71 research outputs found
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
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 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
Automated code generation for discontinuous Galerkin methods
A compiler approach for generating low-level computer code from high-level
input for discontinuous Galerkin finite element forms is presented. The input
language mirrors conventional mathematical notation, and the compiler generates
efficient code in a standard programming language. This facilitates the rapid
generation of efficient code for general equations in varying spatial
dimensions. Key concepts underlying the compiler approach and the automated
generation of computer code are elaborated. The approach is demonstrated for a
range of common problems, including the Poisson, biharmonic,
advection--diffusion and Stokes equations
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
Analysis of pressure-robust embedded-hybridized discontinuous Galerkin methods for the Stokes problem under minimal regularity
We present analysis of two lowest-order hybridizable discontinuous Galerkin
methods for the Stokes problem, while making only minimal regularity
assumptions on the exact solution. The methods under consideration have
previously been shown to produce -conforming and
divergence-free approximate velocities. Using these properties, we derive a
priori error estimates for the velocity that are independent of the pressure.
These error estimates, which assume only -regularity of the exact
velocity fields for any , are optimal in a discrete energy norm.
Error estimates for the velocity and pressure in the -norm are also
derived in this minimal regularity setting. Our theoretical findings are
supported by numerical computations
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