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1 results

  • Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations

  • Ralf Hiptmair, Haijun Wu, Weiying Zheng
  • Journal:
    Numerical Mathematics: Theory, Methods and Applications / Volume 5 / Issue 3 / August 2012
    Published online by Cambridge University Press:
    28 May 2015, pp. 297-332
    Print publication:
    August 2012

  • We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell’s equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their “immediate” 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.