H¹-Galerkin mixed finite element methods for parabolic partial integro-differential equations

Abstract

H¹-Galerkin mixed finite element methods are analysed for parabolic partial integro-differential equations which arise in mathematical models of reactive flows in porous media and of materials with memory effects. Depending on the physical quantities of interest, two methods are discussed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variables is also discussed and it is shown that the H¹-Galerkin mixed finite element approximations have the same rate of convergence as in the classical methods without requiring the LBB consistency condition.