Fast  Solvers and  Domain Decomposition  Preconditioners  for Spectral
Element Discretizations of Problems in H(curl)

Bernhard Hientzsch
November 28, 2001

For problems with piecewise smooth solutions, spectral element methods
hold  great  promise.  They  combine  the  exponential convergence  of
spectral methods  with the  geometric flexibility of  finite elements.
Spectral  elements are well-established  for scalar  elliptic problems
and problems  of fluid  dynamics, and recently  the first  methods for
problems in H(curl) and H(div)  were proposed. In this dissertation we
study spectral element methods for a model problem.  We first consider
Maxwell's equation and  derive the model problem in  H(curl).  Then we
introduce  anisotropic spectral  Nédélec element  discretizations with
variable  numerical integration  for  the model  problem.  We  discuss
their structure,  and their convergence  and approximation properties.
We also obtain results on the norm of the Nédélec interpolants between
Nédélec and Raviart-Thomas spaces  of different degree, needed for the
computation  of the  splitting constant  for the  domain decomposition
preconditioner and the numerical  analysis of nonlinear equations.  We
also prove a Friedrichs-like inequality  for the model problem for the
spectral case.

We  present fast  direct solvers  for the  model problem  on separable
domains,  taking advantage  of the  tensor product  discretization and
fast  diagonalization methods.   We use  those fast  solvers  as local
solvers  in domain  decomposition methods  for problems  that  are too
large to  be solved directly,  or posed on non-separable  domains, and
use  them  to compute  and  subassemble  the  Schur complement  system
corresponding  to the  interface. We  also  apply them  in the  direct
solution of the Schur complement system for general domains.

As  an  example for  the  domain  decomposition  methods that  can  be
implemented  with  these   tools,  we  introduce  overlapping  Schwarz
methods, both one-level and two-level versions.

We extend the  theory for overlapping Schwarz methods  to the spectral
Nédélec element  case.  We  reduce the proof  of the  condition number
estimate  to  three  basic  estimates,  and  present  theoretical  and
numerical results on those estimates. The technique of the proof works
in both the two-dimensional and three-dimensional case.

We also present numerical  results for one-level and two-level methods
in two dimensions.