13,061 research outputs found
Automatic Variationally Stable Analysis for FE Computations: An Introduction
We introduce an automatic variationally stable analysis (AVS) for finite
element (FE) computations of scalar-valued convection-diffusion equations with
non-constant and highly oscillatory coefficients. In the spirit of least
squares FE methods, the AVS-FE method recasts the governing second order
partial differential equation (PDE) into a system of first-order PDEs. However,
in the subsequent derivation of the equivalent weak formulation, a
Petrov-Galerkin technique is applied by using different regularities for the
trial and test function spaces. We use standard FE approximation spaces for the
trial spaces, which are C0, and broken Hilbert spaces for the test functions.
Thus, we seek to compute pointwise continuous solutions for both the primal
variable and its flux (as in least squares FE methods), while the test
functions are piecewise discontinuous. To ensure the numerical stability of the
subsequent FE discretizations, we apply the philosophy of the discontinuous
Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan, by invoking test
functions that lead to unconditionally stable numerical systems (if the kernel
of the underlying differential operator is trivial). In the AVS-FE method, the
discontinuous test functions are ascertained per the DPG approach from local,
decoupled, and well-posed variational problems, which lead to best
approximation properties in terms of the energy norm. We present various 2D
numerical verifications, including convection-diffusion problems with highly
oscillatory coefficients and extremely high Peclet numbers, up to a billion.
These show the unconditional stability without the need for any upwind schemes
nor any other artificial numerical stabilization. The results are not highly
diffused for convection-dominated problems ...Comment: Preprint submitted to Lecture Notes in Computational Science and
Engineering, Springer Verla
Discontinuous Galerkin method for Navier-Stokes equations using kinetic flux vector splitting
Kinetic schemes for compressible flow of gases are constructed by exploiting
the connection between Boltzmann equation and the Navier-Stokes equations. This
connection allows us to construct a flux splitting for the Navier-Stokes
equations based on the direction of molecular motion from which a numerical
flux can be obtained. The naive use of such a numerical flux function in a
discontinuous Galerkin (DG) discretization leads to an unstable scheme in the
viscous dominated case. Stable schemes are constructed by adding additional
terms either in a symmetric or non-symmetric manner which are motivated by the
DG schemes for elliptic equations. The novelty of the present scheme is the use
of kinetic fluxes to construct the stabilization terms. In the symmetric case,
interior penalty terms have to be added for stability and the resulting schemes
give optimal convergence rates in numerical experiments. The non-symmetric
schemes lead to a cell energy/entropy inequality but exhibit sub-optimal
convergence rates. These properties are studied by applying the schemes to a
scalar convection-diffusion equation and the 1-D compressible Navier-Stokes
equations. In the case of Navier-Stokes equations, entropy variables are used
to construct stable schemes
A Conservative Flux Optimization Finite Element Method for Convection-Diffusion Equations
This article presents a new finite element method for convection-diffusion
equations by enhancing the continuous finite element space with a flux space
for flux approximations that preserve the important mass conservation locally
on each element. The numerical scheme is based on a constrained flux
optimization approach where the constraint was given by local mass conservation
equations and the flux error is minimized in a prescribed topology/metric. This
new scheme provides numerical approximations for both the primal and the flux
variables. It is shown that the numerical approximations for the primal and the
flux variables are convergent with optimal order in some discrete Sobolev
norms. Numerical experiments are conducted to confirm the convergence theory.
Furthermore, the new scheme was employed in the computational simulation of a
simplified two-phase flow problem in highly heterogeneous porous media. The
numerical results illustrate an excellent performance of the method in
scientific computing
Superconvergence properties of an upwind-biased discontinuous Galerkin method
In this paper we investigate the superconvergence properties of the
discontinuous Galerkin method based on the upwind-biased flux for linear
time-dependent hyperbolic equations. We prove that for even-degree polynomials,
the method is locally superconvergent at roots of a
linear combination of the left- and right-Radau polynomials. This linear
combination depends on the value of used in the flux. For odd-degree
polynomials, the scheme is superconvergent provided that a proper global
initial interpolation can be defined. We demonstrate numerically that, for
decreasing , the discretization errors decrease for even polynomials
and grow for odd polynomials. We prove that the use of Smoothness-Increasing
Accuracy-Conserving (SIAC) filters is still able to draw out the
superconvergence information and create a globally smooth and superconvergent
solution of for linear hyperbolic equations. Lastly, we
briefly consider the spectrum of the upwind-biased DG operator and demonstrate
that the price paid for the introduction of the parameter is limited
to a contribution to the constant attached to the post-processed error term
High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows
In this paper we present an efficient discretization method for the solution
of the unsteady incompressible Navier-Stokes equations based on a high order
(Hybrid) Discontinuous Galerkin formulation. The crucial component for the
efficiency of the discretization method is the disctinction between stiff
linear parts and less stiff non-linear parts with respect to their temporal and
spatial treatment. Exploiting the flexibility of operator-splitting time
integration schemes we combine two spatial discretizations which are tailored
for two simpler sub-problems: a corresponding hyperbolic transport problem and
an unsteady Stokes problem. For the hyperbolic transport problem a spatial
discretization with an Upwind Discontinuous Galerkin method and an explicit
treatment in the time integration scheme is rather natural and allows for an
efficient implementation. The treatment of the Stokes part involves the
solution of linear systems. In this case a discretization with Hybrid
Discontinuous Galerkin methods is better suited. We consider such a
discretization for the Stokes part with two important features:
H(div)-conforming finite elements to garantuee exactly divergence-free velocity
solutions and a projection operator which reduces the number of globally
coupled unknowns. We present the method, discuss implementational aspects and
demonstrate the performance on two and three dimensional benchmark problems.Comment: 21 pages, 3 figures, 4 tabl
An Implicit Discontinuous Galerkin Finite Element Method for Water Waves
An overview is given of a discontinuous Galerkin finite element method for linear free surface water waves. The method uses an implicit time integration method which is unconditionally stable and does not suffer from the frequently encountered mesh dependent saw-tooth type instability at the free surface. The numerical discretization has minimal dissipation and small dispersion errors in the wave propagation. The algorithm is second order accurate in time and has an optimal rate of convergence O(hp+1) in the L2- norm, both in the potential and wave height, with p the polynomial order and h the mesh size. The numerical discretization is demonstrated with the simulation of water waves in a basin with a bump at the bottom
Analysis and entropy stability of the line-based discontinuous Galerkin method
We develop a discretely entropy-stable line-based discontinuous Galerkin
method for hyperbolic conservation laws based on a flux differencing technique.
By using standard entropy-stable and entropy-conservative numerical flux
functions, this method guarantees that the discrete integral of the entropy is
non-increasing. This nonlinear entropy stability property is important for the
robustness of the method, in particular when applied to problems with
discontinuous solutions or when the mesh is under-resolved. This line-based
method is significantly less computationally expensive than a standard DG
method. Numerical results are shown demonstrating the effectiveness of the
method on a variety of test cases, including Burgers' equation and the Euler
equations, in one, two, and three spatial dimensions.Comment: 25 pages, 7 figure
Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method
This paper discusses the computation of derivatives for optimization problems
governed by linear hyperbolic systems of partial differential equations (PDEs)
that are discretized by the discontinuous Galerkin (dG) method. An efficient
and accurate computation of these derivatives is important, for instance, in
inverse problems and optimal control problems. This computation is usually
based on an adjoint PDE system, and the question addressed in this paper is how
the discretization of this adjoint system should relate to the dG
discretization of the hyperbolic state equation. Adjoint-based derivatives can
either be computed before or after discretization; these two options are often
referred to as the optimize-then-discretize and discretize-then-optimize
approaches. We discuss the relation between these two options for dG
discretizations in space and Runge-Kutta time integration. Discretely exact
discretizations for several hyperbolic optimization problems are derived,
including the advection equation, Maxwell's equations and the coupled
elastic-acoustic wave equation. We find that the discrete adjoint equation
inherits a natural dG discretization from the discretization of the state
equation and that the expressions for the discretely exact gradient often have
to take into account contributions from element faces. For the coupled
elastic-acoustic wave equation, the correctness and accuracy of our derivative
expressions are illustrated by comparisons with finite difference gradients.
The results show that a straightforward discretization of the continuous
gradient differs from the discretely exact gradient, and thus is not consistent
with the discretized objective. This inconsistency may cause difficulties in
the convergence of gradient based algorithms for solving optimization problems
A stabilized cut discontinuous Galerkin framework: I. Elliptic boundary value and interface problems
We develop a stabilized cut discontinuous Galerkin framework for the
numerical solution of el- liptic boundary value and interface problems on
complicated domains. The domain of interest is embedded in a structured,
unfitted background mesh in R d , so that the boundary or interface can cut
through it in an arbitrary fashion. The method is based on an unfitted variant
of the classical symmetric interior penalty method using piecewise
discontinuous polynomials defined on the back- ground mesh. Instead of the cell
agglomeration technique commonly used in previously introduced unfitted
discontinuous Galerkin methods, we employ and extend ghost penalty techniques
from recently developed continuous cut finite element methods, which allows for
a minimal extension of existing fitted discontinuous Galerkin software to
handle unfitted geometries. Identifying four abstract assumptions on the ghost
penalty, we derive geometrically robust a priori error and con- dition number
estimates for the Poisson boundary value problem which hold irrespective of the
particular cut configuration. Possible realizations of suitable ghost penalties
are discussed. We also demonstrate how the framework can be elegantly applied
to discretize high contrast interface problems. The theoretical results are
illustrated by a number of numerical experiments for various approximation
orders and for two and three-dimensional test problems.Comment: 35 pages, 12 figures, 2 table
-discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number
This paper develops some interior penalty -discontinuous Galerkin
(-DG) methods for the Helmholtz equation in two and three dimensions. The
proposed -DG methods are defined using a sesquilinear form which is not
only mesh-dependent but also degree-dependent. In addition, the sesquilinear
form contains penalty terms which not only penalize the jumps of the function
values across the element edges but also the jumps of the first order
tangential derivatives as well as jumps of all normal derivatives up to order
. Furthermore, to ensure the stability, the penalty parameters are taken as
complex numbers with positive imaginary parts. It is proved that the proposed
-discontinuous Galerkin methods are absolutely stable (hence, well-posed).
For each fixed wave number , sub-optimal order error estimates in the broken
-norm and the -norm are derived without any mesh constraint. The
error estimates and the stability estimates are improved to optimal order under
the mesh condition by utilizing these stability and error
estimates and using a stability-error iterative procedure To overcome the
difficulty caused by strong indefiniteness of the Helmholtz problems in the
stability analysis for numerical solutions, our main ideas for stability
analysis are to make use of a local version of the Rellich identity (for the
Laplacian) and to mimic the stability analysis for the PDE solutions given in
\cite{cummings00,Cummings_Feng06,hetmaniuk07}, which enable us to derive
stability estimates and error bounds with explicit dependence on the mesh size
, the polynomial degree , the wave number , as well as all the penalty
parameters for the numerical solutions.Comment: 27 page
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