Analysis of H1 penalized fictitious domain method for parabolic problems

Abstract

This work focuses on establishing optimal a priori error estimates and stability analysis for $H^1$ penalized fictitious domain method for parabolic problems defined over curved complex domains. We embed the given complicated domain Ω into a larger rectangular domain R and extend the governing equation to a rectangular domain R by employing penalty parameter ϵ in the fictitious part $R﹨\Omega$. Considering the inherent characteristic of the $H^1$ penalty, in the variational formulation, we impose the $H^1$ penalty only on the elliptic part and evince that the solution of the new penalized problem converges to the original solution. In order to obtain a numerical solution to the penalized problem, for spatial discretization, we utilize linear-finite elements on structured triangular mesh irrespective of the shape of the domain, and an Euler backward scheme is employed for the discretization of time space. Convergence analysis and discrete stability estimates are derived for semi and fully-discrete schemes. Moreover, numerical experiments are accomplished to validate the theoretical convergence rate and examine the computational efficiency of the proposed method.