Calculation of Error between the Exact Solution and Solution of Parabolic Equation (Heat Equation) by Krylov Approximation Methods

Abstract

Having good estimates in the computation of the approximation to expressions for the form f(A)v is very important in practical applications if we know at what stage the algorithm has to stop i.e avoid the principle of "luckybreak". In this paper we develop an a posteriori upper bound on the Krylov subspace approximation error. We seek the error committed between the exact solution and solution of parabolic equations(heat equation) by Krylov approximation methods.The idea of the method is to approximate the action of the evolution operator on a given state vector by means projection process onto a Krylov subspace. This estimate will allow us not only to theoretically study the behavior of the convergence of the Krylov method as well as its stability but also allow us to give the exact size of the Krylov space according to the fixed stop test and the precisions Wish to establish.