Finite element methods for interface problems

Abstract

The main objective of this thesis is to study the convergence of finite element solutions to the exact solutions of elliptic, parabolic and hyperbolic interface problems in fitted finite element method. The emphasis is on the theoretical aspects of such methods. Due to low global regularity of the true solution it is difficult to apply the classical finite element analysis to obtain optimal order of convergence for interface problems (cf. [5, 11]). In order to maintain the best possible convergence rate, a finite element discretization with straight interface triangles is considered and analyzed. More precisely, we have shown that the finite element solution converges to the exact solution at an optimal rate in L2 and H1 norms for elliptic problems. Then the results are extended for parabolic interface problems and optimal order error estimates in L2(L2) and L2(H1) norms are achieved. Further, optimal L1(H1) and L1(L2) norms error estimates for the parabolic interface problems have been derived under practical regularity assumption of the true solutions. Although various finite element method for elliptic and parabolic interface problems have been proposed and studied in the literature, but finite element treatment of similar hyperbolic problems is mostly missing. In this work, we are able to prove optimal order pointwise-in-time error estimates in L2 and H1 norms for the hyperbolic interface problem with semidiscrete scheme. Fully discrete scheme based on a symmetric difference approximation is also analyzed and optimal H1 norm error is obtained. Finally, numerical results for two dimensional test problems are presented to illustrate our theoretical findings.