Title: Overlapping Schwarz Preconditioners for Spectral Nedelec Elements for a Model Problem in H(curl) 

(NYU-CS-TR834)

Author: Bernhard Hientzsch 

Abstract:
A two-level overlapping domain  decomposition method is analyzed for a
Nedelec spectral element approximation of a model problem appearing in
the solution  of Maxwell's equations.  The  overlap between subdomains
can  consist of  entire spectral  elements or  rectangular  subsets of
spectral elements.  For fixed relative  overlap and overlap  made from
entire  elements,  the condition  number  of  the  method is  bounded,
independently  of  the  mesh  size,  the  number  of  subregions,  the
coefficients and the  degree of the spectral elements.  In the case of
overlap including just  parts of spectral elements, a  bound linear in
the degree of  the elements is proven.  It is  assumed that the coarse
and fine mesh are quasi-uniform  and shape-regular and that the domain
is convex.  Arguments that would  not require quasi-uniformity  of the
coarse  mesh and  convexity of  the  domain are  mentioned.  Our  work
generalizes  results  obtained  for  lower-order Nedelec elements in
Toselli [Numerische  Mathematik (2000) 86:733-752].  Numerical results
for  the two-level  algorithm in  two dimensions  are  also presented,
supporting our analysis.