TY - JOUR
T1 - Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations
AU - Ralf Hiptmair, Haijun Wu & Weiying Zheng
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 297
EP - 332
PY - 2012
DA - 2012/05
SN - 5
DO - http://doi.org/10.4208/nmtma.2012.m1128
UR - https://global-sci.org/intro/article_detail/nmtma/5940.html
KW - MMaxwell's equations, Lagrangian finite elements, edge elements, adaptive multigrid method, successive subspace correction.
AB - <p style="text-align: justify;">We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell&#39;s equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their &quot;immediate&quot; neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to
 the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.</p>