A SPACE-TIME MULTIGRID METHOD FOR PARABOLIC PARTIAL-DIFFERENTIAL EQUATIONS

We consider the solution of parabolic partial differential equations (PDEs). In standard time-stepping techniques multigrid can be used as an iterative solver for the elliptic equations arising at each discrete time step. By contrast, the method presented in this paper treats the whole of the space-time problem simultaneously. Thus the multigrid operations of smoothing and coarse-grid correction are defined on all of the space-time variables of a given grid level. The method is characterized by a coarsening strategy with prolongation and restriction operators which depend at each grid level on the degree of anisotropy of the discretization stencil. Numerical results for the one- and two-dimensional heat equations are presented and are shown to agree closely with predictions from Fourier mode analysis.