132 research outputs found
An analysis of the practical DPG method
In this work we give a complete error analysis of the Discontinuous Petrov
Galerkin (DPG) method, accounting for all the approximations made in its
practical implementation. Specifically, we consider the DPG method that uses a
trial space consisting of polynomials of degree on each mesh element.
Earlier works showed that there is a "trial-to-test" operator , which when
applied to the trial space, defines a test space that guarantees stability. In
DPG formulations, this operator is local: it can be applied
element-by-element. However, an infinite dimensional problem on each mesh
element needed to be solved to apply . In practical computations, is
approximated using polynomials of some degree on each mesh element. We
show that this approximation maintains optimal convergence rates, provided that
, where is the space dimension (two or more), for the Laplace
equation. We also prove a similar result for the DPG method for linear
elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods
are also included.Comment: Mathematics of Computation, 201
Partial expansion of a Lipschitz domain and some applications
We show that a Lipschitz domain can be expanded solely near a part of its
boundary, assuming that the part is enclosed by a piecewise C1 curve. The
expanded domain as well as the extended part are both Lipschitz. We apply this
result to prove a regular decomposition of standard vector Sobolev spaces with
vanishing traces only on part of the boundary. Another application in the
construction of low-regularity projectors into finite element spaces with
partial boundary conditions is also indicated
The auxiliary space preconditioner for the de Rham complex
We generalize the construction and analysis of auxiliary space
preconditioners to the n-dimensional finite element subcomplex of the de Rham
complex. These preconditioners are based on a generalization of a decomposition
of Sobolev space functions into a regular part and a potential. A discrete
version is easily established using the tools of finite element exterior
calculus. We then discuss the four-dimensional de Rham complex in detail. By
identifying forms in four dimensions (4D) with simple proxies, form operations
are written out in terms of familiar algebraic operations on matrices, vectors,
and scalars. This provides the basis for our implementation of the
preconditioners in 4D. Extensive numerical experiments illustrate their
performance, practical scalability, and parameter robustness, all in accordance
with the theory
Spectral discretization errors in filtered subspace iteration
We consider filtered subspace iteration for approximating a cluster of
eigenvalues (and its associated eigenspace) of a (possibly unbounded)
selfadjoint operator in a Hilbert space. The algorithm is motivated by a
quadrature approximation of an operator-valued contour integral of the
resolvent. Resolvents on infinite dimensional spaces are discretized in
computable finite-dimensional spaces before the algorithm is applied. This
study focuses on how such discretizations result in errors in the eigenspace
approximations computed by the algorithm. The computed eigenspace is then used
to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff
distance between the computed and exact eigenvalue clusters are obtained in
terms of the discretization parameters within an abstract framework. A
realization of the proposed approach for a model second-order elliptic operator
using a standard finite element discretization of the resolvent is described.
Some numerical experiments are conducted to gauge the sharpness of the
theoretical estimates
The DPG-star method
This article introduces the DPG-star (from now on, denoted DPG) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG and DPG
methods can be seen as generalizations of and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable
Simulation of Optical Fiber Amplifier Gain Using Equivalent Short Fibers
Electromagnetic wave propagation in optical fiber amplifiers obeys Maxwell
equations. Using coupled mode theory, the full Maxwell system within an optical
fiber amplifier is reduced to a simpler model. The simpler model is made more
efficient through a new scale model, referred to as an equivalent short fiber,
which captures some of the essential characteristics of a longer fiber. The
equivalent short fiber can be viewed as a fiber made using artificial
(unphysical) material properties that in some sense compensates for its reduced
length. The computations can be accelerated by a factor approximately equal to
the ratio of the original length to the reduced length of the equivalent fiber.
Computations using models of two commercially available fibers -- one doped
with ytterbium, and the other with thulium -- show the practical utility of the
concept. Extensive numerical studies are conducted to assess when the
equivalent short fiber model is useful and when it is not
The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow
In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods
An Analysis of the Practical DPG Method
We give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree p on each mesh element. Earlier works showed that there is a trial-to-test operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, T is approximated using polynomials of some degree r \u3e p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r p + N, where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included
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