A C^0 interior penalty method for mth-Laplace equation
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In this paper, we propose a C^0 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 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, An interior penalty method for a sixth-order elliptic equation, IMA J. Numer. Anal., 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.
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