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# Staggered discontinuous Galerkin methods for the incompressible Navier-Stokes equations

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*in*Journal of Computational Physics 302:251-266 · September 2015

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- ... In the SDG method, the stability in L 2 energy is due to a spectro-consistent discretizations with the splitting of the diffusion and the convection term proposed in [19]. The stability in L 2 energy in a numerical method for the convection-diffusion problems is a kind of measure of how well the numerical solution approximates the analytical solution, and has significant effects on the quality of the numerical solution (see, for example, [7], [8]; also see Section 5.3). The ESDG method inherits the stability in L 2 energy from the SDG method due to the same spectro-consistent discretization structure. ...... In particular, when θ = 1/2, it is reduced to ESDG method (22) with a skew-symmetric discretization of the convection term proposed in Section 2. We will compare the discretizations with θ = 0, θ = 1/2 and θ = 1, and observe the advantages brought by the spectro-consistent discretization with the novel splitting of the convection term and the diffusion term. We remark that a similar experiment is performed on the SDG method for incompressible Navier-Stokes equations in [7]. In this experiment, the convection field b = (b 1 , b 2 ) is identical to Experiment 2. ...PreprintFull-text available
- Jun 2018

In this paper, we present an embedded staggered discontinuous Galerkin method for the convection-diffusion equation. The new method combines the advantages of staggered discontinuous Galerkin (SDG) and embedded discontinuous Galerkin (EDG) method, and results in many good properties, namely local and global conservations, free of carefully designed stabilization terms or flux conditions and high computational efficiency. In applying the new method to convection-dominated problems, the method provides optimal convergence in potential and suboptimal convergence in flux, which is comparable to other existing DG methods, and achieves $L^2$ stability by making use of a skew-symmetric discretization of the convection term, irrespective of diffusivity. We will present numerical results to show the performance of the method. - ... Some important examples are provided by Dolejsi et al. [34, 35, 33] for compressible gas dynamics and convection-diffusion equations, as well as the work of [53, 91] for shallow water systems. Concerning staggered-meshes, relevant research has been carried out by Chung et al. in [25, 24] for edge-based staggered meshes, and by Liu et al. for the analysis of a DG finite element method based on the alternative vertex-based staggering approach, see [72, 71]. In 1984, Berger and collaborators presented the adaptive mesh refinement (AMR) approach for finite difference and finite volume schemes for hyperbolic equations, see [11, 10]. ...Article
- Dec 2016
- COMPUT METHOD APPL M

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier-Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space-dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier-Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. The new space-time adaptive staggered DG scheme has been thoroughly verified for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier-Stokes equations on space-time adaptive meshes in two and three space dimensions. - ... A detailed introduction to the SDG method is given by [8,7]. This class of methods has been successfully applied to a wide range of problems including the Maxwell equation [10,6], acoustic wave equation [8], elastic equations [9,15], and incompressible Navier-Stokes equations [3]. In these applications, the approximate solutions obtain some nice properties such as energy conservation, low dispersion error and mass conservation. ...Article
- Oct 2016

In this paper, we present a staggered discontinuous Galerkin (SDG) method for a class of nonlinear elliptic equations in two dimensions. The SDG methods have some distinctive advantages, and have been successfully applied to a wide range of problems including Maxwell equations, acoustic wave equation, elastodynamics and incompressible Navier-Stokes equations. Among many advantages of the SDG methods, one can apply a local post-processing technique to the solution, and obtain superconvergence. We will analyze the stability of the method and derive a priori error estimates. We solve the resulting nonlinear system using the Newton's method, and the numerical results confirm the theoretical rates of convergence and superconvergence. - ... On the other hand, staggered meshes bring the advantages of reducing numerical dissipation in computational fluid dynamics [2, 3, 30] , and numerical dispersion in computational wave propagation [11, 12, 13, 14, 15, 16, 19]. Combining the ideas of DG methods and staggered meshes, a new class of staggered discontinuous Galerkin (SDG) methods for approximations of the incompressible Navier-Stokes equations was proposed [10]. The new class of SDG methods inherits many good properties, including local and global conservations, optimal convergence, and superconvergence through the use of a local postprocessing technique in [24, 25]. ...ArticleFull-text available
- Sep 2016

In this paper, we present a staggered discontinuous Galerkin immersed boundary method (SDG-IBM) for the numerical approximation of fluid-structure interaction. The immersed boundary method is used to model the fluid-structure interaction, while the fluid flow is governed by incompressible Navier-Stokes equations. One advantage of using Galerkin method over the finite difference method with immersed boundary method is that we can avoid approximations of the Dirac Delta function. Another key ingredient of our method is that our solver for incompressible Navier-Stokes equations combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations and pointwise divergence-free velocity field by a local postprocessing technique. Furthermore, energy stability is improved by a skew-symmetric discretization of the convection term. We will present numerical results to show the performance of the method. - ... In all these methods, a collocated grid was used. A novel family of DG schemes on edge-based staggered grids has been presented by Chung et al. in [45, 43, 44, 39], while an interesting analysis of DG methods on vertex-based staggered grids has been outlined in [100, 99]. For a review of spectral DG FEM schemes on collocated grids, the reader is referred to the work of Kopriva and Gassner et al. [92, 93, 11, 70, 73, 74], and references therein, while classical spectral element methods for the Navier-Stokes equations can be found in the work of Canuto et al. [21, 22, 24, 23]. ...In this paper two new families of arbitrary high order accurate spectral DG finite element methods are derived on staggered Cartesian grids for the solution of the inc.NS equations in two and three space dimensions. Pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on a spatially staggered mesh. In the first family, h.o. of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived for the pressure gradient in the momentum equation. The resulting linear system for the pressure is symmetric and positive definite and either block 5-diagonal (2D) or block 7-diagonal (3D) and can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The use of a preconditioner was not necessary. This is a rather unique feature among existing implicit DG schemes for the NS equations. In order to avoid a stability restriction due to the viscous terms, the latter are discretized implicitly. The second family of staggered DG schemes achieves h.o. of accuracy also in time by expressing the numerical solution in terms of piecewise space-time polynomials. In order to circumvent the low order of accuracy of the adopted fractional stepping, a simple iterative Picard procedure is introduced. In this manner, the symmetry and positive definiteness of the pressure system are not compromised. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary h.o. accurate in both space and time. The new numerical method has been thoroughly validated for approximation polynomials of degree up to N=11, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.
- Article
- Jan 2019
- ARCH COMPUT METHOD E

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADER-DG methods on collocated grids for compressible fluids (2) spectral semi-implicit and spectral space–time DG methods on edge-based staggered grids for the incompressible Navier–Stokes equations. In this work, the high-resolution properties of the aforementioned numerical methods are significantly enhanced within a 'cell-by-cell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADER-DG is a novel, communication-avoiding family of algorithms, which achieves high order of accuracy in time not via the standard multi-stage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an element-local predictor stage. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. In the ADER-DG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magneto-hydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand. In this context, we derived two new families of spectral semi-implicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edge-based adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor \(\theta \in [0.5,1]\) for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to \(N = 12\), using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist. - Article
- Apr 2018
- J SCI COMPUT

A hybrid staggered discontinuous Galerkin method is developed for the Korteweg–de Vries equation. The equation is written into a system of first order equations by introducing auxiliary variables. Two sets of finite element functions are introduced to approximate the solution and the auxiliary variables. The staggered continuity of the two finite element function spaces gives a natural flux condition and trace value on the element boundaries in the derivation of Galerkin approximation. On the other hand, to deal with the third order derivative term an hybridization idea is used and additional flux unknowns are introduced. The auxiliary variables can be eliminated in each element and the resulting algebraic system on the solution and the additional flux unknowns is solved. Stability of the semi discrete form is proven for various boundary conditions. Numerical results present the optimal order of \(L^2\)-errors of the proposed method for a given polynomial order. - Article
- Mar 2018
- J SCI COMPUT

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy. - Article
- Mar 2018
- J SCI COMPUT

In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence. - Conference PaperFull-text available
- Aug 2017

The goal of this paper is to establish a staggered scheme for the shallow water model. The model is taken from [1], which include energy dissipation term. The study examines the temporal stability of the uniform flow using monochromatic waves. The result shows that the roll-waves occur and stable if and only if the Froude number is less than four. We simulate roll-waves by using finite difference in staggered grid. For the linear case this method is capable validating this condition. Moreover, momentum conservative scheme is used to handle advection term thus the evolution of roll-waves can be investigated. We found that this method is able to capture shock discontinuity. From the result, we investigate the effect of Froude number, Reynold and the value of σ. We also found that for supercritical Froude number (F > 1) with low Reynold number the shocks appear faster than the higher number. - Article
- Jul 2017
- J SCI COMPUT

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid–structure interaction problems. We will use the immersed boundary method to model the fluid–structure interaction, while the fluid flow is governed by the incompressible Navier–Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid–structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier–Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method. - Article
- Sep 2016
- J COMPUT APPL MATH

A BDDC (Balancing Domain Decomposition by Constraints) algorithm for a staggered discontinuous Galerkin approximation is considered. After applying domain decomposition method, a global linear system on the subdomain interface unknowns is obtained and solved by the conjugate gradient method combined with a preconditioner. To construct a preconditioner that is robust to the coefficient variations, a generalized eigenvalue problem on each subdomain interface is solved and primal unknowns are selected from the eigenvectors using a predetermined tolerance. By the construction of the staggered discontinuous Galerkin methods, the degrees of freedom on subdomain interfaces are shared by only two subdomains, and hence the construction of primal unknowns are simplified. The resulting BDDC algorithm is shown to have the condition number bounded by the predetermined tolerance. A modified algorithm for parameter dependent problems is also introduced, where the primal unknowns are only computed in an offline stage. Numerical results are included to show the performance of the proposed method and to verify the theoretical estimate. - Article
- Aug 2016
- J SCI COMPUT

Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance. - In this paper we propose a novel arbitrary high order accurate semi-implicit space-time DG method for the solution of the three-dimensional incompressible Navier-Stokes equations on staggered unstructured curved tetrahedral meshes. As typical for space-time DG schemes, the discrete solution is represented in terms of space-time basis functions. This allows to achieve very high order of accuracy also in time, which is not easy to obtain for the incompressible Navier-Stokes equations. Similar to staggered finite difference schemes, in our approach the discrete pressure is defined on the primary tetrahedral grid, while the discrete velocity is defined on a face-based staggered dual grid. A very simple and efficient Picard iteration is used in order to derive a space-time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. From numerical experiments we find that the linear system seems to be reasonably well conditioned, since all simulations shown in this paper could be run without the use of any preconditioner. For a piecewise constant polynomial approximation in time and proper boundary conditions, the resulting system is symmetric and positive definite. This allows us to use even faster iterative solvers, like the conjugate gradient method. The proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3D test problems and by comparing the obtained numerical results with available exact analytical solutions, or with other numerical or experimental reference data.

- Article
- Aug 1999
- J COMPUT PHYS

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments. - Quadratic velocity/linear pressure Stokes elements, in: Advances in Computer Methods for Partial Differential Equations—VII
- Jan 1992
- 28-34

- D Arnold
- J Qin

D. Arnold, J. Qin, Quadratic velocity/linear pressure Stokes elements, in: Advances in Computer Methods for Partial Differential Equations—VII, IMACS, 1992, pp. 28–34. - Article
- Jul 2012
- IMA J NUMER ANAL

This paper is concerned with a staggered discontinuous Galerkin method for the curl–curl operator arising from the time-harmonic Maxwell equations. One distinctive feature of the method is that the discrete operators preserve the properties of the differential operators. Moreover, the numerical solution automatically satisfies a discrete divergence-free condition. Stability and optimal convergence of the method are analysed. Numerical experiments for smooth and singular solutions are shown to verify the optimal order of convergence. Furthermore, the method is applied to the corresponding eigenvalue problem. Numerical results for rectangular and L-shaped domains show that our method is able to produce nonspurious eigenvalues. - Discontinuous Galerkin (DG) methods are a cl ass of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any case, stability and optimal convergence of the method are proved. Moreover, a superconvergence result with respect to a discrete H 1-norm for the velocity is proved. Furthermore, a local postprocessing technique is proposed to improve the divergence free property of the velocity approximation and it is proved that the postprocessed velocity retains the original accuracy and is weakly divergence free with respect to pressure test functions. Numerical results are included to validate our theoretical estimates and to present the ability of our method for capturing singular solutions.
- Article
- Dec 2014
- COMPUT MATH APPL

In this paper, a class of FETI-DP preconditioners is developed for a fast solution of the linear system arising from staggered discontinuous Galerkin discretization of the two-dimensional Stokes equations. The discretization has been recently developed and has the distinctive advantages that it is optimally convergent and has a good local conservation property. In order to efficiently solve the linear system, two kinds of FETI-DP preconditioners, namely, lumped and Dirichlet preconditioners, are considered and analyzed. Scalable bounds C(H/h)C(H/h) and C(1+log(H/h))2C(1+log(H/h))2 are proved for the lumped and Dirichlet preconditioners, respectively, with the constant CC depending on the inf–sup constant of the discrete spaces but independent of any mesh parameters. Here H/hH/h stands for the number of elements across each subdomain. Numerical results are presented to confirm the theoretical estimates. - We show, in the framework of steady-state diffusion boundary-value problems, that the staggered discontinuous Galerkin (SDG) method [SIAM J. Numer. Anal., 47 (2009), pp. 3820–3848] can be obtained from a hybridizable discontinuous Galerkin (HDG) method [SIAM J. Numer. Anal., 47 (2009), pp. 1319–1365] by setting its stabilization function to zero at some suitably chosen element faces and by letting it go to infinity at all the remaining others. We then show that this point of view allows the SDG method to immediately acquire new properties all inherited from the HDG methods, namely, their efficient implementation (by hybridization), their postprocessings, and their superconvergence properties.
- Article
- Jan 2012
- IMA J Numer Anal

We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in Cockburn et al. (2010) A new elasticity element made for enforcing weak stress symmetry. Math. Comput., 79, 1331–1349 . In this sequel we exhibit a second family of elements that use the same matrix bubble. This second element uses a stress space smaller than the first while maintaining the same space for rotations (which are the Lagrange multipliers corresponding to a weak symmetry constraint). The space of displacements is of one degree less than the first method. The analysis, while similar to the first, requires a few adjustments as the new Fortin projector may not preserve weak symmetry, but we are able to prove optimal convergence for all the variables. Finally, we present a sufficient condition wherein a mixed method with weakly imposed stress symmetry in fact yields an exactly symmetric stress tensor approximation. - Article
- Feb 2013
- J COMPUT APPL MATH

Some electromagnetic materials exhibit, in a given frequency range, effective dielectric permittivity and/or magnetic permeability which are negative. In the literature, they are called negative index materials, left-handed materials or meta-materials. We propose in this paper a numerical method to solve a wave transmission between a classical dielectric material and a meta-material. The method we investigate can be considered as an alternative method compared to the method presented by the second author and co-workers. In particular, we shall use the abstract framework they developed to prove well-posedness of the exact problem. We recast this problem to fit later discretization by the staggered discontinuous Galerkin method developed by the first author and co-worker, a method which relies on introducing an auxiliary unknown. Convergence of the numerical method is proven, with the help of explicit inf–sup operators, and numerical examples are provided to show the efficiency of the method. - Article
- Feb 2013
- J COMPUT PHYS

In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency. - A method to solve the Navier-Stokes equations for incompressible viscous flows and the convection and diusion of a scalar is proposed in the present paper. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. A recently proposed diusion scheme with interesting theoretical and numerical properties is tested and integrated into the Navier-Stokes solver. Predictions of Poiseuille flows, backward-facing step flows and lid-driven cavity flows are then performed to validate the method. We finally demonstrate the versatility of the method by predicting buoyancy force driven flows of a Boussinesq fluid (natural convection of air in a square cavity with Rayleigh numbers of 103 and 106). © 2000 Éditions scientifiques
- Article
- Jan 1948

The production of a ‘wake’ behind solid bodies has been treated by different authors. S. Goldstein (1) and S. H. Hollingdale (2) have discussed the laminar wake behind a flat plate, while the wake of a two-dimensional grid has been treated for the turbulent case alone using L. Prandtl's ‘Mischungsweg’ theory by E. Anderlik and Gran Olsson (3), (4).(Received May 20 1947) - New implicit finite difference schemes for solving the time-dependent incompressible Navier-Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimates for the discrete solution of the methods are obtained. Employing the operator approach, some requirements on the difference operators of the scheme are formulated in order to derive a scheme which is essentially consistent with the initial differential equations. The operators of the scheme inherit the fundamental properties of the corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To derive the consistent scheme, special approximations for convective terms and div and grad operators are employed. Two variants of time discretization by the operator-splitting technique are considered and compared. It is shown that the derived scheme has a very weak restriction on the time step size. A lid-driven cavity flow has been predicted to examine the stability and accuracy of the schemes for Reynolds number up to 3200 on the sequence of grids with 21 × 21, 41 × 41, 81 × 81 and 161 × 161 grid points.
- Article
- Apr 2012
- MATH COMPUT

We provide an a priori error analysis of a wide class of finite element methods for the Stokes equations. The methods are based on the velocity gradient-velocity-pressure formulation of the equations and include new and old mixed and hybridizable discontinuous Galerkin methods. We show how to reduce the error analysis to the verification of some properties of an elementwise-defined projection and of the local spaces defining the methods. We also show that the projection of the errors only depends on the approximation properties of the projection. We then provide sufficient conditions for the superconvergence of the projection of the error in the approximate velocity. We give many examples of these methods and show how to systematically construct them from similar methods for the diffusion equation. - Article
- Dec 2003
- NUMER MATH

We analyze mixed hp-discontinuous Galerkin finite element methods (DGFEM) for Stokes flow in polygonal domains. In conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders, we prove that the hp-DGFEM leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corners. - In this paper, we discuss the results of a fourth-order, spectro-consistent discretization of the incompressible Navier-Stokes equations. In such an approach the discretization of a (skew-)symmetric operator is given by a (skew-)symmetric matrix. Numerical experiments with spectro-consistent discretizations and traditional methods are presented for a one-dimensional convection-diffusion equation. LES and RANS are challenged by giving a number of examples for which a fourth-order, spectro-consistent discretization of the Navier-Stokes equations without any turbulence model yields better (or at least equally good) results as large-eddy simulations or RANS computations, whereas the grids are comparable. The examples are taken from a number of recent workshops on complex turbulent flows.
- Numerical experiments with discretization methods on nonuniform grids are presented for the convection-diffusion equation. These show that the accuracy of the discrete solution is not very well predicted by the local truncation error. The diagonal entries in the discrete coefficient matrix give a better clue: the convective term should not reduce the diagonal. Also, iterative solution of the discrete set of equations is discussed. The same criterion appears to be favourable.
- Article
- Mar 2007
- J COMPUT PHYS

We present a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier–Stokes equations in convection-dominated flows using triangular and tetrahedral meshes. The scheme is based on a semi-explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the Stokes operator. The nonlinear term is discretized in divergence form by using the local Lax–Friedrichs fluxes; thus, local conservativity is inherent. Spatial discretization of the Stokes operator has employed both equal-order (Pk − Pk) and mixed-order (Pk − Pk−1) velocity and pressure approximations. A second-order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic Helmholtz equation for each component of the velocity and a consistent Poisson equation for the pressure. The consistent Poisson operator is replaced by an equivalent (in stability and convergence) operator, namely that arising from the interior penalty discretization of the standard Poisson operator with appropriate boundary conditions. This yields a simpler and more efficient method, characterized by a compact stencil size. - Article
- Apr 2012
- J Numer Math

This paper is concerned with the staggered discontinuous Galerkin method for convection-diffusion equations. Over the past few decades, staggered type methods have been applied successfully to many problems, such as wave propagation and fluid flow problems. A distinctive feature of these methods is that the physical laws arising from the corresponding partial differential equations are automatically preserved. Nevertheless, staggered methods for convection-diffusion equations are rarely seen in literature. It is thus the main goal of this paper to develop and analyze a class of staggered numerical schemes for the approximation of convection-diffusion equations. We will prove that our new method preserves the underlying physical laws in some discrete sense.Moreover, the stability and convergence of the method are proved. Numerical results are shown to verify the theoretical estimates. - Article
- Sep 2005
- J COMPUT PHYS

In this paper, a compact high order (up to 12th order) numerical method to solve the compressible Navier–Stokes equations will be presented. A staggered arrangement of the variables has been used. It is shown that the method is not only very accurate but numerically also very stable even in the case that not all the energy containing scales in the flow are resolved. This in contrast to standard (collocated) compact finite difference methods. Some results for a turbulent non-reacting and a reacting jet with a Reynolds number of 10,000 and a Mach number of 0.5 are reported. - Article
- Jul 1984
- J COMPUT PHYS

A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible Navier-Stokes equations. In the spectral element discretization, the computational domain is broken into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle. The implementation of the technique is demonstrated on a one-dimensional inflow-outflow advection-diffusion equation, and the method is then applied to laminar two-dimensional (separated) flow in a channel expansion. Comparisons are made with experiment and previous numerical work. - Article
- Dec 1982
- J COMPUT PHYS

The vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10,000 and meshes consisting of as many as 257 × 257 points. For Re = 1000, the (129 × 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional grid-clustering coordinate transformations. - Article
- Feb 2011
- J COMPUT PHYS

We present a hybridizable discontinuous Galerkin method for the numerical solution the incompressible Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical traces and a fully implicit time-stepping method for temporal discretization. The HDG method possesses several unique features which distinguish themselves from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, it allows for pressure, vorticity and stress boundary conditions to be prescribed on different parts of the boundary. Third, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k+1 in the L2-norm, when polynomials of degree k = 0 are used for all components of the approximate solution. And fourth, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0 in the L 2-norm. We present extensive numerical results to demonstrate the accuracy and convergence properties of the method for a wide range of Reynolds numbers and for various polynomial degrees. © 2010 by the American Institute of Aeronautics and Astronautics, Inc. - In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric\slash skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical experiments with matrices from various application areas.
- Article
- Apr 2006
- MATH COMPUT

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 vorticity-velocity formulation of 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 divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free 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 Schur-complement 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. - Article
- Jul 2009
- J SCI COMPUT

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ℘ k 3−℘ k−1 elements whereas the magnetic part of the equations is approximated by discontinuous ℘ k 3−℘ k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments. - Article
- Jan 2005
- SIAM J NUMER ANAL

RESUMEN RESUMEN We will consider both explicit and implicit fully discrete finite volume schemes for solving three-dimensional Maxwell's equations with discontinuous physical coefficients on general polyhedral domains. Stability and convergence for both schemes are analyzed. We prove that the schemes are second order accurate in time. Both schemes are proved to be first order accurate in space for the Voronoi -- Delaunay grids and second order accurate for nonuniform rectangular grids. We also derive explicit expressions for the dependence on the physical parameters in all estimates. - In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.
- In this paper, we analyze a recently d evelopedfinite volume methodfor the time- dependent Maxwell's equations in a three-dimensional polyhedral domain composed of two dielectric materials with different parameter values for the electric permittivity and the magnetic permeability. Convergence anderror estimates of the numerical scheme are establishedfor general nonuniform tetrahedral triangulations of the physical domain. In the case of nonuniform rectangular grids, the scheme converges with second order accuracy in the discrete L2-norm, despite the low regularity of the true solution over the entire domain. In particular, the finite volume method is shown to be superconvergent in the discrete H(curl; Ω)-norm. In addition, the explicit dependence of the error estimates on the material parameters is given.
- Article
- Apr 2002
- SIAM J NUMER ANAL

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L 2 -norm of the errors in the velocities and the pressure. We show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure, for any k 1. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more ecient. Numerical experiments verifying these facts are displayed. - Article
- Oct 2003
- SIAM J NUMER ANAL

We consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that Q(k) - Q(k) and Q(k) - Q(k-1) elements satisfy a generalized inf-sup condition on geometric edge and boundary layer meshes that are refined anisotropically and non quasi-uniformly towards faces, edges, and corners. The discrete inf-sup constant is proven to be independent of the aspect ratios of the anisotropic elements and to decrease as k(-1/2) with the approximation order. We also show that the generalized inf-sup condition leads to a global stability result in a suitable energy norm. - Article
- May 2011
- MATH COMPUT

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree k for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of k+1 in L(2) for any k >= 0. Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k >= 1 and with order 1 for k = 0. Numerical experiments are presented which validate the theoretical results. - Article
- Dec 1965

A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement. The primary dependent variables are the pressure and the velocity components. Also used is a set of marker particles which move with the fluid. The technique is called the marker and cell method. Some examples of the application of this method are presented. All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired. - In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in H(div; Omega) is obtained by simple, element-by-element post-processing. 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 Navier-Stokes 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.
- We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.
- Article
- Sep 1997
- SIAM J NUMER ANAL

this paper, we study the Local Discontinuous Galerkin methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta Discontinuous Galerkin methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, their high-order formal accuracy, and their easy handling of complicated geometries, for convection dominated problems. It is proven that for scalar equations, the Local Discontinuous Galerkin methods are L