Fast Numerical Solution of Time-Periodic Parabolic Problems by a Multigrid Method

Abstract

The discrete solutions of parabolic problems subject to the condition $y( \cdot ,T) = y( \cdot ,0)$ of time periodicity are solutions of large sparse systems. In this paper we propose a multigrid algorithm. It is a very fast iterative method. The algorithm can easily be generalized to nonlinear problems and to conditions of the type $y( \cdot ,0) = A(y( \cdot ,T))$ (A is a nonlinear mapping). The computational work for solving the periodic problem is of the same order as the work for solving an initial value problem ($y( \cdot ,0)$ given). Numerical results are reported for a linear and a nonlinear example.