Subspace correction methods for nonsymmetric parabolic problems

Abstract

A convergence proof for general subspace correction techniques applied to an abstract nonsymmetric parabolic equation is given. One of the main concerns is to give a unified convergence analysis for domain decomposition and multigrid methods for parabolic problems. The analysis is also valid for nonsymmetric problems. For second-order parabolic problems, the convection can dominate the diffusion. The algorithms can be applied to domain decomposition methods with or without the coarse mesh. For applications to multigrid methods, the coarsest mesh does not need to be very coarse. A relation between the coarse mesh size and the time step is needed to get a convergence rate independent of the mesh. The number of iterations at each time step for the algorithms is also estimated. Some numerical experiments are presented for domain decomposition methods with minimum overlap which support the theoretical predictions. The algorithms are able to capture the sharp traveling shocks for convection dominated problems.