Convergence of finite difference method for the generalized solutions of Sobolev equations

Abstract

In this paper, finite difference method is applied to approximate the generalized solutions of Sobolev equations. Using the Steklov mollifier and BrambleHilbert Lemma, a priori error estimates in discrete L² as well as in discrete H¹ norms are derived first for the semidiscrete methods. For the fully discrete schemes, both backward Euler and CrankNicolson methods are discussed and related error analyses are also presented.