192 research outputs found
Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations
We present optimal preconditioners for a recently introduced hybridized
discontinuous Galerkin finite element discretization of the Stokes equations.
Typical of hybridized discontinuous Galerkin methods, the method has
degrees-of-freedom that can be eliminated locally (cell-wise), thereby
significantly reducing the size of the global problem. Although the linear
system becomes more complex to analyze after static condensation of these
element degrees-of-freedom, the pressure Schur complement of the original and
reduced problem are the same. Using this fact, we prove spectral equivalence of
this Schur complement to two simple matrices, which is then used to formulate
optimal preconditioners for the statically condensed problem. Numerical
simulations in two and three spatial dimensions demonstrate the good
performance of the proposed preconditioners
An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations
We present and analyze a new embedded--hybridized discontinuous Galerkin
finite element method for the Stokes problem. The method has the attractive
properties of full hybridized methods, namely an -conforming
velocity field, pointwise satisfaction of the continuity equation and \emph{a
priori} error estimates for the velocity that are independent of the pressure.
The embedded--hybridized formulation has advantages over a full hybridized
formulation in that it has fewer global degrees-of-freedom for a given mesh and
the algebraic structure of the resulting linear system is better suited to fast
iterative solvers. The analysis results are supported by a range of numerical
examples that demonstrate rates of convergence, and which show computational
efficiency gains over a full hybridized formulation
A Hybridizable Discontinuous Galerkin Method for the Navier–Stokes Equations with Pointwise Divergence-Free Velocity Field
We introduce a hybridizable discontinuous Galerkin method for the
incompressible Navier--Stokes equations for which the approximate velocity
field is pointwise divergence-free. The method builds on the method presented
by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We
show that with modifications of the function spaces in the method of Labeur and
Wells it is possible to formulate a simple method with pointwise
divergence-free velocity fields which is momentum conserving, energy stable,
and pressure-robust. Theoretical results are supported by two- and
three-dimensional numerical examples and for different orders of polynomial
approximation
Preconditioning for a pressure-robust HDG discretization of the Stokes equations
We introduce a new preconditioner for a recently developed pressure-robust
hybridized discontinuous Galerkin (HDG) finite element discretization of the
Stokes equations. A feature of HDG methods is the straightforward elimination
of degrees-of-freedom defined on the interior of an element. In our previous
work (J. Sci. Comput., 77(3):1936--1952, 2018) we introduced a preconditioner
for the case in which only the degrees-of-freedom associated with the element
velocity were eliminated via static condensation. In this work we introduce a
preconditioner for the statically condensed system in which the element
pressure degrees-of-freedom are also eliminated. In doing so the number of
globally coupled degrees-of-freedom are reduced, but at the expense of a more
difficult problem to analyse. We will show, however, that the Schur complement
of the statically condensed system is spectrally equivalent to a simple trace
pressure mass matrix. This result is used to formulate a new, provably optimal
preconditioner. Through numerical examples in two- and three-dimensions we show
that the new preconditioned iterative method converges in fewer iterations, has
superior conservation properties for inexact solves, and is faster in CPU time
when compared to our previous preconditioner
An embedded-hybridized discontinuous Galerkin method for the coupled Stokes-Darcy system
We introduce an embedded-hybridized discontinuous Galerkin (EDG-HDG) method
for the coupled Stokes-Darcy system. This EDG-HDG method is a pointwise
mass-conserving discretization resulting in a divergence-conforming velocity
field on the whole domain. In the proposed scheme, coupling between the Stokes
and Darcy domains is achieved naturally through the EDG-HDG facet variables.
\emph{A priori} error analysis shows optimal convergence rates, and that the
velocity error does not depend on the pressure. The error analysis is verified
through numerical examples on unstructured grids for different orders of
polynomial approximation
A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators
The embedded discontinuous Galerkin (EDG) finite element method for the
Stokes problem results in a point-wise divergence-free approximate velocity on
cells. However, the approximate velocity is not H(div)-conforming and it can be
shown that this is the reason that the EDG method is not pressure-robust, i.e.,
the error in the velocity depends on the continuous pressure. In this paper we
present a local reconstruction operator that maps discretely divergence-free
test functions to exactly divergence-free test functions. This local
reconstruction operator restores pressure-robustness by only changing the right
hand side of the discretization, similar to the reconstruction operator
recently introduced for the Taylor--Hood and mini elements by Lederer et al.
(SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error
analysis of the discretization showing optimal convergence rates and
pressure-robustness of the velocity error. These results are verified by
numerical examples. The motivation for this research is that the resulting EDG
method combines the versatility of discontinuous Galerkin methods with the
computational efficiency of continuous Galerkin methods and accuracy of
pressure-robust finite element methods
Proceedings of the FEniCS Conference 2017
Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg
Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods
In this paper we present a geometric multigrid method with Jacobi and Vanka
relaxation for hybridized and embedded discontinuous Galerkin discretizations
of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid
error-propagation operator and show that the multigrid method applied to an
embedded discontinuous Galerkin (EDG) discretization is almost as efficient as
when applied to a continuous Galerkin discretization. We furthermore show that
multigrid applied to an EDG discretization outperforms multigrid applied to a
hybridized discontinuous Galerkin (HDG) discretization. Numerical examples
verify our LFA predictions
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