Finite volume element approximations of nonlocal reactive flows in porous media

In this article, we study finite volume element approximations for two-dimensional parabolic integro-differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher-order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H1- and L2-norm and are superconvergent in a discrete H1-norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L∞- and W1,∞-norms. These results are also new for finite volume element methods for elliptic and parabolic equations. Numer Methods Partial Differential Eq 16: 285-311, 2000.