126 research outputs found
Analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains
This paper presents the first analysis of a space--time hybridizable
discontinuous Galerkin method for the advection--diffusion problem on
time-dependent domains. The analysis is based on non-standard local trace and
inverse inequalities that are anisotropic in the spatial and time steps. We
prove well-posedness of the discrete problem and provide a priori error
estimates in a mesh-dependent norm. Convergence theory is validated by a
numerical example solving the advection--diffusion problem on a time-dependent
domain for approximations of various polynomial degree
An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains
This paper presents a space-time embedded-hybridized discontinuous Galerkin
(EHDG) method for the Navier--Stokes equations on moving domains. This method
uses a different hybridization compared to the space-time hybridized
discontinuous Galerkin (HDG) method we presented previously in (Int. J. Numer.
Meth. Fluids 89: 519--532, 2019). In the space-time EHDG method the velocity
trace unknown is continuous while the pressure trace unknown is discontinuous
across facets. In the space-time HDG method, all trace unknowns are
discontinuous across facets. Alternatively, we present also a space-time
embedded discontinuous Galerkin (EDG) method in which all trace unknowns are
continuous across facets. The advantage of continuous trace unknowns is that
the formulation has fewer global degrees-of-freedom for a given mesh than when
using discontinuous trace unknowns. Nevertheless, the discrete velocity field
obtained by the space-time EHDG and EDG methods, like the space-time HDG
method, is exactly divergence-free, even on moving domains. However, only the
space-time EHDG and HDG methods result in divergence-conforming velocity
fields. An immediate consequence of this is that the space-time EHDG and HDG
discretizations of the conservative form of the Navier--Stokes equations are
energy stable. The space-time EDG method, on the other hand, requires a
skew-symmetric formulation of the momentum advection term to be energy-stable.
Numerical examples will demonstrate the differences in solution obtained by the
space-time EHDG, EDG, and HDG methods
eXtended Hybridizable Discontinous Galerkin (X-HDG) Method for Linear Convection-Diffusion Equations on Unfitted Domains
In this work, we propose a novel strategy for the numerical solution of
linear convection diffusion equation (CDE) over unfitted domains. In the
proposed numerical scheme, strategies from high order Hybridized Discontinuous
Galerkin method and eXtended Finite Element method is combined with the level
set definition of the boundaries. The proposed scheme and hence, is named as
eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the
Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted
domains; i.e, the computational mesh does not need to fit to the domain
boundary; instead, the boundary is defined by a level set function and cuts
through the background mesh arbitrarily. The original unknown structure of HDG
and its hybrid nature ensuring the local conservation of fluxes is kept, while
developing a modified bilinear form for the elements cut by the boundary. At
every cut element, an auxiliary nodal trace variable on the boundary is
introduced, which is eliminated afterwards while imposing the boundary
conditions. Both stationary and time dependent CDEs are studied over a range of
flow regimes from diffusion to convection dominated; using high order XHDG through benchmark numerical examples over arbitrary unfitted domains.
Results proved that XHDG inherits optimal and super
convergence properties of HDG while removing the fitting mesh restriction
A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows
The potential of the hybridized discontinuous Galerkin (HDG) method has been
recognized for the computation of stationary flows. Extending the method to
time-dependent problems can, e.g., be done by backward difference formulae
(BDF) or diagonally implicit Runge-Kutta (DIRK) methods. In this work, we
investigate the use of embedded DIRK methods in an HDG solver, including the
use of adaptive time-step control. Numerical results demonstrate the
performance of the method for both linear and nonlinear (systems of)
time-dependent convection-diffusion equations
A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains
We present a finite element method for the incompressible Navier--Stokes
problem that is locally conservative, energy-stable and pressure-robust on
time-dependent domains. To achieve this, the space--time formulation of the
Navier--Stokes problem is considered. The space--time domain is partitioned
into space--time slabs which in turn are partitioned into space--time
simplices. A combined discontinuous Galerkin method across space--time slabs,
and space--time hybridized discontinuous Galerkin method within a space--time
slab, results in an approximate velocity field that is -conforming and exactly divergence-free, even on time-dependent domains.
Numerical examples demonstrate the convergence properties and performance of
the method
An interface-tracking space-time hybridizable/embedded discontinuous Galerkin method for nonlinear free-surface flows
We present a compatible space-time hybridizable/embedded discontinuous
Galerkin discretization for nonlinear free-surface waves. We pose this problem
in a two-fluid (liquid and gas) domain and use a time-dependent level-set
function to identify the sharp interface between the two fluids. The
incompressible two-fluidd equations are discretized by an exactly mass
conserving space-time hybridizable discontinuous Galerkin method while the
level-set equation is discretized by a space-time embedded discontinuous
Galerkin method. Different from alternative discontinuous Galerkin methods is
that the embedded discontinuous Galerkin method results in a continuous
approximation of the interface. This, in combination with the space-time
framework, results in an interface-tracking method without resorting to
smoothing techniques or additional mesh stabilization terms
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