Abstract. We consider H(curl, Ω)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H 1 (Ω)-context along with local discrete Helmholtz-type decompositions of the edge element space. Key words. Edge elements, local multigrid, stable multilevel splittings, subspace correction theory, regular decompositions of H(curl, Ω), Helmholtz-type decompositions

Quadtree techniques for the construction of structured auxiliary meshes

We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A −∇div in H(div, Ω). For A being a variable coefficient matrix with possible jump discontinuities, we provide a complete a posteriori error analysis which applies to both Raviart–Thomas RTn and Brezzi– Douglas–Marini BDMn elements of any order n in dimensions d =2, 3. We prove a strict reduction of the total error between consecutive iterates, namely a contraction property for the sum of energy error and oscillation, the latter being solution-dependent. We present numerical experiments for RTn with n =0, 1andBDM1 which document the performance of AFEM and corroborate as well as extend the theory.

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