Supraconvergence and supercloseness in Volterra equations

Integro-differential equations of Volterra type arise, naturally, in many applications such as for instance heat conduction in materials with memory, diffusion in polymers and diffusion in porous media. The aim of this paper is to study a finite difference discretization of the mentioned integro-differential equations. Second convergence order with respect to the H1 norm is established which means that the discretization proposed is supraconvergent in finite difference methods language. As the finite difference method can be seen as a piecewise linear finite element method combined with special quadrature formulas, our result establishes the supercloseness of the gradient in the finite element language. Numerical results illustrating the discussed theoretical results are included.