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514
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 120 (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 97 (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 66 (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 53 (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 (13 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.
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 45 (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.
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.
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 45 (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.
A superconvergent LDGhybridizable Galerkin method for secondorder elliptic problems
 Math. Comp
"... Abstract. We identify and study an LDGhybridizable Galerkin method, which is not an LDGmethod, for secondorder elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuousGalerkinmethods using polynomials of degree k ≥ 0 for both the po ..."
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Cited by 37 (10 self)
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Abstract. We identify and study an LDGhybridizable Galerkin method, which is not an LDGmethod, for secondorder elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuousGalerkinmethods using polynomials of degree k ≥ 0 for both the potential as well as the flux, the order of convergence in L2 of both unknowns is k + 1. Moreover, both the approximate potential as well as its numerical trace superconverge in L2like norms, to suitably chosen projections of the potential, with order k+2. This allows the application of elementbyelement postprocessing of the approximate solution which provides an approximation of the potential converging with order k+2 in L2. The method can be thought to be in between the hybridized version of the RaviartThomas and that of the BrezziDouglasMarini mixed methods. 1.
A DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR TIME DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS WITH HIGHER ORDER DERIVATIVES
"... Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applie ..."
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Cited by 37 (9 self)
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Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)th order of accuracy when using piecewise kth degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation. 1.