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In this paper, we propose a C0 interior penalty method for mth-Laplace equation on bounded Lipschitz polyhedral domain in ℝd, 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 Dm 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 Hm-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete Hm-norm. The error estimate under the low regularity assumption of the exact solution is also obtained. Numerical experiments validate our theoretical estimate. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for the interface-resolved finite element approximation in both, the physical and the stochastic dimension. In particular, a fast finite difference scheme is proposed to compute the variance of random solutions by using a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantify the method. [ABSTRACT FROM AUTHOR]