Abstract We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids. 1

We consider the scattering of monochromatic electromagnetic waves at a dielectric object with a non-smooth surface. This paper studies the discretization of this problem by means of coupling finite element methods (FEM) and boundary element methods (BEM). Straightforward symmetric coupling as in [R. Hiptmair, Coupling of finite elements and boundary elements in electromagnetic scattering, SIAM J. Num. Anal. 41 (2003), pp. 919-944] suffers from instabilities at wave numbers related to interior Dirichlet eigenvalues, the so-called spurious resonance phenomenon. A remedy is offered by adopting the idea underlying the widely used combined field integral equations (CFIE). These can be obtained from Robin-type trace operators, which ensure uniqueness of solutions of the associated interior boundary value problem for all frequencies. This implies uniqueness of solutions of the coupled problem. In the spirit of [R. Hiptmair and P. Meury, Stabilized FEM-BEM Coupling for Helmholtz Transmission Problems, SIAM J. Numer. Anal. 44 (2006), pp. 2106-2130], in order to get a coercive variational problem, we have to incorporate a regularizing operator into the modified traces. The discretization of the coupled variational problem is then based on curl-conforming finite elements inside the scatterer, divΓ-conforming boundary elements for the surface currents and curlΓ-conforming boundary elements for an auxiliary function on the boundary. Adapting a Helmholtz-type splitting to the discrete setting, permits us to show asymptotic optimality of the Galerkin-FEM-BEM solution. 1

We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices- the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss-Seidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residual-type a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasi-optimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.

Dedicated to Rolf Leis on the occasion of his 80th birthday For a bounded domain Ω ⊂ R3 with Lipschitz boundary Γ and some relatively open Lipschitz subset Γt 6 = ∅ of Γ, we prove the existence of some c> 0, such that c ||T ||L2(Ω,R3×3) ≤ ||symT ||L2(Ω,R3×3) + ||CurlT ||L2(Ω,R3×3) (0.1) holds for all tensor fields in H(Curl; Ω), i.e., for all square-integrable tensor fields T: Ω → R3×3 with square-integrable generalized rotation CurlT: Ω → R3×3, having vanishing restricted tangential trace on Γt. If Γt = ∅, (0.1) still holds at least for simply connected Ω and for all tensor fields T ∈ H(Curl; Ω) which are L2(Ω)-perpendicular to so(3), i.e., to all skew-symmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood row-wise. For compatible tensor fields T = ∇v, (0.1) reduces to a non-standard variant of the well known Korn’s first inequality in R3, namely c ||∇v||L2(Ω,R3×3) ≤ ||sym∇v||L2(Ω,R3×3) for all vector fields v ∈ H1(Ω,R3), for which ∇vn, n = 1,..., 3, are normal at Γt. On the other hand, identifying vector fields v ∈ H1(Ω,R3) (having the proper boundary conditions) with skew-symmetric tensor fields T, (0.1) turns to Poincaré’s inequality since 2c ||v||L2(Ω,R3) = c ||T ||L2(Ω,R3×3) ≤ ||CurlT ||L2(Ω,R3×3) ≤ 2 ||∇v||L2(Ω,R3). Therefore, (0.1) may be viewed as a natural common generalization of Korn’s first and Poincaré’s inequality. From another point of view, (0.1) states that one can omit compatibility of the tensor field T at the expense of measuring the deviation from compatibility through CurlT. Decisive tools for this unexpected estimate are ar

The standard Adaptive Edge Finite Element Method (AEFEM), using first/second family Nédélec edge elements with any order, for the three dimensionalH(curl)−elliptic problems with variable coefficients is shown to be convergent for the sum of the energy error and the scaled error estimator. The special treatment of the data oscillation and the interior node property are removed from the proof. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi-optimal.