Non conforming domain decomposition:
The non conforming Steklov-Poincaré operator point of view
S. Bertoluzza
One of the common approaches to solve the linear system
arising in the domain decomposition method is to formally reduce it, by a
Schur complement argument, to a lower dimensional linear system whose unknown
is the value of the (discrete) solution on the interface of the decomposition.
Solving such reduced linear system by any iterative technique implies the
need of solving, at each iteration, independent discrete Dirichlet problems in
the subdomains. Such Dirichlet problems constitute the most relevant part of the computational cost
of such an approach and therefore attention needs to be paid in reducing the
actual computational cost of the subdomain solvers. A key observation in this
respect is that what one expects as an output of the iterative procedure is a
(correct order) approximation of the trace of the solution on the interface.
There is no direct need of solving correctly the Dirichlet problems in the
subdomains. The precision with which such problems are solved is only as
relevant as its influence on the error on the trace of the solution on the interface.
Only once the trace of the solution
on the interface has been computed correctly, one
will actually need to retrieve the solution in some or all of the subdomains.
In order to take advantage of this observation it is useful to look at
the Schur complement linear system as non conforming
discretization of the Steklov-Poicaré operator, mapping a function
defined on the interface, to the jump of the normal derivative of its harmonic
lifting (computed subdomain-wise). The non-conformity stems from replacing the
harmonic lifting with its discretization. If we look at the Schur complement
system from this point of view, a straightforward application of the
first Strang Lemma, shows that the discretization in the subdomains needs to be
designed in order to provide a correct order approximation of outer normal
derivative, while there is no direct need to actually provide a good
approximation of the solution in the interior of the subdomains.
The aim of this paper is to formalise the above considerations in the case in
which the starting domain decomposition formulation is the three fields
formulation, and to provide a rigorous error estimate for the trace of the solution
on the the interface, showing that the mesh can actually be chosen to be
sensibly coarser in the interior of the subdomains without affecting the
precision of the interface approximation, resulting in a sensible reduction in
computational cost of the subdomain solvers.
Preprint, 2002
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