Domain decomposition methods are powerful iterative methods
for solving systems of algebraic equations arising from the
discretization of partial differential equations by, e.g., finite
elements. The computational domain is decomposed into overlapping or
nonoverlapping subdomains. The problem is divided into, or assembled
from, smaller subproblems corresponding to these subdomains. In this
dissertation, we focus on domain decomposition methods for mortar
finite elements, which are nonconforming finite element methods that
allow for a geometrically nonconforming decomposition of the
computational domain into subregions and for the optimal coupling of
different variational approximations in different subregions.
We introduce a FETI method for mortar finite elements, and provide numer-
ical comparisons of FETI algorithms for mortar finite elements when different
preconditioners, given in the FETI literature, are considered. We also analyze
the complexity of the preconditioners for the three dimensional versions of the
algorithms.
We formulate a variant of the balancing method for mortar finite
elements, which uses extended local regions to account for the
nonmortar sides of the subre- gions. We prove a polylogarithmic
condition number estimate for our algorithm in the geometrically
nonconforming case. Our estimate is similar to those for other
Neumann{Neumann and substructuring methods for mortar finite elements.
In addition, we establish several fundamental properties of mortar
finite elements: the existence of the nonmortar partition of any
interface, the L^2 stability of the mortar projection for arbitrary
meshes on the nonmortar side, and prove Friedrichs and Poincare
inequalities for geometrically nonconforming mortar elements.