| In this Ph.D. dissertation we introduce two new discontinuous Galerkin (DC) interior penalty methods: the weakly over-penalized nonsymmetric interior penalty (WOP-NIP) method and the weakly over-penalized symmetric interior penalty (WOPSIP) method. We prove quasi-optimal error estimates in the energy norm and the L2 norm for both methods in the conforming P1 DC finite element space and provide numerical results which demonstrate these estimates. Next, we construct a simple block-diagonal preconditioner that reduces the resulting system of linear equations from O(h-4) to O(h-2), which allows us to develop an appropriate smoothing scheme for multigrid algorithms. For each method we are able to establish rigorous convergence results for a W-cycle multigrid algorithm and provide numerical results of the computed contraction numbers for the W-cycle, V-cycle, and F-cycle multigrid algorithms. Finally, we present further numerical results demonstrating the performance of the WOPNIP and WOPSIP methods on the L-shaped domain and on some nonconforming partitions of the unit square. |
