Finite Volume Element Approximations of Nonlocal in Time One-Dimensional Flows in Porous Media

Various finite volume element schemes for parabolic integro-differential equations in 1-D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowest-order (linear and L-splines) finite volume elements, although higher-order volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H1-norms, suboptimal order in the L2-norm and super-convergent order in a discrete H1-norm.