Finite element methods for semilinear elliptic and parabolic interface problems

Authors:
Rajen K. Sinha;Bhupen Deka
Affiliations:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati -- 781 039, India;Department of Mathematical Sciences, Tezpur University, Tezpur -- 784028, Assam, India
Venue:
Applied Numerical Mathematics
Year:
2009

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Abstract

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 H^1-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 L^2(0,T;H^1(@W))-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.