HcurlAFEM.bib

Casc\'on, J. Manuel and Kreuzer, Christian and Nochetto, Ricardo H. and Siebert, Kunibert G.. Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method. Preprint 9, University of Augsburg, :, 2007.

Cascon.J;Kreuzer.C;Nochetto.R;Siebert.K2007

ABSTRACT: We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

Carstensen, C. and Hoppe, R.H.W.. Convergence analysis of an adaptive edge finite element method for the 2D eddy current equations. Journal of Numerical Mathematics, 13:19--32, 2005.

Carstensen.C;Hoppe.R2005b

ABSTRACT: For the 2D eddy currents equations, we design an adaptive edge finite element method (AEFEM) that guarantees an error reduction of the global discretization error in the H (curl)-norm and thus establishes convergence of the adaptive scheme. The error reduction property relies on a residual-type a posteriori error estimator and is proved for discretizations based on the lowest order edge elements of Nédélec's first family. The main ingredients of the proof are the reliability and the strict discrete local efficiency of the estimator as well as the Galerkin orthogonality of the edge element approximation.

Zhimin Chen and Long Wang and W. Zheng. An adaptive multilevel method for time-harmonic Maxwell equations with singularities. SIAM J. Sci. Comput., :, 2007.

Joachim Sch\"oberl. A Posteriori Error Estimates for Maxwell Equations. Preprint, :, 2007.

J. Schoeberl. Commuting Quasi-interpolation operators for mixed finite elements. Preprint2nd European Conference on Computational Mechanics, :854--855, 2001.

Schoeberl.J2001

ABSTRACT: In recent years, conforming finite elements for H(curl) and H(div) spaces have become one of main research topics in numerical analysis. The so called de Rham diagram [3, 4, 2] relates the exact sequence of continuous spaces H exp 1 - > H(curl) - > H(div) - > L exp 2 to their corresponding discrete counterparts. Up to now, only the local nodal interpolation operators, and global Fortin operators have been known to fulfill the commuting diagram property. In this talk, I will present new non-local, Clement-type operators that satisfy the commuting diagram property. The result, in particular, should help to generalize and simplify existing multigrid theories as well as a posteriori error estimates for Maxwell's equations.

R. Hiptmair and J. Xu. Nodal Auxiliary Space Preconditioning in {H(curl)} and {H(div)} spaces. , :, 2006.

Hiptmair.R;Xu.J2006

ABSTRACT: In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of $\Hcurl$- and $\Hdiv$-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by appealing to the abstract theory of auxiliary space preconditioning. The main tool are discrete analogues of so-called regular decomposition results in the function spaces H(curl) and H(div). Our preconditioner for H(curl) space is similar to an algorithm proposed in [{\sc R.~Beck}, {\em Algebraic multigrid by component splitting for edge elements on simplicial triangulations}, Techn. Report SC 99-40, ZIB, Berlin, Germany, 1999.].