Two-grid Method For Maxwell Eigenvalue Problem And Cahn-Hilliard Equation

Fast algorithms of Maxwell eigenvalue problems have been an important research topic in the field of science and engineering computing. In the first part of my thesis, we propose the two-grids for different variation problems, and construct efficient fast algorithms, furthermore prove the error analysis, and test the efficiency of the algorithms by several numerical experiments. In the second part of my thesis, we propose the two-grid method for the Cahn-Hilliard equation. The details are in the following:We propose the two-grid method for edge element discretization system. First we solve the eigenvalue problem on the coarse grid, and then solve an indefinite equation on the fine grid, at last get the eigenvalue by Rayleigh quotient. In the error analysis, we get the error equation through compare the two-grid solution with the finite element solution on the fine grid. Then we divide the error into two orthogonal parts, one part is in the eigen-space, and the other in the orthogonal complement of the eigen-space. We only need to estimate the part in the orthogonal complement of the eigen-space. Through analysis this part and use some lemmas, we got the error estimate. We use the Pminres to solve the equation on the fine grid. The preconditioner is a positive definite system which is solved by HX. The numerical examples are consistent with our theory and also demonstrate the effectiveness of our solver.In the next, we propose the two-grid method for the mixed element discretization system which is added Lagrange multiplier. The saddle type system has more size but does not construct non-physics solution. Similarly, we solve saddle eigenvalue problem on the coarse grid, and then solve an indefinite equation on the fine grid. Through the same technology, we get the error estimate. We use Pgmres to solve the indefinite equation on the fine grid. The preconditioner is a Maxwell boundary problem which does not have the lower order item. We solve the Maxwell boundary value problem through multigrid based on the DMG smoother. Also, we need not solve this problem exactly, only one Vcycle.At last, we propose the two-grid method for the face element discretiza-tion system. The solution is in the divergence free space exactly. We still use the Pgmres to solve the equation on the fine grid. The preconditioner is a stokes equation and mass matrix of edge element. We use the multigrid based on DGS to solve the stokes equation inexactly[70], only three Vcycle.The second part of my thesis is two-grid for Cahn-Hilliard equation. On the coarse grid, this is a mixed formulation problem need to solve, on the fine grid, there are two Poisson equations need to solve. We can use the multigrid method to solve the Poisson equations on the fine grid. The several numerical examples demonstrate the effectiveness of two-grid.