Mixed Finite Element Approximations of Parabolic Integro-Differential Equations with Nonsmooth Initial Data

Abstract

We analyze the semidiscrete mixed finite element methods for parabolic integro-differential equations that arise in the modeling of nonlocal reactive flows in porous media. A priori $L^2$-error estimates for pressure and velocity are obtained with both smooth and nonsmooth initial data. More precisely, a mixed Ritz–Volterra projection, introduced earlier by Ewing et al. in [SIAM J. Numer. Anal., 40 (2002), pp. 1538–1560], is used to derive optimal $L^2$-error estimates for problems with initial data in $H^2\cap H_0^1$. In addition, for homogeneous equations we derive optimal $L^2$-error estimates for initial data just in $L^2$. Here, we use an elementary energy technique and duality argument.