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## Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows (0)

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Venue: | SIAM J. Sci. Comput |

Citations: | 122 - 17 self |

### Citations

1697 |
Mixed and Hybrid Finite Element Methods
- Brezzi, Fortin
- 1991
(Show Context)
Citation Context ...is the inf-sup constant, depending only on Ω, the variational formulation above is well–posed and has a unique solution (u, p) ∈ H 1 0(Ω) 2 ×L 2 0(Ω); see Girault and Raviart [6] or Brezzi and Fortin =-=[4]-=- for details. 2.3 Meshes and Traces Throughout, we assume that the domain Ω can be subdivided into shape–regular affine meshes Th = {K} consisting of parallelograms K; for simplicity, we assume in our... |

1544 | The Mathematical Theory of Finite Element Methods. - Brenner, Scott - 1994 |

1450 |
Finite Element Methods for Navier–Stokes Equations; Theory and Algorithms
- Girault, Raviart
- 1986
(Show Context)
Citation Context ... 0�=v∈H 1 0 (Ω)2 where κ is the inf-sup constant, depending only on Ω, the variational formulation above is well–posed and has a unique solution (u, p) ∈ H 1 0(Ω) 2 ×L 2 0(Ω); see Girault and Raviart =-=[6]-=- or Brezzi and Fortin [4] for details. 2.3 Meshes and Traces Throughout, we assume that the domain Ω can be subdivided into shape–regular affine meshes Th = {K} consisting of parallelograms K; for sim... |

1328 |
Nonlinear functional analysis and its applications IV,
- Zeidler
- 1988
(Show Context)
Citation Context ...ut projecting discontinuous Galerkin solutions onto a conforming space, avoiding constraints which would otherwise lead to more restrictive approximation spaces and meshes. Theorem 8 (Aubin-Lions, in =-=[Z2]-=-). Consider Banach spaces B0, B1, B2 such that B0 ↪→ B1 is compact and B1 ↪→ B2 is continuous. Assume that B0 and B2 are reflexive. Then W := {u ∈ L2(0, T ;B0) : ∂tu ∈ L2(0, T ;B2)} is compactly embed... |

590 | A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techiques - Verfurth - 1996 |

525 | Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems,
- Arnold, Brezzi, et al.
- 2002
(Show Context)
Citation Context ...version of the finite element method. As in [10], the analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini =-=[1]-=- (see also Perugia and Schötzau [18] and Schötzau, Schwab and Toselli [20]), and employing a new hp-version decomposition result discontinuous finite element spaces; the proof of this result will be r... |

435 | Mathematical Analysis and Numerical Methods for Science and Technology. - Dautray, Lions - 2000 |

412 | Geometric Nonlinear Functional Analysis, - Benyamini, Lindenstrauss - 2000 |

211 | Introduction to adaptive methods for differential equations”, Acta Numerica,
- Eriksson, Estep, et al.
- 1995
(Show Context)
Citation Context ...norm of the actual error. In general, to ensure the reliability of the error estimator, CEST must be determined numerically for the underlying problem at hand, cf. Eriksson, Estep, Hansbo and Johnson =-=[5]-=-, for example. 5.1 Example 1 Here, we let Ω ⊂ R2 be the L–shaped domain (−1, 1) 2 \ [0, 1) × (−1, 0]; further, we select ν = 1, f = 0 and enforce appropriate inhomogeneous boundary conditions for u on... |

157 |
On the convergence of shock-capturing streamline diffusion finite element methods for conversation laws,
- Johnson, Szepessy, et al.
- 1990
(Show Context)
Citation Context ..., rn and eVun, by the difference between the dual solution z and its projection (or interpolant) zn. On the other hand, Type II a posterio bounds are in the spirit of the error ysis of Johnson et al. =-=[1]-=-, and do not depend explicitly on the dual solution. These latter error estimates are derived from Type I a posteo bounds by employing standard results from approximation theory to estimate the projec... |

101 | Discontinuous hp-finite element methods for advectiondiffusion-reaction problems.
- Houston, Schwab, et al.
- 2002
(Show Context)
Citation Context ... that the interior penalty stabilization function c has to be chosen suboptimally in k in order to ensure stability. The same problem shows up in the a priori error analysis of DG methods; see, e.g., =-=[12, 18]-=- and the references cited therein. Remark 3.2 Note that, for simplicity, the error in the approximation of the source term f is not taken into account explicitly in Theorem 3.1. However, this can be d... |

92 | A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems.
- Karakashian, Pascal
- 2003
(Show Context)
Citation Context ...or estimation for such approaches. In the context of energy norm error estimation, work has been conducted by Becker, Hansbo and Larson [2], Becker, Hansbo and Stenberg [3] and Karakashian and Pascal =-=[16]-=-, for diffusion problems; Houston, Perugia and Schötzau [7] considered the mixed DG approximation to the time–harmonic Maxwell operator and more recently in [10] we considered the mixed DG approximati... |

71 | Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation
- Jaffre, Johnson, et al.
- 1995
(Show Context)
Citation Context ...[Hi(n)] m for each ns7, we denote by v + (resp., v-) the interior (resp., exterior) trace of v on 0n. The DGFEM for (2.1) is defined as follows: find uns$t,,p such that z .dx} :0 for a]] vns$n,p, cf. =-=[2, 5]-=-, for example. Here, 7/(.,., .) denotes a numerical flux function, assumed to be Lipschitz continuous, consistent and conservative. We emphasize that the choice of the numerical flux function is compl... |

71 | A finite element method for domain decomposition with non-matching grids.
- Becker, Hansbo, et al.
- 2003
(Show Context)
Citation Context ...concerned with a posteriori error estimation for such approaches. In the context of energy norm error estimation, work has been conducted by Becker, Hansbo and Larson [2], Becker, Hansbo and Stenberg =-=[3]-=- and Karakashian and Pascal [16], for diffusion problems; Houston, Perugia and Schötzau [7] considered the mixed DG approximation to the time–harmonic Maxwell operator and more recently in [10] we con... |

53 | Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem.
- Houston, Schotzau, et al.
- 2005
(Show Context)
Citation Context ...norm error estimation, work has been conducted by Becker, Hansbo and Larson [2], Becker, Hansbo and Stenberg [3] and Karakashian and Pascal [16], for diffusion problems; Houston, Perugia and Schötzau =-=[7]-=- considered the mixed DG approximation to the time–harmonic Maxwell operator and more recently in [10] we considered the mixed DG approximation of the Stokes equations. We remark that L 2 –norm or fun... |

49 | Energy norm a posteriori error estimation for discontinuous Galerkin methods.
- Becker, Hansbo, et al.
- 2003
(Show Context)
Citation Context ...nsiderably fewer papers that are concerned with a posteriori error estimation for such approaches. In the context of energy norm error estimation, work has been conducted by Becker, Hansbo and Larson =-=[2]-=-, Becker, Hansbo and Stenberg [3] and Karakashian and Pascal [16], for diffusion problems; Houston, Perugia and Schötzau [7] considered the mixed DG approximation to the time–harmonic Maxwell operator... |

48 | A local regularization operator for triangular and quadrilateral finite elements
- Bernardi, Girault
- 1998
(Show Context)
Citation Context ...le, satisfied if there is a recovery operator R : Sp(T ,Q)→ H1(Ω) such that ‖h−1T (v −Rv)‖+ ‖Rv‖H1(Ω) . ‖v‖T . (7) For details on such recovery operators we refer, for instance, to [BO07], [KP03] and =-=[BG88]-=- and to the references therein. The bound hjTc . h j E(Tc) in (M5) is, together with (M4), the only restriction on the use of hanging nodes in Tc. In the course of the analysis another mesh condition,... |

46 | Functional analysis. 6th edition - Yosida - 1980 |

44 |
Mixed hp-DGFEM for Incompressible Flows,
- Schotzau, Schwab, et al.
- 2003
(Show Context)
Citation Context ...n rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini [1] (see also Perugia and Schötzau [18] and Schötzau, Schwab and Toselli =-=[20]-=-), and employing a new hp-version decomposition result discontinuous finite element spaces; the proof of this result will be reported in the forthcoming paper [9]. For analogous results in the h–versi... |

39 | Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case,
- Houston, Perugia, et al.
- 2005
(Show Context)
Citation Context ...of this result will be reported in the forthcoming paper [9]. For analogous results in the h–version context, we refer to the articles by Karakashian and Pascal [16] and Houston, Perugia and Schötzau =-=[8]-=-. The performance of the proposed error bound within an hp–adaptive mesh refinement procedure will be demonstrated for twodimensional problems with both smooth and singular analytical solutions. In pa... |

30 |
An hp-analysis of the local discontinuous Galerkin method for di!usion problems,
- Perugia, Schotzau
(Show Context)
Citation Context ...d. As in [10], the analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini [1] (see also Perugia and Schötzau =-=[18]-=- and Schötzau, Schwab and Toselli [20]), and employing a new hp-version decomposition result discontinuous finite element spaces; the proof of this result will be reported in the forthcoming paper [9]... |

29 |
E.: hp-discontinuous Galerkin finite element methods for hyperbolic problems: Error analysis and adaptivity.
- Houston, Senior, et al.
- 2002
(Show Context)
Citation Context ...rated by our hp–adaptive algorithm; we remark that the third root of the number of degrees of freedom is chosen on the basis of the a priori error analysis carried out in the article [21], cf., also, =-=[11]-=-. Here, we observe that the error bound over–estimates the true error by a (reasonably) consistent factor; indeed, from Figure 2(b), we see that the computed effectivity indices lie in the range betwe... |

29 | Galerkin methods for miscible displacement problems in porous media - Ewing, Wheeler - 1980 |

26 |
A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems.
- Riviere, Wheeler
- 2003
(Show Context)
Citation Context ...rk that L 2 –norm or functional error estimation for DG discretizations of elliptic problems has been analyzed by Becker, Hansbo and Stenberg [3], Kanschat and Rannacher [15], and Rivière and Wheeler =-=[19]-=-. However, the above articles have focused solely on the h–version of the DG method, where the polynomial degree is kept fixed at some low value. In this paper, we derive an energy norm a posteriori e... |

24 | em A posteriori error analysis for stabilised finite element approximations of transport problems
- Houston, Rannacher, et al.
- 2000
(Show Context)
Citation Context ...uccess, it does not provide guaranteed error control. Moreover, ad hoc refinement strategies may not provide the most economical mesh design for the control of a given error quantity of interest, cf. =-=[2, 3]-=-, for example. The aim of 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... |

22 |
A note on the design of hp-adaptive finite element methods for elliptic partial differential equations,
- Houston, Suli
- 2005
(Show Context)
Citation Context ...r h–refinement/derefinement or p–refinement/derefinement is based on estimating the local smoothness of the (unknown) analytical solution. To this end, we employ the hp–adaptive strategy developed in =-=[13]-=-, where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see, also, [14]. Here, the emphasis will be to demonst... |

22 | A time- discretization procedure for a mixed finite element approximation of miscible displacement in Porous - Jr, Ewing, et al. |

20 |
Sobolev regularity estimation for hp-adaptive finite element methods, in:
- Houston, Senior, et al.
- 2003
(Show Context)
Citation Context ... the hp–adaptive strategy developed in [13], where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see, also, =-=[14]-=-. Here, the emphasis will be to demonstrate that the proposed a posteriori error indicator converges to zero at the same asymptotic rate as the energy norm of the actual error on a sequence of non-uni... |

19 | Mathematical analysis for reservoir models
- Chen, Ewing
- 1999
(Show Context)
Citation Context ...detailed discussions concerning existence, uniqueness, and validity of a maximum principle for weak solutions of (1) to (5). While existence can be established under slightly more general assumptions =-=[CE99]-=- than the ones we employ and specify in (A1) to (A8) below, uniqueness of solutions is only known if u admits additional regularity, e.g., if u ∈ L∞(ΩT ), see [F94]. A discussion of various generalisa... |

16 |
On existence and uniqueness results for a coupled system modeling miscible displacement in porous
- Feng
- 1995
(Show Context)
Citation Context ... slightly more general assumptions [CE99] than the ones we employ and specify in (A1) to (A8) below, uniqueness of solutions is only known if u admits additional regularity, e.g., if u ∈ L∞(ΩT ), see =-=[F94]-=-. A discussion of various generalisations of our mathematical model can be found in [F02]. The major goal of this paper is to contribute to closing the gap between analytical and numerical results for... |

16 | A combined mixed finite element and discontinuous Galerkin method for miscible displacement problems in porous media.
- Sun, Riviere, et al.
- 2001
(Show Context)
Citation Context ...methods, see [SRW02, RW02]. These approaches are motivated by the convection dominated character of the concentration equation (1). While error estimates are available in the case of strong solutions =-=[SRW02]-=-, whose existence is largely open, weak accumulation of approximations at weak solutions under minimum regularity assumptions has not been investigated yet. Since the construction of solutions in [F94... |

15 |
An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian–Lagrangian localized adjoint method and mixed finite element methods
- Wang, Liang, et al.
(Show Context)
Citation Context ...dent viscosity is described by µ(c) = µ(0)(1+(M1/4−1) c)−4, where µ(0) is the viscosity of oil andM = µ(0)/µ(1) is the mobility ratio; the rescaled values for the parameters used below are taken from =-=[WLELQ00]-=-. The nonlinear system of equations arising in each time step is solved with a fixed-point iteration. Numerical Example 1 (Qualitative Behaviour). The ‘Quarter of Five Spot’ benchmark [WLELQ00] models... |

12 | Local error analysis of the interior penalty discontinuous Galerkin method for second order elliptic problems,
- Kanschat, Rannacher
- 2002
(Show Context)
Citation Context ... the Stokes equations. We remark that L 2 –norm or functional error estimation for DG discretizations of elliptic problems has been analyzed by Becker, Hansbo and Stenberg [3], Kanschat and Rannacher =-=[15]-=-, and Rivière and Wheeler [19]. However, the above articles have focused solely on the h–version of the DG method, where the polynomial degree is kept fixed at some low value. In this paper, we derive... |

12 |
Interpolation of compact operators by the methods of Calderon and Gustavsson-Peetre
- Cwikel, Kalton
- 1995
(Show Context)
Citation Context ...n with exponent θ ∈ (0, 1) is denoted by [X0, X1]θ. We remark that if X1 is reflexive then also [X0, X1]θ is a reflexive space for θ ∈ (0, 1), cf. [BL00, p. 449]. We shall make use of Theorem 11 from =-=[CK95]-=-: Theorem 7. Consider the Banach couples (X0, X1) and (Y0, Y1) and a linear mapping L : X0+X1 → Y0+Y1 such that L|X0 : X0 → Y0 is compact and L|X1 : X1 → Y1 is continuous. Suppose there is a Banach sp... |

9 | hp–Interpolation of non-smooth functions
- Melenk
(Show Context)
Citation Context ...ext, we prove an auxiliary result; to this end we first quote the following approximation property for the velocity which follows from employing the (conforming) hp–Clément interpolant constructed in =-=[17]-=-. Lemma 4.4 For any v ∈ H 1 0 (Ω)2 , there exists vh ∈ Vc h such that � K∈Th � 2 kKh −2 K �v − vh� 2 0,K + �∇(v − vh)� 2 1 1 − 0,K + �k 2 h 2 (v − vh)� 2 � 2 0,∂K ≤ CI �∇v� 2 0,Ω , with an interpolati... |

9 | Discontinuous Galerkin methods for Friedrichs systems with irregular solutions,
- Jensen
- 2004
(Show Context)
Citation Context ...r analysis shows that (M4) and (M5) as well as the projection onto Dh ensure sufficient control on the diffusive flux to guarantee convergence to a weak solution. Applying standard techniques [HSS02] =-=[J05]-=- and (15) one can rewrite Bcq to resemble the traditional, fully primal formulation (13) more closely: Bcq(ch, wh;uh) = ( uh∇h ch, wh ) + 1/2 ( (qI + qIh + q P − qPh ) ch, wh )− ∑ K∈T j ( (uh · nK)−[c... |

7 |
Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons
- Schötzau, Wihler
(Show Context)
Citation Context ...ce of meshes generated by our hp–adaptive algorithm; we remark that the third root of the number of degrees of freedom is chosen on the basis of the a priori error analysis carried out in the article =-=[21]-=-, cf., also, [11]. Here, we observe that the error bound over–estimates the true error by a (reasonably) consistent factor; indeed, from Figure 2(b), we see that the computed effectivity indices lie i... |

7 |
Some remarks about the density of smooth functions in weighted Sobolev spaces
- Piat, Cassano
- 1994
(Show Context)
Citation Context ... T ;W (u)) : ∂tv ∈ L2(0, T ;W (u)∗)}, W (u) = {v ∈ H1(Ω) : ‖D(u)1/2 ∇v‖ <∞} (35) in [CE99]. We take a different route here, partially due to the issue whether smooth functions are dense in (35), e.g. =-=[PC94]-=-, partially to remain within the framework of Bochner spaces. In light of the previous section and [F94], we select instead W := {w ∈ L2(0, T ;S) : ∂tw ∈ L2(0, T ;W p+1,4(Ω)∗)}. Theorem 9. Let (ui, pi... |

3 |
Recent developments on modeling and analysis of flow of miscible fluids in porous media, Fluid flow and transport in porous media
- Feng
(Show Context)
Citation Context ...(A8) below, uniqueness of solutions is only known if u admits additional regularity, e.g., if u ∈ L∞(ΩT ), see [F94]. A discussion of various generalisations of our mathematical model can be found in =-=[F02]-=-. The major goal of this paper is to contribute to closing the gap between analytical and numerical results for the model problem. Popular methods for solving (1) to (5) numerically employ non-conform... |

2 | Coupling locally conservative methods for single phase flow. - Riviere, Wheeler - 2002 |

1 | Mixed finite element methods for compressible miscible displacement problems in reservoir studies, Functional analysis with current applications - Ali, Pani - 1996 |

1 |
Variational convergence of IP-DGFEM
- Buffa, Ortner
- 2007
(Show Context)
Citation Context ... (M4) is, for example, satisfied if there is a recovery operator R : Sp(T ,Q)→ H1(Ω) such that ‖h−1T (v −Rv)‖+ ‖Rv‖H1(Ω) . ‖v‖T . (7) For details on such recovery operators we refer, for instance, to =-=[BO07]-=-, [KP03] and [BG88] and to the references therein. The bound hjTc . h j E(Tc) in (M5) is, together with (M4), the only restriction on the use of hanging nodes in Tc. In the course of the analysis anot... |