AMA1 V. Dolejsi: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes
CONV P. Angot, V. Dolejsi, M.Feistauer, J. Felcman: Analysis of a combined barycentric finite volume - nonconforming finite element method for nonlinear convection - diffusion problems
HORS J. Felcman, V. Dolejsi: Investigation of order of accuracy for higher order schemes for finite volumes method on triangular meshes
THES V. Dolejsi: Combined finite volume - finite element methods for compressible flow on unstructured meshes (partly written in French)
HABIL V. Dolejsi: Adaptive higher order methods for compressible flow, Prague, 2003
TURB V. Dolejsi, M.Feistauer, J. Felcman: Algebraic turbulence models for unstructured meshes
PROF M.Feistauer, J. Felcman, V. Dolejsi: Numerical Simulation of Compressible Viscous Flow through Cascades of Profiles
VISC M.Feistauer, J. Felcman: Theory and applications of numerical schemes for nonlinear convection-diffusion problems and compressible viscous flow
AMA2 V. Dolejsi: Anisotropic Mesh Adaptation Technique for Viscous Flow Simulation
AMA3 V. Dolejsi: Anisotropic mesh adaptation and its application for scalar diffusion equations
DGM1 V. Dolejsi, M. Feistauer, C. Schwab: A Finite Volume Discontinuous Galerkin Scheme for Nonlinear Convection-Diffusion Problems
DGM2 V. Dolejsi, M. Feistauer, C. Schwab: On some aspects of the Discontinuous Galerkin finite element method for conservation laws
DGM3 V. Dolejsi, M. Feistauer and C. Schwab: On Discontinuous Galerkin Methods for Nonlinear Convection-Diffusion Problems and Compressible Flow
DGM5 V. Dolejsi, M. Feistauer: On the Discontinuous Galerkin Method for the Numerical Solution of Compressible High-Speed Flow
DGM6 V. Dolejsi: On the Discontinuous Galerkin Method for the Numerical Solution of the Euler and the Navier-Stokes Equations
DGM7 V. Dolejsi, M. Feistauer: A Semi-Implicit Discontinuous Galerkin Finite Element Method for the Numerical Solution of Inviscid Compressible Flow
DGM8 V. Dolejsi, M. Feistauer, V. Sobotikova: Analysis of the Discontinuous Galerkin Method for Nonlinear Convection--Diffusion Problems
DGM9 V. Dolejsi, M. Feistauer: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems
DGM10 V. Dolejsi, M. Feistauer: Analysis of Semi-Implicit DGFEM for Nonlinear Convection-Diffusion Problems
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Abstract We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.
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Abstract The finite volumes method (FVM) is very used in computational flow problems. The basic scheme of FVM which uses for approximation of integral over an edge of a finite volume only the values from adjacent elements is only of first order of accuracy. Many authors tried to increase the order of accuracy using a higher order reconstruction for approximation of integral over an edge. We have tested several methods of higher order for problems of compressible inviscid flows and determined the order of accuracy for Sod's channel where the analytical solution is known. The computation is performed for three different triangulation with seizes of mesh $h,\ h/2$ and $h/4$ and the determination of the order of accuracy is done using the method of the least squares.
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Abstract
The paper presents the use of an adaptive mesh refinement method
devised for flow problems with shocks. The goal is to detect
automatically the regions where the local structures such as shocks
and contact discontinuities occur. During the computational algorithm
these regions are refined with the aim to improve the quality
of the previously computed approximation in a further computation.
This can be accomplished with the aid of a suitable error indicator
or by a shock indicator.
Here the numerical study of several shock indicators is presented.
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Abstract
We present an efficient adaptive method for unstructured triangular meshes
especially developed for the numerical
simulation of viscous compressible flow.
This method uses the anisotropic mesh adaptation (AMA) technique
introduced in \cite{AMA-98}, \cite{AMA-99} with some additional
improvement reflecting physical properties of viscous high speed flow.
We summarize the AMA method with its geometrical interpretation and we
describe the numerical solver, which uses the combination of finite volume
and finite element methods.
We show that the direct application
of AMA technique is not suitable for the viscous flow simulation and
we propose some modifications, namely for capturing boundary layers and
a wake. Moreover an additional smoothing technique increasing the regularity
of the triangulation is presented.
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Abstract
We present an efficient mesh adaptation algorithm, which can be
successfully
applied to numerical solutions of a
wide range of 2D problems of physics and
engineering described by partial differential equations.
We are interested in the numerical solution of a
general boundary value
problem discretized on triangular grids.
We formulate the necessary condition for the properties of the
triangulation on which the discretization error
is below the prescribed tolerance
and control this necessary condition by the interpolation
error. For the sufficiently smooth function, we recall the strategy how
to
construct the mesh on which the interpolation error is below the prescribed
tolerance. Solving the boundary value problem we apply this strategy to
the smoothed approximate solution.
The novelty of the method lies in the
smoothing procedure that, followed by the anisotropic mesh adaptation (AMA)
algorithm, leads to the significant improvement of the numerical results.
We apply AMA to the numerical solution
of an elliptic equation, where the exact solution is known and
demonstrate the practical aspects of the adaptation procedure:
how to control the ratio between the longest and the shortest edge
of the triangulation and how to control the transition of the coarsest part
of the mesh to the finest one if the two length scales of all the triangles
are clearly different.
An example of the use of AMA for the physically relevant numerical
simulation of a geometrically challenging industrial problem
(inviscid transonic flow around NACA0012 profile) is presented.
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Abstract
The paper is concerned with the numerical solution of nonlinear
conservation laws and nonlinear convection-diffusion problems.
We discuss
two versions of this method: a) Finite volume discontinuous
Galerkin method, which is a generalization of the combined finite
volume - finite element method. Its advantage is the use of only
one mesh (in contrast to the combined finite volume - finite element
schemes). However, it is of the first order only. b) Further, the pure
discontinuous Galerkin finite element method of higher order is
considered. In
this case a new limiting is developed
to avoid spurious oscillations in the vicinity of
shocks.
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Abstract
The paper is concerned with the discontinuous Galerkin finite element
method for the numerical solution of nonlinear
conservation laws and nonlinear convection-diffusion problems with
emphasis on applications to the simulation of compressible flow.
We discuss
two versions of this method: a) Finite volume discontinuous
Galerkin method, which is a generalization of the combined finite
volume - finite element method. Its advantage is the use of only
one mesh (in contrast to the combined finite volume - finite element
schemes). However, it is of the first order only. b) Further, the pure
discontinuous Galerkin finite element method of higher order is
considered combined with a technique avoiding spurious oscillations in the
vicinity of shock waves.
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Abstract
The paper deals with an application of the discontinuous Galerkin
finite element (DG FE)
method to the numerical solution of
the system of hyperbolic equations. We extend our results from
DGM1, DGM2,
where two versions of the DG FE method were applied to
the scalar convection-diffusion equation. In order to avoid spurious
oscillations near discontinuities we develop a new limiting which
is based on the control of interelement jumps and
switches from piecewise linear to piecewise constant approximations.
Isoparametric finite elements are used near a curved boundary of nonpolygonal
computational domain in order to achieve a physically admissible and
sufficiently accurate numerical solution.
Numerical examples of transonic flow through the GAMM channel and
around the
NACA0012 airfoil are presented. Finally, we mention some theoretical
results obtained for a modified DG FE method applied to a nonlinear
convection-diffusion problem.
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Abstract
The paper deals with the use of the discontinuous Galerkin finite element
method (DG FE) for the numerical solution of compressible flows.
We start with a scalar convection-diffusion equation
and present a discretization with the aid of the antisymmetric
variant of DG FE. We also mention some theoretical results.
Then we extend the scheme to the system of the Euler and
Navier-Stokes equations. The use of order limiting in vicinity of
discontinuities and steep gradients
and the use of superparametric finite elements are discussed.
Several numerical examples are presented.
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Abstract
We deal with the numerical solution of an inviscid compressible flow
with the aid of the discontinuous Galerkin finite
element method.
Since the explicit discretization of the time derivative requires
a high restriction of the time step,
we propose a semi-implicit time discretization approach.
The homogeneity of inviscid fluxes allows us
a simple linearization of the Euler equations which leads to
a linear algebraic system. Numerical experiments performed for
the Ringleb flow problem verify a higher order of
accuracy of the presented method and
demonstrate lower CPU-time costs in comparison with an explicit method.
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Abstract
The subject-matter is the analysis
of the discontinuous Galerkin finite element method applied to a nonlinear
convection-diffusion problem. In the contrary to the standard FEM the
requirement of the conforming properties is omitted. This allows us to consider
general polyhedral elements with mutually disjoint interiors. We do not
require their convexity, but assume only that they are uniformly star-shaped.
We present an error analysis for the case of a nonsymmetric discretization of
diffusion terms. Theoretical results are accompanied by numerical experiments.
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M.Feistauer, J. Felcman, V. Dolejsi:
Numerical Simulation of
Compressible Viscous Flow through Cascades of Profiles
ZAMM, 76:301-304, 1996.
M.Feistauer, J. Felcman:
Theory and applications of numerical
schemes for nonlinear convection-diffusion problems and compressible
viscous flow
In Proc. of the MAFELAP96 Conf., Willey 1996.
V. Dolejsi:
Anisotropic Mesh Adaptation Technique for Viscous
Flow Simulation
East-West Journal of Numerical Mathematics,
9(1):1-24,2001.
V. Dolejsi:
Anisotropic mesh adaptation and its application
for scalar diffusion equations
Numerical Methods for Partial Differential Equations
(submitted)
V. Dolejsi, M. Feistauer, C. Schwab:
A Finite Volume Discontinuous
Galerkin Scheme for Nonlinear Convection-Diffusion Problems
Calcolo (submitted), also Preprint ETH Zurich, January 2001.
V. Dolejsi, M. Feistauer, C. Schwab:
On some aspects of the Discontinuous Galerkin
finite element method for conservation laws
Mathematics and Computers in Simulation (submitted),
also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2001/5, 2001.
V. Dolejsi, M. Feistauer, C. Schwab:
On Discontinuous Galerkin Methods
for Nonlinear Convection-Diffusion Problems
and Compressible Flow
Mathematica Bohemica, (to appear)
also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2001/9, 2001.
V. Dolejsi, M. Feistauer:
On the Discontinuous Galerkin Method for the
Numerical Solution of Compressible High-Speed Flow
Proceeding of ENUMATH'01 conference, (submitted)
also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2002/1, 2001.
V. Dolejsi:
On the Discontinuous Galerkin Method
for the Numerical Solution of the Euler
and the Navier-Stokes Equations
Int. J. Numer. Methods Fluids (submitted).
Also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2002/2, 2002
V. Dolejsi, M. Feistauer:
A Semi-Implicit Discontinuous Galerkin
Finite Element Method for the Numerical Solution of
Inviscid Compressible Flow
J. Comp. Phys. (submitted).
Also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2003/2, 2003
V. Dolejsi, M. Feistauer, V. Sobotikova:
Analysis of the Discontinuous Galerkin
Method for Nonlinear Convection-Diffusion Problems
Comput. Methods Appl. Mech. Eng. (submitted).
Also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2004/4, 2003
V. Dolejsi, M. Feistauer:
Error estimates of the discontinuous
Galerkin method for nonlinear nonstationary convection-diffusion
problems
Numerical Functional Analysis and Optimization, (submitted).
Also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2004/5, 2004
V. Dolejsi, M. Feistauer:
Analysis of Semi-Implicit DGFEM
for Nonlinear Convection-Diffusion Problems
Numerische Mathematik, (submitted).
Also The Preprint Series of the School of Mathematics,
Charles University Prague, No. MATH-KNM-2004/1, 2004