Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data |
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Authors: | Kai Wang & Na Wang |
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Affiliation: | Department of Mathematics,Southern University of Science and Technology,Shenzhen 518005,China;Applied and Computational Mathematics Division,Beijing Computational Science Research Center,Beijing 100193,China |
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Abstract: | This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $mathcal{O}(t_n^{-k}tau^k+t_n^{-1}h^2|log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data. |
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Keywords: | Parabolic interface problem Finite element method Backward difference formulae Error estimate Nonsmooth initial data. |
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