On the numerical solution of a class of nonlinear parabolic problems with Volterra operators by a Rothe-Galerkin finite element method

Abstract

This paper deals with a class of nonlinear initial boundary value problems of parabolic type with Volterra operators in both the integro-differential equation and in the natural boundary condition. We give a constructive proof of the existence and uniqueness of an exact weak (variational) solution, under rather weak conditions on the data. To this end we extend the method of discretization in time (method of semi-discretization, Rothe method), developed e.g. by Rektorys in 1982 and Kacur in 1985, to this type of parabolic problem. Upon this method we superpose a Galerkin finite element method to derive a fully discrete approximation technique. We prove the convergence of the discretization schemes and obtain error estimates for the approximate solutions. In the important special case that the differential operator is potential, linear approximation schemes are analyzed. Finally, we present an example of a heat transfer problem leading to a parabolic problem of the type considered in the paper.