Numerische Mathematik

, Volume 52, Issue 4, pp 427-458

First online:

The hierarchical basis multigrid method

  • Randolph E. BankAffiliated withDepartment of Mathematics, University of California at San Diego
  • , Todd F. DupontAffiliated withDepartment of Mathematics, University of Chicago
  • , Harry YserentantAffiliated withFachbereich Mathematik, Universität Dortmund

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We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.

Subject Classifications

AMS(MOS): 65F10 65F35 65N20 65N30 CR:G1.8