A C⁰ interior penalty method for mth-Laplace equation

In this paper, we propose a C⁰ interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in Double-struck capital 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.