Analysis of the Fictitious Domain Method with L 2 - Penalty for Elliptic and Parabolic Problems

Analysis of the fictitious domain method with L 2-penalty for elliptic and parabolic problems Abstract. The fictitious domain method with L 2-penalty for elliptic and parabolic problems are considered, respectively. The regularity theorems and a priori estimates for L 2-penalty problems are given. We derive error estimates for penalization and finite element interpolation with P 1-element. Numerical experiments are performed to confirm the theoretical results. 1. Introduction. The purpose of this paper is to establish a mathematical study of the fictitious domain method for elliptic and parabolic problems. The fictitious domain method is well known to be based on a reformulation of the original problem in a larger spatial domain, called the fictitious domain, with a simple shape. Then, the fictitious domain can be discretized by a uniform mesh, independent of the original boundary. The advantage of this approach is that we can avoid the time-consuming construction of a boundary-fitted mesh. Furthermore, this approach will be useful to solve time-dependent moving-boundary problems. In our previous reports ([14, 15]), we developed a mathematical theory for the H 1-penalty fictitious domain method for elliptic and parabolic problems. The aim of this paper is to establish rigorous estimates of the errors induced by L 2 penalization and finite element interpolation. We examine the L 2 penalization by studying the H 2 regularity and estimates of the L 2-penalty problem, which is a different approach from [1], where the L 2 penalization for Navier-Stokes equation is considered without numerical analysis. Thanks to our regularity and estimate results, the finite element analysis becomes easy to treat. Our error estimates in the H 1 norm of L 2 penalization for elliptic and parabolic problems maintain the sharpness of those for Navier-Stokes problems in [1]; moreover, we show the error estimates of L 2 norm. The convergence of L 2 penalization for elliptic and parabolic problems has been proved in [7]; however, no error estimate has been found, neither the finite element analysis. Our analysis method presented here can also be applied to Stokes and Navier-Stokes problems with little difficulty. The rest of this paper is arranged as follow. In Sect. 2, we consider the elliptic problem. We first show the error estimates for L 2 penalization, then we turn to the finite element approximation. And Sect. 3 is devoted to the parabolic problem, as the same way to the elliptic case. The numerical experiments to validate …