275 research outputs found
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
A Combined Preconditioning Strategy for Nonsymmetric Systems
We present and analyze a class of nonsymmetric preconditioners within a
normal (weighted least-squares) matrix form for use in GMRES to solve
nonsymmetric matrix problems that typically arise in finite element
discretizations. An example of the additive Schwarz method applied to
nonsymmetric but definite matrices is presented for which the abstract
assumptions are verified. A variable preconditioner, combining the original
nonsymmetric one and a weighted least-squares version of it, is shown to be
convergent and provides a viable strategy for using nonsymmetric
preconditioners in practice. Numerical results are included to assess the
theory and the performance of the proposed preconditioners.Comment: 26 pages, 3 figure
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin finite element methods in -type norms
We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for -version discontinuous Galerkin finite element methods in -type norms, which arise in applications to fully nonlinear Hamilton--Jacobi--Bellman partial differential equations. We show that for a symmetric model problem, the condition number of the preconditioned system is at most of order , 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 result for this class of methods that explicitly accounts for the dependence of the condition number on , and its sharpness is shown numerically. The key analytical tool is an original optimal order approximation result between fine and coarse discontinuous finite element spaces.\ud
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We then go beyond the model problem and show computationally that these methods lead to efficient and competitive solvers in practical applications to nonsymmetric, fully nonlinear Hamilton--Jacobi--Bellman equations
A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods
In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented
Uniform subspace correction preconditioners for discontinuous Galerkin methods with -refinement
In this paper, we develop subspace correction preconditioners for
discontinuous Galerkin (DG) discretizations of elliptic problems with
-refinement. These preconditioners are based on the decomposition of the DG
finite element space into a conforming subspace, and a set of small
nonconforming edge spaces. The conforming subspace is preconditioned using a
matrix-free low-order refined technique, which in this work we extend to the
-refinement context using a variational restriction approach. The condition
number of the resulting linear system is independent of the granularity of the
mesh , and the degree of polynomial approximation . The method is
amenable to use with meshes of any degree of irregularity and arbitrary
distribution of polynomial degrees. Numerical examples are shown on several
test cases involving adaptively and randomly refined meshes, using both the
symmetric interior penalty method and the second method of Bassi and Rebay
(BR2).Comment: 24 pages, 9 figure
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