Single Step Fully Discrete Schemes for the Homogeneous Equation

Abstract

In this chapter we consider single step fully discrete methods for the initial boundary value problem for the homogeneous heat equation, and show analogues of our previous stability and error estimates in the spatially semidiscrete case for both smooth and nonsmooth data. Our approach is to first study the discretization of an abstract parabolic equation in a Hilbert space setting with respect to time by using rational approximations of the exponential, which allows the standard Euler and Crank-Nicolson procedures as special cases, and then to apply the results obtained to the spatially discrete problem investigated in the preceding chapters. The analysis uses eigenfunction expansions related to the elliptic operator occurring in the parabolic equation, which we assume positive definite.