A $C^0$ interior penalty method for $m$th-Laplace equation

Chen, Huangxin; Li, Jingzhi; Qiu, Weifeng

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 2081-2103

In this paper, we propose a C⁰ interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝ$$, where m and d can be any positive integers. The standard H¹-conforming piecewise r-th order polynomial space is used to approximate the exact solution u, where r can be any integer greater than or equal to m. Unlike the interior penalty method in Gudi and Neilan [IMA J. Numer. Anal. 31 (2011) 1734–1753], we avoid computing D$$ of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete H$$-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete H$$-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate.

Classification : 65N30, 65L12
Keywords: $$⁰ interior penalty, $$th-Laplace equation, stabilization, error estimates

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