Error analysis for a Crouzeix–Raviart approximation of the p-Dirichlet problem

Alex Kaltenbach

doi:10.1515/jnma-2022-0106

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Error analysis for a Crouzeix–Raviart approximation of the p-Dirichlet problem

Alex Kaltenbach

Published/Copyright: August 21, 2023

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From the journal Journal of Numerical Mathematics Volume 32 Issue 2

Abstract

In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a (p, δ)-structure for some p ∈ (1, ∞) and δ ⩾ 0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ ⩾ 0, medius error estimates, i.e., best-approximation results, and a primal–dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.

Keywords: p-Dirichlet problem; Crouzeix–Raviart element; a priori error analysis; medius error analysis; a posteriori error analysis

MSC 2010: 49M29; 65N15; 65N50

Acknowledgment

The author is grateful for the stimulating discussions with S. Bartels.

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A Appendix

In this appendix, we give a proof of the inequalities (3.1).

Proposition A.1

Let ψ : ℝ_⩾0 → ℝ_⩾0 be an N-function such that ψ ∈ Δ₂ ∩ ∇₂. Then, for every vh∈SD1,cr(Th), m ∈ {0, 1}, a ⩾ 0, and T ∈ 𝓣_h, we have that

∫−Tψa(hTm|∇hm(vh−Ihavvh)|)dx⩽cav∑S∈Sh(T)∖ΓN∫−Sψa(|[[vh]]S|)ds

where c_av > 0 depends only on Δ₂(ψ), Δ₂(ψ^*) > 0 and the chunkiness ω₀ > 0.

Proof

Owing to [8, Lem. A.2] together with [30, Lem. 12.1], there exists a constant c_av > 0, depending only on the chunkiness ω₀ > 0, such that

hTm∥∇hm(vh−Ihavvh)∥L∞(T;Rdm)⩽c¯av∑S∈Sh(T)∖ΓN∫−S|[[vh]]S|ds. (A.1)

Using in (A.1) the Δ₂-condition and convexity of ψ_a : ℝ_⩾0 → ℝ_⩾0, a ⩾ 0, in particular, Jensen’s inequality, and that sup_h>0 sup_{T∈𝓣_h} card(𝓢_h(T) ∖ Γ_N) ⩽ c_𝓣, where c_𝓣 > 0 depends only on the chunkiness ω₀ > 0, we find that

∫−Tψa(hTm|∇hm(vh−Ihavvh)|)dx⩽Δ2(ψa)⌈c¯avcT⌉ψa(1card(Sh(T)∖ΓN)∑S∈Sh(T)∖ΓN∫−S|[[vh]]S|ds)⩽Δ2(ψa)⌈c¯avcT⌉1card(Sh(T)∖ΓN)∑S∈Sh(T)∖ΓN∫−Sψa(|[[vh]]S|)ds.

Eventually, using that sup_a⩾0 Δ₂(ψ_a) < ∞, cf. [22, Lem. 22], we conclude the assertion.□

Corollary A.1

Let ψ : ℝ_⩾0 → ℝ_⩾0 be an N-function such that ψ ∈ Δ₂ ∩ ∇₂. Then, for every v_h ∈ SD1,cr (𝓣_h), m ∈ {0, 1}, a ⩾ 0, and T ∈ 𝓣_h, we have that

where c̃_av > 0 depends only on Δ₂(ψ), Δ₂(ψ^*) > 0 and the chunkiness ω₀ > 0.

Proof

Follows from Proposition A.1, if we exploit that [[vh]]S=[[∇hvh]]S⋅(idRd−xS) on S for all S ∈ 𝓢_h and v_h ∈ SD1,cr (𝓣_h) and the discrete trace inequality [30, Lem. 12.8].□

Received: 2022-10-22

Revised: 2023-02-25

Accepted: 2023-03-25

Published Online: 2023-08-21

Published in Print: 2024-06-25

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Articles in the same Issue

https://doi.org/10.1515/jnma-2022-0106

Keywords for this article

p-Dirichlet problem; Crouzeix–Raviart element; a priori error analysis; medius error analysis; a posteriori error analysis