Error analysis of the diffuse domain finite element method for second order parabolic equations

In this paper, we analyze the diffuse domain finite element method (DDFE) to solve a class of second-order parabolic partial differential equations defined in general irregular domains. The proposed method first applies the diffuse domain method (DDM) with a phase-field function to extend the target parabolic equation to a similar problem defined over a larger rectangular domain that contains the original physical domain. The transformed equation is then discretized by using the finite element method with continuous piecewise multilinear basis functions in space and the BDF2 scheme in time to produce a fully discrete numerical scheme. Based on the weighted Sobolev spaces, we prove the convergence of the DDM solution to the original solution as the interface thickness parameter goes to zero, with the corresponding approximation errors under the $L^2$ and $H^1$ norms. Furthermore, the optimal error estimate for the fully discrete DDFE scheme is also obtained under the $H^1$ norm. Various numerical experiments are finally carried out to validate the theoretical results and demonstrate the performance of the proposed method.