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Kernel Preserving Multigrid Methods for Convection-Diffusion Equations

We propose a kernel preserving multigrid approach for solving convection-diffusion equations. The multigrid methods use Petrov-Galerkin coarse grid correction and linear interpolation. The restriction operator is constructed by preserving the kernel of the convection-diffusion operator. The construction considers constant and variable coefficient problems as well as cases where the convection term is not known explicitly. For constant convection-diffusion problems, we prove that the resulting Petrov-Galerkin coarse grid correction has small phase errors and the coarse grid matrix is almost an M-matrix. We demonstrate numerically the effectiveness of the multigrid methods by solving a constant convection problem, a recirculating flow problem, and a real application problem for pricing Asian options.