Block multigrid preconditioner for higher order finite volume method

Abstract

In this thesis I introduce block iterative methods for higher order finite volume methods(FVMs) to the second order elliptic partial differential equations. FVMs were basically developed only for the lowest order case. However, new higher order FVMs were introduced by Cai, Douglas and Park[5], recently. The main idea in their research is that the linear system derived by the hybridization with Lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for pressure variable. And the reduced system is obtained by an appropriate quadrature rule which satisfying a certain approximation order of accuracy for integration. Since the linear system comes from the higher order method, it is still not only large but also sparse. Besides, it is not diagonally dominant. The conjugate gradient(CG) method is a natural choice to solve the resulting system, but it seems slow, possibly due to the non-diagonal dominance of the system. On the other hand, the linear system has a special structure which is closely related with the higher order FVMs. This structure is a certain block structure corresponding to the order of approximation for pressure variable. For this reason, I propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order FVM and prove their convergence. Especially, with a proper ordering, each block subproblem can be solved by fast methods such as multigrid(MG) methods. The numerical experiments verify the propose of block iterative method to solve the resulting linear system after reordering, and also show that these block iterative methods are much faster than CG.