Multigrid methods for Maxwell\u27s equations

In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments

Multigrid methods for parameter dependent problems

Multigrid methods for parameter dependent problems are discussed. The contraction numbers of the algorithms are proved within a unifying framework to be bounded away from one, independent of the parameter and the mesth levels. Examples include the pure displacement and pure traction boundary value problems in planar linear elasticity, the Timoshenko beam problem, and the Reissner-Mindlin plate problem

Multigrid methods for saddle point problems: Darcy systems

We design and analyze multigrid methods for the saddle point problems resulting from Raviart–Thomas–Nédélec mixed finite element methods for the Darcy system in porous media flow. Uniform convergence of the W-cycle algorithm in a nonstandard energy norm is established. Extensions to general second order elliptic problems are also addressed