An $hp$-Version Discontinuous Galerkin Method for Integro-Differential Equations of Parabolic Type
SIAM Journal on Numerical Analysis
Supraconvergence and supercloseness in Volterra equations
Applied Numerical Mathematics
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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.