A priori $$\varvec{L^{\infty }(L^2)}$$ error estimates for finite element approximations to parabolic integro-differential equations with discontinuous coefficients

Abstract

Finite element treatment for parabolic integro-differential equations with discontinuous coefficients are presented in this work. Due to low global regularity of the solutions, the error analysis technique of the standard finite element method is difficult to adopt for interface problems. In this paper, convergence of continuous time Galerkin method for the spatially discrete scheme and backward difference scheme in time direction are discussed in $$L^{\infty }(L^2)$$ norm for fitted finite element method with straight interface triangles. More precisely, optimal error estimates are derived in $$L^{\infty }(L^2)$$ norm when initial data $$u_0\in H^3\cap H_0^1(\Omega )$$ . The interface is assumed to be smooth for our purpose.