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## A CLASS OF DISCONTINUOUS PETROV-GALERKIN METHODS. PART I: THE TRANSPORT EQUATION

Citations: | 41 - 13 self |

### Citations

2573 |
The Finite Element Method for Elliptic Problems, North-Holland,
- Ciarlet
- 1978
(Show Context)
Citation Context ...es (15) with w = 0 and µ = 0, then integrating into K from the outflow boundary along the ⃗ β-direction, we find that v ≡ 0. □ One can now define a new finite element formally in the sense of Ciarlet =-=[7]-=- as follows. Let Σ denote the set of linear functionals ℓq, ℓη defined by ℓq(v) = ( ⃗ β · ⃗ ∇ v, q)K, for all q ∈ Pp(K), ℓη(v) = 〈v, η〉F , for all η ∈ Pp+1(F ), for all faces F ⊆ ∂outK. Then, with Σ a... |

593 |
A posteriori error estimation in finite element analysis,
- Ainsworth, Oden
- 2000
(Show Context)
Citation Context ...e using the same algorithm except for a different load to the local problem, comp. (1.17). Notice a similarity with the implicit a-posteriori error estimation techniques using equilibriated residuals =-=[20]-=-. The main difference lies in the fact that, in the case of the DPG method, there is no need to equlibriate the element residuals. Scope of this paper. The present work is a continuation of [1] and fo... |

430 |
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework,
- Cockburn, Shu
- 1989
(Show Context)
Citation Context ...rical flux of Peter Lax whose definition requires typically stronger regularity assumptions on the solution than implied by the energy setting, see e.g. the original contributions of Cockburn and Shu =-=[10, 11]-=-. For instance, use of traces for the standard DG method for the convection problem is illegal at the continuous level - the L2 functions do not admit traces (see the discussion in [8]). The numerical... |

300 | The local discontinuous galerkin method for time-dependent convection-diffusion systems
- Cockburn, Shu
- 1998
(Show Context)
Citation Context ... alternatives for advective problems and their perturbations. In particular, for our primary example of the convection dominated diffusion problem, DG methods have remained an active area of research =-=[13, 15, 20, 21, 22, 23, 31]-=-. While earlier research concentrated on tailoring numerical fluxes for upwinding, maintaining high approximation order, and addition of stabilization terms [4, 15, 22], there is a resurgence of inter... |

257 |
Triangular mesh methods for the neutron transport equation
- Reed, Hill
- 1973
(Show Context)
Citation Context ...thods. Since our contribution fits in the latter, we shall now review previous works in this category in detail. The well known first papers proposing and analyzing the original DG method for (1) are =-=[14, 15, 18]-=-. To distinguish this method from our DG method, we will call the original DG method the “upwind DG method” and denote it by UDG, while we call ours the “discontinuous Petrov-Galerkin method” and deno... |

203 |
On a finite element method for solving the neutron transport equation, in: Mathematical Aspects of finite Elements in Partial Differential Equations, edited by C.
- Lesaint, Raviart
- 1974
(Show Context)
Citation Context ...thods. Since our contribution fits in the latter, we shall now review previous works in this category in detail. The well known first papers proposing and analyzing the original DG method for (1) are =-=[14, 15, 18]-=-. To distinguish this method from our DG method, we will call the original DG method the “upwind DG method” and denote it by UDG, while we call ours the “discontinuous Petrov-Galerkin method” and deno... |

178 |
Survey lectures on the mathematical foundations of the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations
- Babuska, Aziz
- 1972
(Show Context)
Citation Context ...v in the unit balls of U and V , resp. (we will tacitly use such notations throughout). Additionally we assume that {v ∈ V : b(u, v) = 0 ∀ u ∈ U} = {0}. (2.4) Under these conditions, it is well known =-=[2]-=- that problem (2.1) has a unique solution for any ℓ ∈ V ′ (primes are used to denote dual spaces). Let us also recall the famous result of Babuˇska on approximation of (2.1) by the following Galerkin ... |

144 |
A multi-dimensional upwind scheme with no crosswind diffusion
- Huges, Brooks
- 1979
(Show Context)
Citation Context ...y by Mitchell and Griffiths in the context of finite difference methods – see [30] – but it was fully realized in the famous Streamline Upwind Petrov Galerkin Method (SUPG) of Hughes et al., see e.g. =-=[24, 25]-=-. Demkowicz was supported in part by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615], and by a research contract with Boeing. Gopalakrishnan ... |

142 |
Finite element methods for linear hyperbolic problems”, Computer methods in applied mechanics and engineering,
- Johnson, Nävert, et al.
- 1984
(Show Context)
Citation Context ...y by Mitchell and Griffiths in the context of finite difference methods – see [30] – but it was fully realized in the famous Streamline Upwind Petrov Galerkin Method (SUPG) of Hughes et al., see e.g. =-=[24, 25]-=-. Demkowicz was supported in part by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615], and by a research contract with Boeing. Gopalakrishnan ... |

127 |
An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation.
- Johnson, Pitkaranta
- 1986
(Show Context)
Citation Context ...thods. Since our contribution fits in the latter, we shall now review previous works in this category in detail. The well known first papers proposing and analyzing the original DG method for (1) are =-=[14, 15, 18]-=-. To distinguish this method from our DG method, we will call the original DG method the “upwind DG method” and denote it by UDG, while we call ours the “discontinuous Petrov-Galerkin method” and deno... |

105 | First-order system least squares for secondorder partial differential equations: Part II
- Cai, Manteuffel, et al.
(Show Context)
Citation Context ...rse of the Riesz map defined by RV : V ′ ↦→ V by (RV (ℓ), v)V = 〈ℓ, v〉V . Thus our method is indeed of the least squares type. It is also related to the so-called “negative-norm least squares method” =-=[7, 8, 10]-=-. To see this, first note that (2.8) implies that T = RV ◦ B where B : U ↦→ V ′ is the operator generated by the bilinear form , i.e., 〈Bu, v〉V = b(u, v) for all u ∈ U, v ∈ V . Then (2.12) can equival... |

102 |
The finite difference method in partial differential equations
- Mitchell, Griffiths
- 1980
(Show Context)
Citation Context ..., in particular, refers to the original contribution of Petrov [33]. The idea of Petrov-Galerkin method was exploited early by Mitchell and Griffiths in the context of finite difference methods – see =-=[30]-=- – but it was fully realized in the famous Streamline Upwind Petrov Galerkin Method (SUPG) of Hughes et al., see e.g. [24, 25]. Demkowicz was supported in part by the Department of Energy [National Nu... |

100 | Unified hybridization of discontinuous Galerkin mixed and continuous Galerkin methods for second order elliptic problems.
- Cockburn, Gopalakrishnan, et al.
- 2009
(Show Context)
Citation Context ...s, possess provably good approximation properties. Our method also introduces a new flux unknown on the element interfaces. This is in line with the recent developments on hybridized DG (HDG) methods =-=[9]-=-. HDG methods that extend the ideas in [9] to the case of convection can be found in recent works [10, 16]. These methods are constructed by defining a independent flux variable on the element □4 L. ... |

100 | Discontinuous hp finite element methods for advection–diffusion problems,
- Houston, Schwab, et al.
- 2000
(Show Context)
Citation Context .... You might say, that the stabilization comes from the element interior. Remarkably, the bubble techniques may lead to the same discrete equations as the SUPG method [15]. Except for DPG methods (see =-=[16, 17, 18]-=-, majority of stabilized methods is restricted to low order elements. The common property of all stabilized methods is a proper modification of the bilinear and linear form at the discrete level witho... |

99 |
Error-bounds for the finite element method.
- Babuska
- 1971
(Show Context)
Citation Context ...t follows from the construction of the optimal test functions that the discrete inf-sup constant inf ‖uhp‖E=1 sup ‖vhp‖=1 |b(uhp, vhp)| (1.14) 2 is also equal to one. Consequently, Babuška’s Theorem =-=[3]-=- implies that ‖u− uhp‖E ≤ inf whp∈Uhp ‖u− whp‖E (1.15) i.e., the method delivers the best approximation error in the energy norm. The construction of optimal test functions implies also that the globa... |

87 | A least-squares approach based on a discrete minus one inner product for first order systems. submitted
- Bramble, Lazarov, et al.
(Show Context)
Citation Context ...rse of the Riesz map defined by RV : V ′ ↦→ V by (RV (ℓ), v)V = 〈ℓ, v〉V . Thus our method is indeed of the least squares type. It is also related to the so-called “negative-norm least squares method” =-=[7, 8, 10]-=-. To see this, first note that (2.8) implies that T = RV ◦ B where B : U ↦→ V ′ is the operator generated by the bilinear form , i.e., 〈Bu, v〉V = b(u, v) for all u ∈ U, v ∈ V . Then (2.12) can equival... |

81 |
Choosing bubbles for advection–diffusion problems,
- Brezzi, Russo
- 1994
(Show Context)
Citation Context ...e decades. In particular, in the case of “bubble methods”, stabilization comes from introducing and condensing out additional bubble shape functions (i.e. functions vanishing on the element boundary) =-=[13, 14]-=-. You might say, that the stabilization comes from the element interior. Remarkably, the bubble techniques may lead to the same discrete equations as the SUPG method [15]. Except for DPG methods (see ... |

79 |
Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion–Reaction and Flow Problems. 2nd ed.
- Roos, Stynes, et al.
- 2008
(Show Context)
Citation Context ...tion of “robust” discretization schemes for singularly perturbed problems in general, and for convection-dominated problems in particular, has been a subject of an intensive research for decades, see =-=[25]-=- for a recent extensive review of related work. Even the definition of what we mean by the ”robustness” is not easy. Intuitively, given a mesh, we expect the numerical solution to behave uniformly in ... |

69 |
Arbitrary discontinuities in finite elements.
- Belytschko, Moes, et al.
- 2001
(Show Context)
Citation Context ...d indeed, need not even be continuous. We are also not the first to consider such functions with discontinuities within a finite element. Such elements are routinely used in X-FEM and similar methods =-=[2]-=- for difficult simulations like crack propagation. However, we use discontinuities solely for stability purposes, and solely in test spaces. Our trial spaces, being standard polynomial spaces, possess... |

62 |
A stable finite element for the Stokes equation, Calcolo
- DN, Brezzi, et al.
- 1984
(Show Context)
Citation Context ...e decades. In particular, in the case of “bubble methods”, stabilization comes from introducing and condensing out additional bubble shape functions (i.e. functions vanishing on the element boundary) =-=[13, 14]-=-. You might say, that the stabilization comes from the element interior. Remarkably, the bubble techniques may lead to the same discrete equations as the SUPG method [15]. Except for DPG methods (see ... |

56 | A multilevel discontinuous Galerkin method.
- Gopalakrishnan, Kanschat
- 2003
(Show Context)
Citation Context ... alternatives for advective problems and their perturbations. In particular, for our primary example of the convection dominated diffusion problem, DG methods have remained an active area of research =-=[13, 15, 20, 21, 22, 23, 31]-=-. While earlier research concentrated on tailoring numerical fluxes for upwinding, maintaining high approximation order, and addition of stabilization terms [4, 15, 22], there is a resurgence of inter... |

53 |
hp-Finite Element Methods for Singular Perturbations,
- Melenk
- 2003
(Show Context)
Citation Context ...of) reset δ = δ/2 end of loop through mesh refinements The strategy reflects our old experiments with boundary layers [21], and the rigorous approximability results of Schwab and Suri [22] and Melenk =-=[23]-=- on optimal hp discretizations of boundary layers: we proceed with h-refinements until the diffusion scale is reached and then continue with p-refinements. We start with an easy case of = 10−3. Fi... |

46 |
An implicit highorder hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations,”
- Nguyen, Peraire, et al.
- 2009
(Show Context)
Citation Context ...he element interfaces. This is in line with the recent developments on hybridized DG (HDG) methods [9]. HDG methods that extend the ideas in [9] to the case of convection can be found in recent works =-=[10, 16]-=-. These methods are constructed by defining a independent flux variable on the element □4 L. DEMKOWICZ AND J. GOPALAKRISHNAN interfaces which can solved for first, after which the internal variables ... |

44 | The p and hp versions of the finite element method for problems with boundary layers,
- Schwab, Suri
- 1996
(Show Context)
Citation Context ...ew Ndof = old Ndof) reset δ = δ/2 end of loop through mesh refinements The strategy reflects our old experiments with boundary layers [21], and the rigorous approximability results of Schwab and Suri =-=[22]-=- and Melenk [23] on optimal hp discretizations of boundary layers: we proceed with h-refinements until the diffusion scale is reached and then continue with p-refinements. We start with an easy case... |

37 |
hp-version discontinuous Galerkin methods for hyperbolic conservation laws,
- Bey, Oden
- 1996
(Show Context)
Citation Context ...m [8] that “the mechanisms that induce the loss of h 1/2 in the order of convergence of the L 2 -norm of the error are not very well known yet”. An hp analysis of the UDG scheme was first provided in =-=[3]-=-. They considered the regular perturbation (3) under the assumption that 0 < c0 ≤ α(⃗x) ∀⃗x ∈ Ω (4) Because of this assumption, they are able to control the L 2 (Ω)-norm of the solution. They also int... |

34 |
A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation
- PETERSON
- 1991
(Show Context)
Citation Context ... the optimal rate of convergence of the UDG method [8, 19] with respect to h. Nonetheless, on general meshes, the suboptimal rate of convergence cannot be improved, as shown by a numerical example in =-=[17]-=- using a particular quasiuniform mesh and a smooth exact solution. To express the sentiment of many, we quote from [8] that “the mechanisms that induce the loss of h 1/2 in the order of convergence of... |

33 |
Error-bounds for finite element method
- Babuˇska
- 1971
(Show Context)
Citation Context ...) family of methods for the transport equation which has provably optimal convergence rates on very general meshes. The design of our method is guided by a generalization of Céa lemma due to Babuˇska =-=[1, 5]-=-. We only need a simple version of the result, which we now describe using the following notations (all our spaces are over R): Let X, Xh ⊂ X, and Vh be Banach spaces and let ah(·, ·) be a bilinear fo... |

31 |
Special finite element methods for a class of second order elliptic problems with rough coefficients
- Babuˇska, Caloz, et al.
- 1994
(Show Context)
Citation Context ... space that has an (almost) optimal stability constant in the energy norm. The concept of determining optimal test functions numerically is not new and resurfaces from time to time in literature, see =-=[3, 11, 18, 19, 35, 36]-=-, just to mention a few. Its connection with the least squares Galerkin method is also previously known [8, Remark 2.4] (and we will explain this at length in Section 2). The novelty in our approach i... |

30 | An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables. - Demkowicz, Oden - 1986 |

26 |
Stabilized hp-finite element methods for firstorder hyperbolic problems,
- Houston, Schwab, et al.
- 2000
(Show Context)
Citation Context ... Because of this assumption, they are able to control the L 2 (Ω)-norm of the solution. They also introduced a stabilization parameter into the original upwind DG method. A few years later, the paper =-=[12]-=- extended the results of [3] in several directions, providing a unified theory for an hp version of the streamline diffusion method, as well as the upwind DG method. Their analysis did not assume (4),... |

26 |
and hp-finite element methods. Numerical Mathematics and Scientific Computation.
- Schwab
- 1998
(Show Context)
Citation Context ...ind diffusion for the DPG method with the optimal test functions. 5. Concluding remarks 5.1. Summary. We proved optimal error estimates for uh in h and p. Using the known best approximation estimates =-=[20]-=- for a quasiuniform mesh of meshsize h, we obtain as a corollary of Theorem 3.3 that C‖u − uh‖Ω ≤ hs1 p s1 |u|H s 1 (Ω) + ( 1 | ⃗ )hs2−1/2 β|h1/2 ps2−1/2 |u|Hs2 (Ω) (50) with some C independent of h a... |

24 |
Mixed and Hybrid Finite Element Methods, no
- Brezzi, Fortin
- 1991
(Show Context)
Citation Context ...) family of methods for the transport equation which has provably optimal convergence rates on very general meshes. The design of our method is guided by a generalization of Céa lemma due to Babuˇska =-=[1, 5]-=-. We only need a simple version of the result, which we now describe using the following notations (all our spaces are over R): Let X, Xh ⊂ X, and Vh be Banach spaces and let ah(·, ·) be a bilinear fo... |

22 |
An optimal-order error estimate for the discontinuous Galerkin method
- Richter
- 1988
(Show Context)
Citation Context ... best approximation error of the finite element space is O(h s ). For some special classes of meshes however, many authors have observed (and proved) the optimal rate of convergence of the UDG method =-=[8, 19]-=- with respect to h. Nonetheless, on general meshes, the suboptimal rate of convergence cannot be improved, as shown by a numerical example in [17] using a particular quasiuniform mesh and a smooth exa... |

19 |
Estimation of iterated matrices, with application to the Von Neumann condition
- Kato
- 1960
(Show Context)
Citation Context ...n‖U. wn∈Un In the early paper [2], we find this result with the constant M/γn replaced by 1 + M/γn. It is now well known that in the Hilbert space setting, one can remove the “1” in the constant (see =-=[27]-=-, [37, Theorem 2], or [16, Theorem 4]). The starting point of our analysis is the definition of an alternative norm, which we call the energy norm on the trial space U. It is defined by ‖u‖E From this... |

17 |
The discontinuous Petrov-Galerkin method for elliptic problems.
- Bottasso, Micheletti, et al.
- 2002
(Show Context)
Citation Context ...ve discontinuities inside the mesh elements. Many researchers have put the above principle to good use. In fact, even the abbreviation we use for our new method “DPG method”, has been previously used =-=[4, 6]-=- for other methods. The theme in these works is the search for stable test spaces using bubbles or other polynomials. Our test space functions, in contrast, need not be polynomial on an element, and i... |

17 | A class of discontinuous Petrov–Galerkin methods. Part IV: the optimal test norm and time-harmonic wave propagation in 1D
- Zitelli, Muga, et al.
- 2011
(Show Context)
Citation Context ...wledge at the time of the WONAPDE meeting (January 2010) where this work was presented. Since then we have made a significant progress on both theoretical and numerical sides. Among other results, in =-=[26]-=- we came up with a fundamental concept of the optimal test norm. With the use of the optimal test norm, the corresponding energy norm is close to the original norm in the trial space, i.e. the L2-norm... |

16 | Pasciak, A new approximation technique for div-curl systems
- Bramble, E
(Show Context)
Citation Context ...rse of the Riesz map defined by RV : V ′ ↦→ V by (RV (ℓ), v)V = 〈ℓ, v〉V . Thus our method is indeed of the least squares type. It is also related to the so-called “negative-norm least squares method” =-=[7, 8, 10]-=-. To see this, first note that (2.8) implies that T = RV ◦ B where B : U ↦→ V ′ is the operator generated by the bilinear form , i.e., 〈Bu, v〉V = b(u, v) for all u ∈ U, v ∈ V . Then (2.12) can equival... |

13 | An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM), Adv Water Res 25
- Russell, Celia
- 2002
(Show Context)
Citation Context ... space that has an (almost) optimal stability constant in the energy norm. The concept of determining optimal test functions numerically is not new and resurfaces from time to time in literature, see =-=[3, 11, 18, 19, 35, 36]-=-, just to mention a few. Its connection with the least squares Galerkin method is also previously known [8, Remark 2.4] (and we will explain this at length in Section 2). The novelty in our approach i... |

11 | Babuska ⇔ Brezzi ? - Demkowicz - 2006 |

9 | 2008a) Optimal convergence of the original DG method for the transport-reaction equation on special meshes
- COCKBURN, DONG, et al.
(Show Context)
Citation Context ... best approximation error of the finite element space is O(h s ). For some special classes of meshes however, many authors have observed (and proved) the optimal rate of convergence of the UDG method =-=[8, 19]-=- with respect to h. Nonetheless, on general meshes, the suboptimal rate of convergence cannot be improved, as shown by a numerical example in [17] using a particular quasiuniform mesh and a smooth exa... |

9 | Some observations on Babuˇska and Brezzi theories - Xu, Zikatanov |

9 |
A new finite element method for solving compressible Navier-Stokes equations based on an operator splitting method and hp adaptivity
- Demkowicz, Oden, et al.
- 1990
(Show Context)
Citation Context ...-refine the element endif endif end of loop through elements if (new Ndof = old Ndof) reset δ = δ/2 end of loop through mesh refinements The strategy reflects our old experiments with boundary layers =-=[21]-=-, and the rigorous approximability results of Schwab and Suri [22] and Melenk [23] on optimal hp discretizations of boundary layers: we proceed with h-refinements until the diffusion scale is reache... |

7 |
Further considerations on residual free bubbles for advective-diffusive equations
- Brezzi, Franca, et al.
- 1998
(Show Context)
Citation Context ... on the element boundary) [13, 14]. You might say, that the stabilization comes from the element interior. Remarkably, the bubble techniques may lead to the same discrete equations as the SUPG method =-=[15]-=-. Except for DPG methods (see [16, 17, 18], majority of stabilized methods is restricted to low order elements. The common property of all stabilized methods is a proper modification of the bilinear a... |

6 |
Analysis of the DPG method for the Poisson problem
- Demkowicz, Gopalakrishnan
(Show Context)
Citation Context ... enriched space. The individual element contributions eK serve as element error indicators and form the basis for adaptive mesh refinement. Comparison with DG and other Stabilized Methods We refer to =-=[1, 8, 9]-=- for a partial review of related DG literature. The DG methods are based on the concept of numerical flux of Peter Lax whose definition requires typically stronger regularity assumptions on the soluti... |

6 |
A robust a posteriori error estimate for hp-adaptive dg methods for convection–diffusion equations
- Zhu, Schötzau
- 2010
(Show Context)
Citation Context .... You might say, that the stabilization comes from the element interior. Remarkably, the bubble techniques may lead to the same discrete equations as the SUPG method [15]. Except for DPG methods (see =-=[16, 17, 18]-=-, majority of stabilized methods is restricted to low order elements. The common property of all stabilized methods is a proper modification of the bilinear and linear form at the discrete level witho... |

5 | Applied Functional Analysis for Science and Engineering - Oden, Demkowicz - 1996 |

5 |
A multiscale formulation of the discontinuous Petrov-Galerkin method for advectivediffusive problems
- Bottasso, Micheletti, et al.
(Show Context)
Citation Context ...as little practical value. The situation changes, if the above methodology is applied in context of the Discontinuous Petrov Galerkin (DPG) method introduced by Bottasso, Micheletti, Sacco and Causin =-=[4, 5, 6, 7]-=-. The starting point of the method is a system of first-order differential equations. The equations are multiplied by test functions, integrated over the domain, and then integrated by parts. Contrary... |

4 | A discontinuous Petrov-Galerkin method with Lagrangian multipliers for second order elliptic problems
- Causin, Sacco
(Show Context)
Citation Context ...ve discontinuities inside the mesh elements. Many researchers have put the above principle to good use. In fact, even the abbreviation we use for our new method “DPG method”, has been previously used =-=[4, 6]-=- for other methods. The theme in these works is the search for stable test spaces using bubbles or other polynomials. Our test space functions, in contrast, need not be polynomial on an element, and i... |

4 |
Mesh orientation and anisotropic refinement in the streamline diffusion method, in Finite element methods (Jyväskylä
- Zhou, Rannacher
- 1993
(Show Context)
Citation Context ... method, this observation was made in [12], where DG is compared with the streamline diffusion method. While the streamline diffusion solution can also be improved by aligning meshes with shock lines =-=[21]-=-, its convergence rate remains limited by the regularity of theDPG METHOD 23 (a) Log-log plots of ‖u − uh‖ for various h and p values (b) Log-log plots of flux errors ‖φ − φh‖ 1 β ,Eh Figure 6. The h... |

4 | hp-finite element methods for advection-diffusion-reaction problems - Discontinuous |

4 |
Variational Methods in
- Mikhlin
- 1964
(Show Context)
Citation Context ...o known as the Bubnov-Galerkin method), in which one uses different trial and test spaces. For a detailed historical review on the Galerkin method, we refer to the introduction in the book of Mikhlin =-=[29]-=- who, in particular, refers to the original contribution of Petrov [33]. The idea of Petrov-Galerkin method was exploited early by Mitchell and Griffiths in the context of finite difference methods – ... |

3 | A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusion problems.
- Egger, Schoberl
- 2009
(Show Context)
Citation Context ...he element interfaces. This is in line with the recent developments on hybridized DG (HDG) methods [9]. HDG methods that extend the ideas in [9] to the case of convection can be found in recent works =-=[10, 16]-=-. These methods are constructed by defining a independent flux variable on the element □4 L. DEMKOWICZ AND J. GOPALAKRISHNAN interfaces which can solved for first, after which the internal variables ... |

2 | Analysis of finite element methods for linear hyperbolic problems, in Discontinuous Galerkin Methods
- Falk
(Show Context)
Citation Context ... (1) is obtained by the addition of a small viscosity term with second derivatives. This is harder to analyze. Within the domain of finite element methods for (1), there are two broad categories (see =-=[11]-=- for a review). One is the very popular The work of the first author was supported by DOE through Predictive Engineering Science (PECOS) Center at ICES (PI: Bob Moser), and by a research contract with... |

2 |
Application of the Method of Galerkin to a Problem Involving the Stationary Flow of a Viscous Fluid
- PETROV
- 1940
(Show Context)
Citation Context ...al and test spaces. For a detailed historical review on the Galerkin method, we refer to the introduction in the book of Mikhlin [29] who, in particular, refers to the original contribution of Petrov =-=[33]-=-. The idea of Petrov-Galerkin method was exploited early by Mitchell and Griffiths in the context of finite difference methods – see [30] – but it was fully realized in the famous Streamline Upwind Pe... |

1 |
Flux-upwind stabilization of the discontinuous Petrov-Galerkin formulation with Lagrange multipliers for advection-diffusion problems, M2AN
- Causin, Sacco, et al.
(Show Context)
Citation Context ... by using a DG framework. We are not the first to consider Petrov-Galerkin methods in the DG framework. In fact, schemes christened “DPG methods” (discontinuous Petrov-Galerkin methods) already exist =-=[5, 6, 11, 12]-=-. In [5], a method of the mixed form for the Laplace’s equation with test spaces enriched with bubbles are considered. An error analysis, higher order generalizations and hybridization aspects of such... |

1 |
On Galerkin methods
- Wendland
- 1996
(Show Context)
Citation Context ... space that has an (almost) optimal stability constant in the energy norm. The concept of determining optimal test functions numerically is not new and resurfaces from time to time in literature, see =-=[3, 11, 18, 19, 35, 36]-=-, just to mention a few. Its connection with the least squares Galerkin method is also previously known [8, Remark 2.4] (and we will explain this at length in Section 2). The novelty in our approach i... |

1 | proceedings of the 4th USENIX Workshop on Hot Topics in Cloud Computing (HotCloud) 2012. [2] The rise of RaaS: the Resource-as-a-Service - In |

1 |
Computing with hp Finite Elements. I.One
- Demkowicz
- 2006
(Show Context)
Citation Context ...f the second type, the order for L2-conforming triangles is always p − 2, with p being the order of the corresponding H1-conforming element. For details on exact sequences and hp discretizations, see =-=[24]-=-. 4. 2D Numerical Experiments We consider now the 2D convection-dominated diffusion problem 1 σ −∇u = 0 in Ω −divσ + div(βu) = f in Ω (4.56) accompanied with boundary condition for velocity u, u ... |