Finite element methods for semilinear elliptic and parabolic interface problems

The purpose of this paper is to study the finite element methods for second-order semilinear elliptic and parabolic interface problems in two dimensional convex polygonal domains. Optimal order error estimate in the H1-norm is proved for the semilinear elliptic interface problem when the grid lines follow the actual interface. An extension to the semilinear parabolic interface problem is considered, and both semidiscrete and fully discrete schemes are discussed. The convergence of the semidiscrete solution to the exact solution is shown to be of order O(h) in the L2(0,T;H1(Ω))-norm. Further, a fully discrete scheme based on backward Euler method is analyzed and optimal energy-norm error estimate is established. The interface is assumed to be of arbitrary shape but is smooth.