Analysis of a Fully Discrete Finite Element Method for Parabolic Interface Problems with Nonsmooth Initial Data (Journal of Computational Mathematics, 2022)
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,…,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 $\mathcal{O}(t_n^{-k}τ^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data.
Keywords: Parabolic interface problem; Finite element method; Backward difference formulae; Error estimate; Nonsmooth initial data.