estimates in (A) lead to

(A.13) h E 1 2 ⁢ || | 𝜼 E | || 0 , E ≲ ∑ T ∈ ω E || | Ψ - Φ h | || h , T ⁢ ( 1 + ϵ - 2 ⁢ ( 1 + || | Ψ | || 1 , T 2 + || | Ψ - Φ h | || h , T ⁢ ( || | Φ h | || 1 + || | Ψ | || 1 , T ) ) ) .

A combination of (A.7) and A.13 completes the proof of ( i ) in Lemma 3.8.

(ii) For Φ h = I h ⁢ Ψ in (A.6), Lemma 3.6 (v) and (A.1) yield

B T ⁢ ( I h ⁢ Ψ , I h ⁢ Ψ , I h ⁢ Ψ , 𝝆 T ) - B T ⁢ ( Ψ , Ψ , Ψ , 𝝆 T ) ≲ ϵ - 2 ⁢ || | Ψ | || 1 + α , T 3 ⁢ ( h T 2 ⁢ α ⁢ || | ∇ ⁡ 𝝆 T | || 0 , T + h T 1 + α ⁢ || | 𝝆 T | || 0 , T )

(A.14) ≲ ϵ - 2 ⁢ || | Ψ | || 1 + α , T 3 ⁢ ( h T 2 ⁢ α + h T 2 + α ) ⁢ h T - 1 ⁢ || | 𝜼 T | || 0 , T .

Substitute (A.4), (A.5), (A) into (A) and utilize Lemma 3.3 to arrive at

h T ⁢ || | 𝜼 T | || 0 , T ≲ || | ∇ ⁡ ( I h ⁢ Ψ - Ψ ) | || 0 , T + ϵ - 2 ⁢ || | I h ⁢ Ψ - Ψ | || 0 , T + ϵ - 2 ⁢ h T 2 ⁢ α ⁢ || | Ψ | || 1 + α 3

(A.15) ≲ h T α ⁢ ( 1 + ϵ - 2 ⁢ h T α ⁢ ( 1 + || | Ψ | || 1 + α 2 ) ) ⁢ || | Ψ | || 1 + α .

A choice of Φ h = I h ⁢ Ψ in (A.12), Lemma 3.6 (v) and (A.2) yield

∑ T ∈ ω E ( B T ⁢ ( I h ⁢ Ψ , I h ⁢ Ψ , I h ⁢ Ψ , 𝝆 E ) - B T ⁢ ( Ψ , Ψ , Ψ , 𝝆 E ) ) ≲ ϵ - 2 ⁢ ∑ T ∈ ω E || | Ψ | || 1 + α , T 3 ⁢ ( h T 2 ⁢ α ⁢ || | ∇ ⁡ 𝝆 E | || 0 , T + h T 1 + α ⁢ || | 𝝆 E | || 0 , T )

(A.16) ≲ ϵ - 2 ⁢ h E - 1 2 ⁢ || | 𝜼 E | || 0 , E ⁢ ∑ T ∈ ω E || | Ψ | || 1 + α , T 3 ⁢ ( h T 2 ⁢ α + h E ⁢ h T 2 + α ) .

Substitute (A.10), (A.11), (A) into (A) and employ Lemma 3.3 to obtain

(A.17) h E 1 2 ⁢ || | 𝜼 E | || 0 , E ≲ ∑ T ∈ ω E h T α ⁢ ( 1 + ϵ - 2 ⁢ h T α ⁢ ( 1 + || | Ψ | || 1 + α 2 ) ) ⁢ || | Ψ | || 1 + α .

A combination of (A) and (A.17) concludes the proof of (ii) in Lemma 3.8. ∎

The proof of Lemma 3.9, respectively Lemma 3.10, follows analogous to the proof of Lemma 3.8 with the choice of

𝝆 T := { ( Δ ⁢ ( I h ⁢ 𝝃 ) + 2 ⁢ ϵ - 2 ⁢ ( | I h ⁢ Ψ | 2 ⁢ Θ h + 2 ⁢ ( I h ⁢ Ψ ⋅ Θ h ) ⁢ I h ⁢ Ψ - Θ h ) ) ⁢ b T in ⁢ T , 0 in ⁢ Ω ∖ T ,

𝝆 E := { [ ∇ ⁡ ( I h ⁢ 𝝃 ) ⁡ ν ] ⁢ b E in ⁢ ω E , 0 in ⁢ Ω ∖ ω E ,

respectively

𝝆 T := { ( G h + Δ ⁢ ( I h ⁢ 𝝌 ) - 2 ⁢ ϵ - 2 ⁢ ( | I h ⁢ Ψ | 2 ⁢ I h ⁢ 𝝌 + 2 ⁢ ( I h ⁢ Ψ ⋅ I h ⁢ 𝝌 ) ⁢ I h ⁢ Ψ - I h ⁢ 𝝌 ) ) ⁢ b T in ⁢ T , 0 in ⁢ Ω ∖ T ,

𝝆 E := { [ ∇ ⁡ ( I h ⁢ 𝝌 ) ⁡ ν ] ⁢ b E in ⁢ ω E , 0 in ⁢ Ω ∖ ω E .

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Received: 2020-11-16

Accepted: 2020-12-03

Published Online: 2020-12-16

Published in Print: 2021-01-01

Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem

Abstract

A Appendix

Lemma A.1 ([1, 34]).

Proof of Lemma 3.8.

References

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