# Finite element approximation of the $p(\cdot)$-Laplacian

Breit D, Diening L, Schwarzacher S (2015)
SIAM Journal on Numerical Analysis 53(1): 551-572.

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We study a~priori estimates for the Dirichlet problem of the $p(\cdot)$-Laplacian, $-\mathrm{div}(|\nabla v|^{p(\cdot)-2} \nabla v) = f.$ We show that the gradients of the finite element approximation with zero boundary data converges with rate $O(h^\alpha)$ if the exponent $p$ is $\alpha$-H\"{o}lder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $|\nabla v|^{\frac{p-2}{2}} \nabla v$.
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Breit D, Diening L, Schwarzacher S. Finite element approximation of the $p(\cdot)$-Laplacian. SIAM Journal on Numerical Analysis. 2015;53(1):551-572.
Breit, D., Diening, L., & Schwarzacher, S. (2015). Finite element approximation of the $p(\cdot)$-Laplacian. SIAM Journal on Numerical Analysis, 53(1), 551-572. doi:10.1137/130946046
Breit, D., Diening, L., and Schwarzacher, S. (2015). Finite element approximation of the $p(\cdot)$-Laplacian. SIAM Journal on Numerical Analysis 53, 551-572.
Breit, D., Diening, L., & Schwarzacher, S., 2015. Finite element approximation of the $p(\cdot)$-Laplacian. SIAM Journal on Numerical Analysis, 53(1), p 551-572.
D. Breit, L. Diening, and S. Schwarzacher, “Finite element approximation of the $p(\cdot)$-Laplacian”, SIAM Journal on Numerical Analysis, vol. 53, 2015, pp. 551-572.
Breit, D., Diening, L., Schwarzacher, S.: Finite element approximation of the $p(\cdot)$-Laplacian. SIAM Journal on Numerical Analysis. 53, 551-572 (2015).
Breit, Dominic, Diening, Lars, and Schwarzacher, Sebastian. “Finite element approximation of the $p(\cdot)$-Laplacian”. SIAM Journal on Numerical Analysis 53.1 (2015): 551-572.
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arXiv 1311.5121