UC Irvine

Finite element exterior calculus (FEEC) is a framework to design and understand finite element discretizations for a wide variety of systems of partial differential equations. The applications are already made to the Hodge Laplacian, Maxwell’s equations, the equations of elasticity, elliptic eigenvalue problems and etc.. In this thesis, we propose fast solvers for several numerical schemes based on the discretization of this approach and present theoretical analysis. Specifically, in the first part, we propose efficient block diagonal and block triangular preconditioners for solving the discretized linear system of the vector Laplacian by mixed finite element methods. A variable V-cycle multigrid method with the standard point-wise Gauss-Seidel smoother is proved to be a good preconditioner for the Schur complement. The major benefit of our approach is that the point-wise Gauss-Seidel smoother is more algebraic and can be easily implemented as a ‘black-box’ smoother. The multigrid solver for the Schur complement will be further used to build preconditioners for the original saddle point systems. In the second part, we propose a discretization method for the Darcy-Stokes equations under the framework of FEEC. The discretization is shown to be uniform with respect to the perturbation parameter. A preconditioner for the discrete system is also proposed and shown to be efficient. In the last part, we focus on the stochastic Stokes equations. The stochastic saddle-point linear systems are obtained by using finite element discretization under the framework of FEEC in physical space and generalized polynomial chaos expansion in random space. We prove the existence and uniqueness of the solutions to the continuous problem and its corresponding stochastic Galerkin discretization. Optimal error estimates are also derived. We construct block-diagonal/triangular preconditioners for use with the generalized minimum residual method and the bi-conjugate gradient stabilized method. An optimal multigrid solver is applied to efficiently solve the diagonal blocks that correspond to deterministic discrete Stokes systems. To demonstrate the efficiency and robustness of the discretization methods and proposed preconditioners, various numerical examples also are provided.