In this talk, we present an energy minimization approach to constructing robust interpolation and restriction operators for multigrid for solving convection dominated problems. Numerical results for PDEs in 1D and 2D, in particular, entering flow and recirculating flow problems will be presented.
Multigrid for solving elliptic partial differential equations (PDEs) with smooth coefficients has been proven, both numerically and theoretically, to be a successful and powerful techniques. For PDEs whose coefficients are discontinuous or oscillatory, sophisticated interpolation techniques have been developed, for instance, algebraic multigrid, black box multigrid, etc. Recently, Wan, Chan and Smith proposed a robust construction of interpolation operators by minimizing the energy norm of the coarse grid basis functions subject to the constraint that the coarse grid basis functions preserve the null space of the underlying PDE. Fast multigrid convergence is resulted for several types of nonsmooth coefficient elliptic PDEs on Cartesian as well as general triangular meshes.
In this talk, we extend the energy minimization approach to non-elliptic PDEs; specifically, convection diffusion equations. Since the discretization matrix is nonsymmetric in general, the energy norm induced by the symmetric positive definite matrix in the case of elliptic PDEs no longer applicable. In other words, one cannot simply take the matrix and construct energy minimizing basis function using the induced matrix norm. Our idea is to construct two sets of basis functions, one by minimizing the energy norm of the elliptic operator, and the other set by minimizing the L2 norm, subject to appropriate constraints. In this way, the norms are well-defined so that norm minimization makes sense. More importantly, we can capture the elliptic and hyperbolic parts of the underlying PDE by these two set of basis functions. Then they are used in defining the interpolation and restriction operators for multigrid. Numerical results for solving the entering and recirculating flow problems are presented to demonstrate the effectiveness of the proposed approach.