Results 1 
6 of
6
Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids
 In Multiscale, Nonlinear and Adaptive Approximation
, 2009
"... Abstract We give an overview of multilevel methods, such as Vcycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasiuniform meshes and extend them to graded meshes and completely unstructured grids. We firs ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
Abstract We give an overview of multilevel methods, such as Vcycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasiuniform 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
Stabilized FEMBEM Coupling for Maxwell Transmission Problems
, 2007
"... We consider the scattering of monochromatic electromagnetic waves at a dielectric object with a nonsmooth 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. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We consider the scattering of monochromatic electromagnetic waves at a dielectric object with a nonsmooth 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. 919944] suffers from instabilities at wave numbers related to interior Dirichlet eigenvalues, the socalled 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 Robintype 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 FEMBEM Coupling for Helmholtz Transmission Problems, SIAM J. Numer. Anal. 44 (2006), pp. 21062130], 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 curlconforming finite elements inside the scatterer, divΓconforming boundary elements for the surface currents and curlΓconforming boundary elements for an auxiliary function on the boundary. Adapting a Helmholtztype splitting to the discrete setting, permits us to show asymptotic optimality of the GalerkinFEMBEM solution. 1
OPTIMAL MULTILEVEL METHODS FOR GRADED BISECTION GRIDS
"... 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 pe ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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 quasiuniform grids, for which the multilevel theory is wellestablished.
Uniform Convergence of . . . FOR THE TIMEHARMONIC MAXWELL EQUATION
, 2010
"... For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the timeharmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge ..."
Abstract
 Add to MetaCart
For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the timeharmonic 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 GaussSeidel 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 residualtype a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasioptimal 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.
Poincare ́ meets Korn via Maxwell: Extending Korn’s First Inequality to Incompatible
, 2014
"... 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 f ..."
Abstract
 Add to MetaCart
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 squareintegrable tensor fields T: Ω → R3×3 with squareintegrable 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 skewsymmetric constant tensors. Here, both operations, Curl and tangential trace, are to be understood rowwise. For compatible tensor fields T = ∇v, (0.1) reduces to a nonstandard variant of the well known Korn’s first inequality in R3, namely c ∇vL2(Ω,R3×3) ≤ sym∇vL2(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 skewsymmetric tensor fields T, (0.1) turns to Poincaré’s inequality since 2c vL2(Ω,R3) = c T L2(Ω,R3×3) ≤ CurlT L2(Ω,R3×3) ≤ 2 ∇vL2(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
Convergence of adaptive edge finite element methods for H(curl)−elliptic problems
"... 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 spec ..."
Abstract
 Add to MetaCart
(Show Context)
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 quasioptimal.