Stability and a posteriori error analysis of discontinuous Galerkin methods for linearized elasticity

We consider discontinuous Galerkin finite element methods for the discretization of linearized elasticity problems in two space dimensions. Inf–sup stability results on the continuous and the discrete level are provided. Furthermore, we derive upper and lower a posteriori error bounds that are robust with respect to nearly incompressible materials, and can easily be implemented within an automatic mesh refinement procedure. The theoretical results are illustrated with a series of numerical experiments.     

A finite volume discontinuous Galerkin scheme¶for nonlinear convection–diffusion problems

A popular method for the discretization of conservation laws is the finite volume (FV) method, used extensively in CFD, based on piecewise constant approximation of the solution sought. However, the FV method has problems with the approximation of diffusion terms. Therefore, in several works [17–19, 1, 12, 16, 2], a combination of the FV and FE methods is used. To this end, it is necessary to construct various combinations of simplicial FE meshes with suitable associated FV grids. This is rather complicated from the point of view of the mesh refinement, particularly in 3D problems [20, 21]. It is desirable to use only one mesh. The combination of FV and FE discretizations on the same triangular grid is proposed in [39]. Another possibility is to use the DG method (see [7] or [9] (and the references there) for a general survey). Here we shall use a compromise between the DG FE method and the FV method using piecewise linear discontinuous finite elements over the grid ? _h and piecewise constant approximation of convective terms on the same grid. Dedicated to Professor Ivo Babuška on the occasion of his 75th birthday Received: May 2001 / Accepted: September 2001     

An adaptive discontinuous Galerkin method for viscoplastic analysis

This paper describes an h-adaptive, space-time discontinuous Galerkin finite element method for quasi-static viscoplastic response in a time-varying domain. The equations of equilibrium and the evolution equations in the viscoplastic material model comprise an elliptic/hyperbolic system. We focus here on two aspects of the model: stabilization of the hyperbolic subproblem and residual-based error estimates and adaptive algorithms for viscoplastic analysis.     

Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a Gårding type inequality. Optimal order L ₂ norm a priori error estimates are derived for an adjoint consistent interior penalty method.     

The direct discontinuous Galerkin (DDG) viscous flux scheme for the high order spectral volume method

The direct discontinuous Galerkin (DDG) method was developed by Liu and Yan to discretize the diffusion flux. It was implemented for the discontinuous Galerkin (DG) formulation. In this paper, we perform four tasks: (i) implement the direct discontinuous Galerkin (DDG) scheme for the spectral volume method (SV) method, (ii) design and implement two variants of DDG (called DDG2 and DDG3) for the SV method, (iii) perform a Fourier type analysis on both methods when solving the 1D diffusion equation and combine the above with a non-linear global optimizer, to obtain modified constants that give significantly smaller errors (in 1D), (iv) use the above coefficients as starting points in 2D. The dissipation properties of the above schemes were then compared with existing flux formulations (local discontinuous Galerkin, Penalty and BR2). The DDG, DDG2 and DDG3 formulations were found to be much more accurate than the above three existing flux formulations. The accuracy of the DDG scheme is heavily dependent on the penalizing coefficient for the odd ordered schemes. Hence a loss of accuracy was observed even for mildly non-uniform grids for odd ordered schemes. On the other hand, the DDG2 and DDG3 schemes were mildly dependent on the penalizing coefficient for both odd and even orders and retain their accuracy even on highly irregular grids. Temporal analysis was also performed and this yielded some interesting results. The DDG and its variants were implemented in 2D (on triangular meshes) for Navier–Stokes equations. Even the non-optimized versions of the DDG displayed lower errors than the existing schemes (in 2D). In general, the DDG and its variants show promising properties and it indicates that these approaches have a great potential for higher dimension flow problems.     

Expression templates implementation of continuous and discontinuous Galerkin methods

Efficiency and flexibility are often mutually exclusive features in a code. This still prompts a large part of the Scientific Computing community to use traditional procedural languages. In the last years, however, new programming techniques have been introduced allowing for a high level of abstraction without loss of performance. In this paper we present an application of the Expression Templates technique introduced in (Veldhuizen in Expression templates. C++ Report magazine, vol 7, pp 26–31, 1995) to the assembly step of a finite element computation. We show that a suitable implementation, such that the compiler has the role of parsing abstract operations, allows for user-friendliness. Moreover, it gains in performance with respect to more traditional techniques for achieving this kind of abstraction. Both the cases of conforming and discontinuous Galerkin finite element discretization are considered. The proposed implementation is finally applied to a number of problems entailing different kind of complications.     

Application of a discontinuous Galerkin finite element method to special relativistic hydrodynamic models

A Runge–Kutta discontinuous Galerkin (RKDG) finite element method is proposed for solving the special relativistic hydrodynamic (SRHD) equations and as a limiting case the ultra-relativistic hydrodynamic (URHD) equations. The latter model is obtained by ignoring the rest-mass energy when the internal energy of fluid particles is sufficiently large. Several test problems of SRHD and URHD models are carried out. For validation, the results of DG-method are compared with the staggered central scheme. The numerical results verify the accuracy of the proposed method qualitatively and quantitatively.     

Coupling of continuous and discontinuous Galerkin methods for transport problems

In this paper, we formulate a coupled discontinuous/continuous Galerkin method for the numerical solution of convection–diffusion (transport) equations, where convection may be dominant. One motivation for this approach is to use a discontinuous method where the solution is rough, e.g., in regions of high gradients, and use a continuous method where the solution is smooth. In this approach, the domain is decomposed into two regions, and appropriate transmission conditions are applied at the interface between regions. In one region, a local discontinuous Galerkin method is applied, and in the other region a standard continuous Galerkin method is employed. Stability and a priori error estimates for the coupled method are derived, and numerical results in one space dimension are given for smooth problems and problems with sharp fronts.     

Bubble stabilized discontinuous Galerkin methods on conforming and non-conforming meshes

The aim of this paper is to discuss the properties of the bubble stabilized discontinuous Galerkin method with respect to mesh geometry. First we show that on certain non-conforming meshes the bubble stabilized discontinuous Galerkin method allows for hanging nodes/edges. Then we consider the case of conforming meshes and present a post-processing algorithm based on the Crouzeix-Raviart method to obtain the Bubble Stabilized Discontinuous Galerkin (BSDG) method. Although finally the post-processed solution does not coincide with the BSDG-solution in general, they satisfy the same (approximation) properties and are close to each other. Moreover, the post-processed solution has continuous flux over the edges.     

欧拉方程的隐式间断有限元算法研究

针对Euler方程,设计了适合间断Galerkin有限元方法的LU-SGS、GMRES以及修正LU-SGS隐式算法。采用Roe通量以及Van Albada限制器技术实现了经典LU-SGS、GMRES算法,引入高阶项误差补偿,发展了修正LU-SGS算法。以NACA0012、RAE2822翼型为例验证分析了算法的可靠性和高效性。结果表明修正LU-SGS算法存储量较少,程序实现方便,而且计算效率是LU-SGS算法的2.5倍以上,接近于循环GMRES算法。     

The discontinuous Galerkin finite element method for the 2D shallow water equations

A high-order finite element method, total variational diminishing (TVD) Runge–Kutta discontinuous Galerkin method is investigated to solve free-surface problems in hydraulic dynamics. Some cases of circular dam and rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method. The numerical simulations are given on structured rectangular mesh for regular domain and on unstructured triangular mesh for irregular domain, respectively.     

Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics

A discontinuous Galerkin (DG) finite element method (FEM) for the solution of linear elasto-dynamic problems is revisited and modified. The new DG FEM is based on a method originally proposed by [Comput. Methods Appl. Mech. Engrg. 84 (1990) 327] and recently adapted by [Comput. Methods Appl. Mech. Engrg. 191(46) (2002) 5315] for the solution of dynamic solid–solid phase transitions. As the FEM formulations in both the cited works have been found not to be unconditionally stable in cases where the underlying FEM grid is completely unstructured, this paper offers a modification of these formulations yielding a single-field DG FEM that is unconditionally stable without any restrictions on the grid structure. Furthermore, an energy conserving variant of the formulation is also suggested.     

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.     

Simulation of a viscous compressible flow past a circular cylinder with high-order discontinuous Galerkin methods

This paper is devoted to the simulation of viscous compressible flows with high-order accurate discontinuous Galerkin methods on bidimensional unstructured meshes. The formulation for the solution of the Navier–Stokes equations is due to Oden et al. [An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, The University of Texas at Austin, 1997; J Comput Phys 1998;146:491–519]. It involves a weak imposition of continuity conditions on the state variables and on fluxes across interelement boundaries. It does not make use of any auxiliary variables and then the cost for the implementation is reasonable. The method is coupled with a limiting procedure recently developed by the authors to suppress oscillations near large gradients. The limiter is totally free of problem dependence and maintains the convergence order for errors measured in the L¹-norm. This paper presents detailed numerical results of a viscous compressible flow past a circular cylinder at a Reynolds number of 100 for the cases of subsonic and supersonic regimes. The proposed simulations suggest that the method is very robust and is able to produce very accurate results on unstructured meshes.     

Continuous and discontinuous finite element methods for a peridynamics model of mechanics

In contrast to classical partial differential equation models, the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems. The peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges. Through a series of simple, one-dimensional computational experiments, we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates. One issue addressed is the effect of the relative sizes of the horizon and the grid. For problems with smooth solutions, we find that continuous and discontinuous piecewise-linear approximations result in the same accuracy as that obtained by continuous piecewise-linear approximations for classical models. Piecewise-constant approximations are less robust and require the grid size to be small with respect to the horizon. We then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions. In case a grid point is placed at the locations of the jump discontinuities, such results are directly obtained. In the general case, we show that such results can be obtained through a simple, automated, abrupt, local refinement of elements containing the discontinuity. In order to reduce the number of degrees of freedom while preserving accuracy, we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities.     

A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems

Four discontinuous Galerkin (DG) methods are proposed to enrich the resource of modeling elasticity problems as they are volume locking-free and allow hanging nodes in meshing. A detailed finite element formulation of these DG methods is presented. For implementation, we coded a three-dimensional nodal-based DG program in which the conventional nodal-based pure displacement finite element codes are fully exploited. The robustness and accuracy of each DG method are demonstrated and compared with mixed methods through solving a rubber beam problem. The coupled use of DG with continuous elements is proposed for some practical applications.     

Mixed discontinuous Galerkin analysis of thermally nonlinear coupled problem

A stabilized mixed discontinuous Galerkin (SMDG) method based on Brezzi–Hughes–Marini–Masud [F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Masud, Mixed discontnuous Galerkin methods for Darcy flow, J. Sci. Comput. 22 (2005) 119–145.] is proposed to solve a thermally coupled nonlinear elliptic system modeling a large class of engineering problems. A fixed point algorithm is adopted to solve the nonlinear systems. Convergence analysis and error estimates are presented for equal order linear or bilinear discontinuous Lagrangian finite element interpolations for all fields. Numerical results are presented confirming the predicted convergence rates and illustrating the performance of the proposed formulation solving problems with globally stable and blowing up solutions.     

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.