Showing papers by "Garth N. Wells published in 2018"
TL;DR: It is shown 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.
Abstract: 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 34(2):A889---A913, 2012). 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.
80 citations
TL;DR: In this article, optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations are presented. But their performance is limited to two dimensions.
Abstract: 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.
41 citations
Posted Content•
TL;DR: In this article, optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations are presented. But their performance is limited to two dimensions.
Abstract: 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.
14 citations
Posted Content•
TL;DR: In this paper, the authors present a new embedded hybridized discontinuous Galerkin finite element method for the Stokes problem, which has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field, pointwise satisfaction of the continuity equation and error estimates for the velocity that are independent of the pressure.
Abstract: 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 $H({\rm div})$-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.
12 citations
DOI•
14 Aug 2018
TL;DR: This release is specifically created to document the version of Firedrake used in a particular set of experiments, and is not to cite this as a general source for Firedrake or any of its dependencies.
Abstract: Version of Firedrake used in 'A domain-specific language for the static condensation and hybridization of finite element methods'.
This release is specifically created to document the version of
Firedrake used in a particular set of experiments. Please do not cite
this as a general source for Firedrake or any of its
dependencies. Instead, refer to
http://www.firedrakeproject.org/publications.html
1 citations