Incomplete Iterative Solution of the Algebraic Systems at the Time Levels

In the fully discrete methods for the solution of parabolic equations which we have studied so far, a finite diminensional system of linear algebraic equations has to be solved at each time level of the time stepping procedure, and our analysis has always assumed that these systems are solved exactly. Because in applications these systems are of high dimension, direct methods are most often not appropriate, and iterative methods have to be used. Since the linear system to be solved at an individual time level is a discretization of an elliptic partial differential equation (with the step size occurring as a small parameter), methods normally used for elliptic problems are natural to apply here. In practice, only a moderate finite number of iterations can be carried out at each time level, and it thus becomes interesting to determine how many steps of the iterative algorithm are needed to guarantee that no loss occurs in the order of accuracy compared to the basic procedure in which the systems are solved exactly. For a successful iterative strategy it is also important to make a proper choice of the starting approximation at each time step.