747 research outputs found
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
Space Decompositions and Solvers for Discontinuous Galerkin Methods
We present a brief overview of the different domain and space decomposition
techniques that enter in developing and analyzing solvers for discontinuous
Galerkin methods. Emphasis is given to the novel and distinct features that
arise when considering DG discretizations over conforming methods. Connections
and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
This paper analyses the following question: let , be
the Galerkin matrices corresponding to finite-element discretisations of the
exterior Dirichlet problem for the heterogeneous Helmholtz equations
. How small must and be (in terms of -dependence) for
GMRES applied to either or
to converge in a -independent number of
iterations for arbitrarily large ? (In other words, for to be
a good left- or right-preconditioner for ?). We prove results
answering this question, give theoretical evidence for their sharpness, and
give numerical experiments supporting the estimates.
Our motivation for tackling this question comes from calculating quantities
of interest for the Helmholtz equation with random coefficients and .
Such a calculation may require the solution of many deterministic Helmholtz
problems, each with different and , and the answer to the question above
dictates to what extent a previously-calculated inverse of one of the Galerkin
matrices can be used as a preconditioner for other Galerkin matrices
Preconditioning of block tridiagonal matrices
Preconditioning methods via approximate block factorization for block tridiagonal matrices are studied. Bounds for the resulting condition numbers are given, and two methods for the recursive construction of the approximate Schur complements are presented. Illustrations for elliptic problems are also given, including a study of sensitivity to jumps in the coefficients and of a suitably motidied Poincaré-Steklov operator on the continuous level
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations
We analyse a class of nonoverlapping domain decomposition preconditioners for
nonsymmetric linear systems arising from discontinuous Galerkin finite element
approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial
differential equations. These nonsymmetric linear systems are uniformly bounded
and coercive with respect to a related symmetric bilinear form, that is
associated to a matrix . In this work, we construct a
nonoverlapping domain decomposition preconditioner , that is based
on , and we then show that the effectiveness of the preconditioner
for solving the} nonsymmetric problems can be studied in terms of the condition
number . In particular, we establish the
bound , where
and are respectively the coarse and fine mesh sizes, and and
are respectively the coarse and fine mesh polynomial degrees. This represents
the first such result for this class of methods that explicitly accounts for
the dependence of the condition number on ; our analysis is founded upon an
original optimal order approximation result between fine and coarse
discontinuous finite element spaces. Numerical experiments demonstrate the
sharpness of this bound. Although the preconditioners are not robust with
respect to the polynomial degree, our bounds quantify the effect of the coarse
and fine space polynomial degrees. Furthermore, we show computationally that
these methods are effective in practical applications to nonsymmetric, fully
nonlinear HJB equations under -refinement for moderate polynomial degrees
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