. We are concerned with the design and analysis of a multigrid algorithm for H(div; ## --elliptic linear variational problems. The discretization is based on H(div; --conforming Raviart--Thomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as curls of finite element functions in so-called Nedelec spaces. It turns out that a combined nodal multilevel decomposition of both the Raviart--Thomas and Nedelec finite element spaces provides the foundation for a viable multigrid method. Its Gau--Seidel smoother involves an extra stage where solenoidal error components are tackled. By means of elaborate duality techniques we can show the asymptotic optimality in the case of uniform refinement. Numerical experiments confirm that the typical multigrid efficiency is actually achieved for model...

. The iterative substructuring methods, also known as Schur complement methods, form one of two important families of domain decomposition algorithms. They are based on a partitioning of a given region, on which the partial differential equation is defined, into non-overlapping substructures. The preconditioners of these conjugate gradient methods are then defined in terms of local problems defined on individual substructures and pairs of substructures, and, in addition, a global problem of low dimension. An iterative method of this kind is introduced for the lowest order Raviart-Thomas finite elements in three dimensions and it is shown that the condition number of the relevant operator is independent of the number of substructures and grows only as the square of the logarithm of the number of unknowns associated with an individual substructure. The theoretical bounds are confirmed by a series of numerical experiments. Key words. Raviart-Thomas finite elements, domain decomposition, ...

Abstract. Iterative substructuring methods, also known as Schur complement methods, form an important family of domain decomposition algorithms. They are preconditioned conjugate gradient methods where solvers on local subregions and a solver on a coarse mesh are used to construct the preconditioner. For conforming finite element approximations of H 1,itisknownthat the number of conjugate gradient steps required to reduce the residual norm by a fixed factor is independent of the number of substructures, and that it grows only as the logarithm of the dimension of the local problem associated with an individual substructure. In this paper, the same result is established for similar iterative methods for low-order Nédélec finite elements, which approximate H(curl; Ω) in two dimensions. Results of numerical experiments are also provided. 1.

Abstract. Time harmonic Maxwell equations in lossless media lead to a second order differential equation for electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec spaces can be employed to get approximate solutions numerically. The problem of preconditioning the indefinite matrix arising from this method is discussed here. Specifically, two overlapping Schwarz methods will be shown to yield uniform preconditioners. 1.

. A class of FETI methods for the edge element approximation of vector eld problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, a family of FETI methods for edge element approximations is proposed. It is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures. The estimate is also independent of the jumps of both of the coecients of the original problem. Numerical results validating the theoretical bounds are given. The method and its analysis can be easily generalized to Raviart{Thomas element approximations in two and three dimensions. Key words. Edge elements, Maxwell's equations, domain decomposition, FETI, preconditioners, heterogeneous coecients AMS subjec...

Abstract. This paper is concerned with the saddle-point problems arising from edge element discretizations of Maxwell’s equations in a general three dimensional nonconvex polyhedral domain. A new augmented technique is first introduced to transform the problems into equivalent augmented saddlepoint systems so that they can be solved by some existing preconditioned iterative methods. Then some substructuring preconditioners are proposed, with very simple coarse solvers, for the augmented saddle-point systems. With the preconditioners, the condition numbers of the preconditioned systems are nearly optimal; namely, they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size.

We consider domain decomposition preconditioners for the linear algebraic equations which result from finite element discretization of problems involving the bilinear form #(, + (curl defined on a polyhedral domain# Here (, denotes the inner product in 2(#7 3 and # is a positive number. We use Nedelec's curl-conforming finite elements to discretize the problem. Both additive and multiplicative overlapping Schwarz preconditioners are studied. Our results are uniform with respect to the mesh size and # under standard assumptions concerning the overlapping subdomains.

The URLs given were last checked and found valid in November 2001. To Alla, for all her help and care To Lady Mathematics, for all the fun and moments of enlightenment iv Acknowledgments First and foremost I want to thank my advisor and friend, Olof Widlund, for proposing the thesis subject, and for all his support and help in the last six years, four of them as his student. I also want to thank Yu Chen and Jonathan Goodman for their willingness to serve as readers on short notice. I thank all the faculty, staff, and students of the Courant Institute who contributed to create a warm and motivating atmosphere. I thank all the professors who transmitted some of their excitement about mathematics to me, I especially want to mention John Rinzel and Bud Mishra. I also want to thank all who fed my neverending interest in all things mathematical and who were as curious as me about mathematics, physics and all that. Thank you, Sávio and Franz. I have had the privilege of becoming very good friends with four fantastic people during my several

In this paper, we study some Schwarz methods of Neumann-Neumann type for some vector field problems, discretized with the lowest order Raviart-Thomas and Nedelec finite elements. We consider a hybrid Schwarz peconditioner consisting of a coarse component, which involves the solution of the original problem on a coarse mesh, and local ones, which involve the solution of Neumann problems on the elements of the coarse triangulation, also called substructures. We show that the condition number of the corresponding method is independent of the number of substructures and grows logarithmically with the number of unknowns associated with an individual substructure. It is also independent of the jumps of both the coecients of the original problem. The numerical results presented validate our theoretical bound.

Abstract. In this paper, we describe an approximation technique for divcurl systems based in (L 2 (Ω) 3) where Ω is a domain in R 3. We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate inf-sup condition. This results in a very weak formulation where the solution space is (L 2 (Ω)) 3 and the data reside in various negative norm spaces. Subsequently, we consider finite element approximations based on this weak formulation. The main approach of this paper involves the development of “stable pairs ” of discrete test and trial spaces. With this approach, we enlarge the test space so that the discrete inf-sup condition holds and we use a negative-norm least-squares formulation to reduce to a uniquely solvable linear system. This leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low Sobolev regularity (e.g., (H s (Ω)) 3 with 0 < s < 1/2). The resulting algebraic equations are symmetric, positive definite and well conditioned. A second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned. Some numerical results are also presented. 1.