Finite element exterior calculus (FEEC) is an 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, and elliptic eigenvalue problems.
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 an efficient block diagonal preconditioner 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 $A$. 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 block
preconditioners, various numerical examples also are provided.