MIXED FINITE ELEMENT APPROXIMATIONS OF PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONSMOOTH INITIAL DATA

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 L²-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 L²-error estimates for problems with initial data in $H^{2}\cap H_{0}^{1}$ . In addition, for homogeneous equations we derive optimal L²-error estimates for initial data just in L². Here, we use an elementary energy technique and duality argument.