Fourier analysis and truncation error estimates of multigrid methods and conservative Jacobians on hexagonal grids

Multigrid methods are fast solvers for elliptic partial differential equations (PDEs), which leads to lots of applications in numerical computations, such as the multigrid solver on geodesic grids. Multigrid methods use approximations on grids of different mesh sizes. Comparing the approximate truncation error on two consequent grids leads to a higher order approximation to the truncation error on the coarse grid, which is known as relative trunction error estimates. Hexagonal grids offer some advantages compared with rectangular grids, and hexagonal grids approximate to geodesic grids. This dissertation investigates the multigrid methods on hexagonal grids.;Quantitative insights for the multigrid solver on geodesic grids can be obtained by local Fourier analysis on hexagonal grid. Using oblique coordinates to express the grids and a dual basis for the Fourier modes, the analysis proceeds essentially the same as for rectangular grids. The framework for one- and two-grid analysis is given, and numerical results confirm the analysis.;An accurate formulation of the truncation error estimates for linear problems limited to non-staggered grids has been investigated. We extend this formulation to non-linear problems on staggered grids. The analysis is valid on both rectangular and hexagonal grids. We apply the method to the Cauchy-Riemann equations, Stokes equations and Navier-Stokes equations on rectangular grids.;The discretizations of the Jacobian operator on hexagonal grids are investigated. We prove that there is no compact (nearest-neighbor) conservative discretization of the Jacobian of order greater than two on the uniform rectangular grid or hexagonal grid. In addition, a fourth-order non-compact conservative discretization on the hexagonal grid is derived by extrapolation. Furthermore, conservative discretizations for boundary points on a hexagon domain are also studied. Numerical results from solving the nondivergent barotropic vorticity equation, which is expressed by using the Jacobian, verify the conservation properties of the discretizations. Multigrid method is applied when solving the Poisson equation for updating the streamfunction. With a flow through the boundary, we test the boundary formulations.