Using the Cayley transform and finite elements to solve linear time dependent problems

Abstract

In this paper the approach of using the Cayley transformation to evaluate the solutionof a linear parabolic equation is coupled with standard finite-element methods. The potential advantage of this approach is that the solution at a fixed time is written in terms of a linear combination of solutions obtained by solving a time-independent problem where the coefficients in the linear combination are functions of time. In comparison with standard discretization methods this approach requires fewer “pseudotime steps”, i.e., fewer terms in the linear combination of solutions, to obtain the solution at a fixed time.