A C⁰ interior penalty method for mth-Laplace equation

¹ School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian 361005, P.R. China
² Department of Mathematics and National Center for Applied Mathematics Shenzhen and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, P.R. China
³ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, P.R. China

^* Corresponding author: weifeqiu@cityu.edu.hk

Received: 14 November 2021
Accepted: 8 August 2022

Abstract

In this paper, we propose a C⁰ interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝ^d, 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^m 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^m-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^m-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate.