TY - JOUR
T1 - Local Multigrid in H(Curl)
AU - Ralf Hiptmair & Weiying Zheng
JO - Journal of Computational Mathematics
VL - 5
SP - 573
EP - 603
PY - 2009
DA - 2009/10
SN - 27
DO - http://doi.org/10.4208/jcm.2009.27.5.012
UR - https://global-sci.org/intro/article_detail/jcm/8591.html
KW - Edge elements, Local multigrid, Stable multilevel splittings, Subspace correction theory, Regular decompositions of $\boldsymbol{H}(curl, Ω)$, Helmholtz-type decompositions, Local
mesh refinement.
AB - <p style="text-align: justify;">We consider $\boldsymbol{H}$(<strong>curl</strong>, $Ω$)-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.</p>