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

TL;DR

This paper introduces a new $C^{0}$ interior penalty method for solving the $m$th-Laplace equation that simplifies implementation and achieves optimal convergence, validated by numerical experiments.

Contribution

The paper develops a $C^{0}$ interior penalty method for the $m$th-Laplace equation that avoids complex derivative computations, making it easier to implement while maintaining optimal convergence.

Findings

Method achieves stability and optimal convergence in discrete $H^{m}$-norm.

Numerical experiments confirm theoretical error estimates.

Avoids computing high-order derivatives on elements and interfaces.

Abstract

In this paper, we propose a $C^{0}$ interior penalty method for $m$th-Laplace equation on bounded Lipschitz polyhedral domain in $\mathbb{R}^{d}$, where $m$ and $d$ can be any positive integers. The standard $H^{1}$-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 [T.~Gudi and M.~Neilan, {\em An interior penalty method for a sixth-order elliptic equation}, IMA J. Numer. Anal., \textbf{31(4)} (2011), pp. 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. Numerical experiments validate our theoretical estimate.