Flux conservation principle on construction of residual restriction operators for multigrid method

Multigrid method is one of the most efficient iterative algorithms to accelerate the convergence rate of both linear and non-linear algebraic equations. It has been indicated by many researches that the actual performance of multigrid method is influenced by various factors, among which the most important one is the implementation of residual restriction. For different discretized forms of the same partial differential governing equation, it is proposed in the present paper that the multigrid method could behave most effectively only in the case that the implementation of residual restriction satisfies the flux conservation principle of the finite volume method. Taking the correction scheme (CS scheme) of the multigrid method into consideration, when the flux conservation principle is well satisfied, residual restriction operators for different algebraic equations are derived. Corresponding numerical studies are made via four cases of pure diffusion and convective-diffusion problems. It is found that the acceleration of the convergence rate of the multigrid method is most desirable when the residual restriction satisfies the flux conservation principle.