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Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows
 SIAM J. Sci. Comput
"... this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is ..."
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Cited by 122 (17 self)
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this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is devoted to the derivation of weighted a posteriori error bounds for linear functionals of the solution. Finally, in Section 4 we present some numerical examples to demonstrate the performance of the resulting adaptive finite element algorithm. 2 Model problem and discretisation Given an open bounded polyhedral domain fl in lI n, n _> 1, with boundary 0fl, we consider the following problem: find u: fl > lI m, m _> 1, such that div(u) = 0 in , (2.1) where, ,: m __> mxn is continuously differentiable. We assume that the system of conservation laws (2.1) may be supplemented by appropriate initial/boundary conditions. For example, assuming that B(u, y) := EiL1 biVu'(u) has m real eigenvalues and a complete set of linearly independent eigenvectors for all y = (yl,, Yn) C n; then at inflow/outflow boundaries, we require that B(u, n)(u g) = 0, where n denotes the unit outward normal vector to 0fl, B(u, n) is the negative part of B(u, n) and g is a (given) realvalued function. To formulate the discontinuous Galerkin finite element method (DGFEM, for short) for (2.1), we first introduce some notation. Let 7 = {n} be an admissible subdivision of fl into open element domains n; here h is a piecewise constant mesh function with h(x) = diam(n) 2 Houston e al. when x is in element n. For p Iq0, we define the following finite element space n,  {v [L()]": vl [%(n)] " Vn }, where Pp(n) denotes the set of polynomials of degree at most p over n. Given that v [Hi(n)] m for each n...
UNIFIED HYBRIDIZATION OF DISCONTINUOUS GALERKIN, MIXED AND CONTINUOUS GALERKIN METHODS FOR SECOND ORDER ELLIPTIC PROBLEMS
"... Abstract. We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixeddual finite element methods including hybridized mixed, continuous Galerkin, nonconforming and a new wide cla ..."
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Cited by 100 (18 self)
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Abstract. We introduce a unifying framework for hybridization of finite element methods for second order elliptic problems. The methods fitting in the framework are a general class of mixeddual finite element methods including hybridized mixed, continuous Galerkin, nonconforming and a new wide class of hybridizable discontinuous Galerkin methods. The main feature of the methods in this framework is that their approximate solutions can be expressed in an elementbyelement fashion in terms of an approximate trace satisfying a global weak formulation. Since the associated matrix is symmetric and positive definite, these methods can be efficiently implemented. Moreover, the framework allows, in a single implementation, the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled. Finally, the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying new, extremely localized and simple mortaring techniques, as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom. 1.
Reduced basis method for finite volume approximations of parametrized linear evolution equations
 M2AN, Math. Model. Numer. Anal
"... The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RBmethods have so far mainly been applied to finite element sch ..."
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Cited by 64 (24 self)
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The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RBmethods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for general evolution problems and the derivation of rigorous aposteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized. This is the basis for a rapid online computation in case of multiplesimulation requests. We introduce a new offline basisgeneration algorithm based on our a posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convectiondiffusion problem demonstrate the efficient applicability of the approach. 1
SpaceTime Discontinuous Galerkin Finite Element Methods
"... In these notes an introduction is given to spacetime discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The spacetime DG discretization is explained in detail, including the definition of the numerical fluxes and stabilizati ..."
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Cited by 52 (4 self)
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In these notes an introduction is given to spacetime discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The spacetime DG discretization is explained in detail, including the definition of the numerical fluxes and stabilization operators necessary to maintain stable and nonoscillatory solutions. In addition, a pseudotime integration method for the solution of the algebraic equations resulting from the DG discretization and the relation between the spacetime DG method and an arbitrary Lagrangian Eulerian approach are discussed. Finally, a brief overview of some applications to aerodynamics is given.
A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and NavierStokes Equations
"... In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and NavierStokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuit ..."
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Cited by 49 (12 self)
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In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and NavierStokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, but offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method. I.
SubCell Shock Capturing for Discontinuous Galerkin Methods, AIAA paper
, 2006
"... A shock capturing strategy for higher order Discontinuous Galerkin approximations of scalar conservation laws is presented. We show how the original explicit artificial viscosity methods proposed over fifty years ago for finite volume methods, can be used very effectively in the context of high ord ..."
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Cited by 46 (5 self)
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A shock capturing strategy for higher order Discontinuous Galerkin approximations of scalar conservation laws is presented. We show how the original explicit artificial viscosity methods proposed over fifty years ago for finite volume methods, can be used very effectively in the context of high order approximations. Rather than relying on the dissipation inherent in Discontinuous Galerkin approximations, we add an artificial viscosity term which is aimed at eliminating the high frequencies in the solution, thus eliminating Gibbstype oscillations. We note that the amount of viscosity required for stability is determined by the resolution of the approximating space and therefore decreases with the order of the approximating polynomial. Unlike classical finite volume artificial viscosity methods, where the shock is spread over several computational cells, we show that the proposed approach is capable of capturing the shock as a sharp, but smooth profile, which is typically contained within one element. The method is complemented with a shock detection algorithm which is based on the rate of decay of the expansion coefficients of the solution when this is expressed in a hierarchical orthonormal basis. For the Euler equations, we consider and discuss the performance of several forms of the artificial viscosity term. I.
Multigrid Solution for HighOrder Discontinuous Galerkin . . .
, 2004
"... A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. Thi ..."
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Cited by 45 (16 self)
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A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the pmultigrid solver, which relies on an elementline Jacobi smoother. The elementline Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2D scalar convectiondiffusion shows that the elementline Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higherorder is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.
Stabilization mechanisms in discontinuous Galerkin finite element methods
 Comput. Methods Appl. Mech. Engrg
, 2006
"... In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabili ..."
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Cited by 44 (6 self)
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In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes–Franca type residualbased stabilizations.
Spacetime discontinuous Galerkin method for nonlinear water waves, preprint for
 J. Comput. Phys.,
, 2006
"... Summary. A spacetime discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The spacetime DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolut ..."
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Cited by 44 (13 self)
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Summary. A spacetime discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The spacetime DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution. The dispersion and dissipation errors of the scheme are investigated and the algorithm is demonstrated with the simulation of waves generated by a wave maker.
Plane wave discontinuous Galerkin methods
, 2007
"... Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency ..."
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Cited by 40 (8 self)
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Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. Among them the ultra weak variational formulation (UWVF) of Cessenat and Despres [O. Cessenat and B. Despres, Application of an ultra weak variational formulation of elliptic PDEs to the twodimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299.]. We identify the UWVF as representative of a class of Trefftztype discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the hversion of these plane wave discontinuous Galerkin methods. To that end, we develop new inverse and approximation estimates for plane waves in two dimensions and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion. Key words. Wave propagation, finite element methods, discontinuous Galerkin methods, plane waves, ultra weak variational formulation, duality estimates, numerical dispersion AMS subject classifications. 65N15, 65N30, 35J05