[Paper Review] Finite element approximation of the $p(\cdot)$-Laplacian

Dominic Breit, Lars Diening+1 more|arXiv (Cornell University)|Nov 20, 2013

Advanced Mathematical Modeling in Engineering34 references3 citations

TL;DR

This paper establishes a priori error estimates for the finite element approximation of the variable exponent $p(\cdot)$-Laplacian, showing that the gradient error in the quasi-norm converges at rate $O(h^{\alpha})$ when the exponent $p(\cdot)$ is $\alpha$-Hölder continuous. The analysis relies on weighted norm techniques and properties of variable exponent Lebesgue and Sobolev spaces.

ABSTRACT

We study a~priori estimates for the Dirichlet problem of the $p(\cdot)$-Laplacian, \[-\mathrm{div}(| abla v|^{p(\cdot)-2} abla v) = f. \] We show that the gradients of the finite element approximation with zero boundary data converges with rate $O(h^α)$ if the exponent $p$ is $α$-Hölder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $| abla v|^{\frac{p-2}{2}} abla v$.

Motivation & Objective

• To establish a priori error estimates for the finite element approximation of the $p(\cdot)$-Laplacian with variable exponent $p(\cdot)$.
• To quantify the convergence rate of the finite element gradient approximation in the quasi-norm, which is essential for nonlinear problems with variable growth.
• To bridge the gap in the literature by providing the first explicit convergence rate for finite element methods in variable exponent nonlinear PDEs.
• To extend classical finite element convergence theory to the setting of variable exponent spaces using weighted norms and Orlicz-type estimates.
• To analyze the dependence of the convergence rate on the regularity of the exponent $p(\cdot)$, specifically its Hölder continuity.

Proposed method

• Use of the quasi-norm $\| |\nabla v|^{\frac{p-2}{2}} \nabla v - |\nabla v_h|^{\frac{p-2}{2}} \nabla v_h \|_{L^2}$ to measure the error in the finite element approximation.
• Application of best approximation and interpolation estimates in generalized Lebesgue and Sobolev spaces with variable exponents.
• Employment of the key estimate for variable exponent spaces involving log-Hölder continuity of $p(\cdot)$ to control oscillations in the exponent.
• Utilization of the $\rho$-function framework for $N$-functions, with $\rho(t) = \int_0^t (\kappa + s)^{p(x)-2} s \, ds$, to model the $p(\cdot)$-Laplacian structure.
• Use of shift-invariant estimates and change-of-shift lemmas (e.g., Lemma A.5) to control differences in the nonlinearities arising from the variable exponent.
• Establishment of equivalence between the nonlinear gradient difference and the $\rho_{|\nabla v|}$-norm, enabling the use of weighted energy estimates.

Experimental results

Research questions

• RQ1 What is the convergence rate of the finite element approximation for the $p(\cdot)$-Laplacian when the exponent $p(\cdot)$ is only Hölder continuous?
• RQ2 How can the quasi-norm, which accounts for the nonlinear structure of the $p(\cdot)$-Laplacian, be used to derive optimal error estimates?
• RQ3 To what extent does the regularity of the exponent $p(\cdot)$, measured in Hölder continuity, influence the convergence rate of the finite element method?
• RQ4 Can the classical finite element convergence theory be extended to variable exponent problems using Orlicz space techniques and weighted norms?
• RQ5 What role does log-Hölder continuity of $p(\cdot)$ play in ensuring boundedness of the maximal operator and enabling the error analysis?

Key findings

• The finite element approximation of the $p(\cdot)$-Laplacian converges with rate $O(h^{\alpha})$ in the quasi-norm when the exponent $p(\cdot)$ is $\alpha$-Hölder continuous.
• The error is measured in the $L^2$-norm of $|\nabla v|^{\frac{p-2}{2}} \nabla v$, which is the natural energy variable for the $p(\cdot)$-Laplacian.
• The convergence rate is sharp and depends directly on the Hölder regularity of the exponent $p(\cdot)$, with higher regularity yielding faster convergence.
• The analysis relies on the log-Hölder continuity of $p(\cdot)$ to ensure boundedness of the maximal operator in $L^{p(\cdot)}$, which is essential for the key estimate.
• The results extend classical finite element convergence theory to variable exponent problems by introducing a weighted energy framework based on $N$-functions.
• The framework allows for the treatment of nonlinearities with variable growth via shift-invariant estimates and equivalence of norms in Orlicz-type spaces.