4 research outputs found
Cascadic Multilevel Algorithms for Symmetric Saddle Point Systems
In this paper, we introduce a multilevel algorithm for approximating
variational formulations of symmetric saddle point systems. The algorithm is
based on availability of families of stable finite element pairs and on the
availability of fast and accurate solvers for it symmetric positive definite
systems. On each fixed level an efficient solver such as the gradient or the
conjugate gradient algorithm for inverting a Schur complement is implemented.
The level change criterion follows the cascade principle and requires that the
iteration error be close to the expected discretization error. We prove new
estimates that relate the iteration error and the residual for the constraint
equation. The new estimates are the key ingredients in imposing an efficient
level change criterion. The first iteration on each new level uses information
about the best approximation of the discrete solution from the previous level.
The theoretical results and experiments show that the algorithms achieve
optimal or close to optimal approximation rates by performing a non-increasing
number of iterations on each level. Even though numerical results supporting
the efficiency of the algorithms are presented for the Stokes system, the
algorithms can be applied to a large class of boundary value problems,
including first order systems that can be reformulated at the continuous level
as symmetric saddle point problems, such as the Maxwell equations.Comment: 15 pages, 4 table
Saddle Point Least Squares Preconditioning of Mixed Methods
We present a simple way to discretize and precondition mixed variational
formulations. Our theory connects with, and takes advantage of, the classical
theory of symmetric saddle point problems and the theory of preconditioning
symmetric positive definite operators. Efficient iterative processes for
solving the discrete mixed formulations are proposed and choices for discrete
spaces that are always compatible are provided. For the proposed discrete
spaces and solvers, a basis is needed only for the test spaces and assembly of
a global saddle point system is avoided. We prove sharp approximation
properties for the discretization and iteration errors and also provide a sharp
estimate for the convergence rate of the proposed algorithm in terms of the
condition number of the elliptic preconditioner and the discrete
and constants of the pair of discrete spaces.Comment: Submitted to CAMWA on 5/17/1
A note on stability and optimal approximation estimates for symmetric saddle point systems
We establish sharp well-posedness and approximation estimates for variational
saddle point systems at the continuous level. The main results of this note
have been known to be true only in the finite dimensional case. Known spectral
results from the discrete case are reformulated and proved using a functional
analysis view, making the proofs in both cases, discrete and continuous, less
technical than the known discrete approaches. We focus on analyzing the special
case when the form is bounded, symmetric, and coercive, and
the mixed form is bounded and satisfies a standard
or LBB condition. We characterize the spectrum of the symmetric
operators that describe the problem at the continuous level. For a particular
choice of the inner product on the product space of , we prove
that the spectrum of the operator representing the system at continuous level
is . As
consequences of the spectral description, we find the minimal length interval
that contains the ratio between the norm of the data and the norm of the
solution, and prove explicit approximation estimates that depend only on the
continuity constant and the continuous and the discrete condition
constants.Comment: 14 pages. A slightly different version of this note was originally
submitted to Numerische Mathematik on August 21, 2013. No figure
A nonconforming saddle point least squares approach for elliptic interface problems
We present a non-conforming least squares method for approximating solutions
of second order elliptic problems with discontinuous coefficients. The method
is based on a general Saddle Point Least Squares (SPLS) method introduced in
previous work based on conforming discrete spaces. The SPLS method has the
advantage that a discrete condition is automatically satisfied for
standard choices of test and trial spaces. We explore the SPLS method for
non-conforming finite element trial spaces which allow higher order
approximation of the fluxes. For the proposed iterative solvers, inversion at
each step requires bases only for the test spaces. We focus on using projection
trial spaces with local projections that are easy to compute. The choice of the
local projections for the trial space can be combined with classical gradient
recovery techniques to lead to quasi-optimal approximations of the global flux.
Numerical results for 2D and 3D domains are included to support the proposed
method.Comment: The original manuscript was submitted to De Gruyter, Computational
Methods in Applied Mathematics on December 19, 201