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A locally conservative LDG method for the incompressible NavierStokes equations
 Math. Comp
"... Abstract. In this paper a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the ..."
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Cited by 38 (13 self)
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Abstract. In this paper a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergencefree approximate velocity in H(div; Ω) is obtained by simple, elementbyelement postprocessing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible NavierStokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers. 1.
A note on discontinuous Galerkin divergencefree solutions of the NavierStokes equations
 J. Sci. Comput
"... We present a class of discontinuous Galerkin methods for the incompressible NavierStokes equations yielding exactly divergencefree solutions. Exact incompressibility is achieved by using divergenceconforming velocity spaces for the approximation of the velocities. The resulting methods are locall ..."
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Cited by 25 (5 self)
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We present a class of discontinuous Galerkin methods for the incompressible NavierStokes equations yielding exactly divergencefree solutions. Exact incompressibility is achieved by using divergenceconforming velocity spaces for the approximation of the velocities. The resulting methods are locally conservative, energystable, and optimally convergent. We present a set of numerical tests that confirm these properties. The results of this note naturally expand the work in [15].
Local Discontinuous Galerkin Methods for HighOrder TimeDependent Partial Differential Equations
, 2010
"... Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, ..."
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Cited by 12 (1 self)
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Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, less restriction in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and local data structure and the resulting high parallel efficiency. In this paper, we give a general review of the local DG (LDG) methods for solving highorder timedependent partial differential equations (PDEs). The important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, is highlighted. Some of the applications of the LDG methods for highorder timedependent PDEs are also be discussed.
Hybridized, globally divergencefree LDG methods. Part I: The Stokes problem, submitted
, 2004
"... Abstract. We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticityvelocity formulation of the Stokes equations and by applying a new hybr ..."
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Cited by 9 (3 self)
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Abstract. We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticityvelocity formulation of the Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergencefree approximate velocity without having to construct globally divergencefree finitedimensional spaces; only elementwise divergencefree basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schurcomplement matrix for this approximate pressure is of order h−2 in the mesh size h. Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates. 1.
An Equalorder DG Method for the Incompressible NavierStokes Equations
, 2008
"... We introduce and analyze a discontinuous Galerkin method for the incompressible NavierStokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equalorder approach is ensured by a pressure stabiliz ..."
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Cited by 8 (1 self)
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We introduce and analyze a discontinuous Galerkin method for the incompressible NavierStokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equalorder approach is ensured by a pressure stabilization term. A simple elementbyelement postprocessing procedure is used to provide globally divergencefree velocity approximations. For small data, we prove the existence and uniqueness of discrete solutions and carry out an error analysis of the method. A series of numerical results are presented that validate our theoretical findings. 1
Detailed Water with Coarse Grids: Combining Surface Meshes and Adaptive Discontinuous Galerkin
"... Figure 1: A simulation in a 25 × 25 × 25 grid generates thin splashes and sheets down to 1/1200 the domain width. We present a new adaptive fluid simulation method that captures a high resolution surface with precise dynamics, without an inefficient fine discretization of the entire fluid volume. P ..."
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Cited by 3 (0 self)
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Figure 1: A simulation in a 25 × 25 × 25 grid generates thin splashes and sheets down to 1/1200 the domain width. We present a new adaptive fluid simulation method that captures a high resolution surface with precise dynamics, without an inefficient fine discretization of the entire fluid volume. Prior adaptive methods using octrees or unstructured meshes carry large overheads and implementation complexity. We instead stick with coarse regular Cartesian grids, using detailed cut cells at boundaries, and discretize the dynamics with a padaptive Discontinuous Galerkin (DG) method. This retains much of the data structure simplicity of regular grids, more efficiently captures smooth parts of the flow, and offers the flexibility to easily increase resolving power where needed without geometric refinement.
PhaseField Modeling of Droplet Movement using the Discontinuous Finite Element Method
"... Abstract In this paper, a discontinuous finite element method is presented for the fourthorder nonlinear CahnHilliard equation that models multiphase flows together with the NavierStokes equations. A flux scheme suitable for the method is proposed and analyzed together with numerical results. Th ..."
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Abstract In this paper, a discontinuous finite element method is presented for the fourthorder nonlinear CahnHilliard equation that models multiphase flows together with the NavierStokes equations. A flux scheme suitable for the method is proposed and analyzed together with numerical results. The model is applied to simulate the droplet movement and numerical results are presented.
SCIENTIA Series A: Mathematical Sciences, Vol. 1 (1988,111117
"... Abstract. In 1978 Rousseau and Sheehan showed that the bookstar Ramsey number r(K(1, 1, m), K1,n1) = 2n 1 for n> 3m 3. We show that this result is true when the star is replaced by an arbitrary tree on n vertices. I. Preliminaries. Let G, and G2 be simple graphs without isolated vertices. T ..."
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Abstract. In 1978 Rousseau and Sheehan showed that the bookstar Ramsey number r(K(1, 1, m), K1,n1) = 2n 1 for n> 3m 3. We show that this result is true when the star is replaced by an arbitrary tree on n vertices. I. Preliminaries. Let G, and G2 be simple graphs without isolated vertices. The Ramsey number r(G1,G2) is the smallest positive integer p such that coloring each edge of Kp one of two colors there is either a copy of G1 in the first color or a copy of G2 in the second color. By tradition, we shall let the colors be R (red) and B (blue) with the resulting edgeinduced subgraphs denoted ()Z) and (B) respectively. Throughout the paper a colored Kp will always refer to one in which each edge is colored red or blue. It is well known for a connected graph G2 that (1) r(G1, G2) ? (X(G1) 1) (P(G2) 1) + s(G1), P(G2)> 3 (G1), where x(GI) is the chromatic number of G1, p(G2) the order of G2, and s(G1) the chromatic surplus of G1. Here the chromatic surplus is the smallest number of vertices in a color class under any X(G1)coloring of the vertices of G1. Inequality (1) follows by
Http://www.pims.math.ca/publications/preprints/
 Math. Comp
, 2005
"... In this paper, a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: Its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the incompre ..."
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In this paper, a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: Its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergencefree approximate velocity in H(div; # is obtained by a simple, elementbyelement postprocessing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible NavierStokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers. 1.
LOCAL AND POINTWISE ERROR ESTIMATES OF THE LOCAL DISCONTINUOUS GALERKIN METHOD APPLIED TO STOKES PROBLEM
"... Abstract. We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877899] to prove pointwise estimates for the Laplace equation, we ..."
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Abstract. We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries. 1.