## Almost optimal convergence of FEM-FDM for a linear parabolic interface problem

### Abstract

The solution of a second-order linear parabolic interface problem by the finite element method is discussed. Quasi-uniform triangular elements are used for the spatial discretization while the time discretization is based on a four-step implicit scheme. The integrals involved are evaluated by numerical quadrature, and it is assumed that the mesh cannot be fitted to the interface. With low regularity assumption on the solution across the interface, the stability of the method is established, and an almost optimal convergence rate of $O\left(k^4+h^2\left(1+\frac{1}{|\log h|}\right)\right)$ in the $L^2(\Omega)$-norm is obtained. In terms of matrices arising in the scheme, we show that the scheme preserves the maximum principle under certain conditions. Numerical experiments are presented to support the theoretical results.

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### Key words

finite element method, interface, almost optimal, parabolic equation, implicit scheme

### AMS subject classifications

65N06, 65N15, 65N30

### Links to the cited ETNA articles

 [11] Vol. 36 (2009-2010), pp. 149-167 István Faragó, János Karátson, and Sergey Korotov: Discrete maximum principles for the FEM solution of some nonlinear parabolic problems

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