Fast domain decomposition solvers for discrete problems with chaotically sub-domain wise variable orthotropism

Vadim Korneev

Oct. 1, 2012, 3:30 p.m. S2 416

The second order elliptic equation is considered in the domain, composed of some
shape and size irregular rectangles, which are nests of the orthogonal nonuniform de-
composition mesh. The matrix of coefficients of the elliptic operator, written in the
divergent form, is diagonal and its nonzero coefficients in each subdomain are arbitrary
positive numbers. The orthogonal finite element mesh satisfies only one condition:
it is uniform on each subdomain. No other conditions on the coefficients of the elliptic
equation and on variable step sizes of the discretization and decomposition meshes are
imposed. For the resulting discrete finite element problem, we present the DD (domain
decomposition) preconditioner of the Dirichlet-Dirichlet type, in which d.o.f.’s at nodes
of the decomposition mesh are split from others, and it is assumed that the contribution
of the subproblem, related to these d.o.f.’s, to the computational cost of the DD solver
is secondary. Essential components of DD preconditioner are the same as in the paper
of Korneev, Poborchii & Salgado (2007). However, here we remove the weak dependence
of the relative condition number of the DD preconditioner on some measure of the local
orthotropism of the discretization. We show that the DD solver has linear complexity,
independently of the aspect ratios of the three types of orthotropism listed above. The
result became possible due to the special way of the interface preconditioning by means
of the inexact solver employing the preconditioner-multiplicator and the preconditioner-
solver. The interface preconditioning makes also the main difference from other authors
works on fast DD solvers for similar problems, e.g., of Khoromskij & Wittum (1999,2004)
and Kwak, Nepomnyaschikh & Pyo (2004).