On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection–diffusion equations

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Highlights

  • A stable DG FEM is considered which gives solutions with sharp layers.

  • Post-processing methods for reducing spurious oscillations are studied.

  • Modifications and extensions of methods from the literature are proposed.

  • First systematic numerical studies of these slope limiting techniques are performed.

  • Considerable reductions of spurious oscillations are often observed.

Abstract

A standard discontinuous Galerkin (DG) finite element method for discretizing steady-state convection–diffusion equations is known to be stable and to compute sharp layers in the convection-dominated regime, but also to show large spurious oscillations. This paper studies post-processing methods for reducing the spurious oscillations, which replace the DG solution in a vicinity of layers by a constant or linear approximation. Three methods from the literature are considered and several generalizations and modifications are proposed. Numerical studies with the post-processing methods are performed at two-dimensional examples.

Introduction

Convection–diffusion equations model the transport of a physical species like temperature (energy balance) or concentration (mass balance). In practice, the convective transport by the velocity field is usually much stronger than the diffusive transport. This situation is called convection-dominated regime. Characteristic features of the solution of a convection–diffusion equation in this regime are layers, which are thin regions where the gradient of the solution possesses a very large norm. These small spatial scales are present for both, solutions of time-dependent and steady-state convection–diffusion equations. Since solutions of the time-dependent equation often do not possess small scales with respect to time, such that the major features are the small scales with respect to space, we will concentrate here on the discussion of the steady-state problem.

In the presentation of the discretization and the numerical methods, a reactive term will be included, since the methods can be applied also to convection–diffusion–reaction equations. The steady-state convection–diffusion–reaction equation is given by εΔu+bu+cu=f in Ω,u=g on ΓD,εun=0 on ΓN,where ΩRd, d{2,3}, is a bounded domain with polyhedral Lipschitz boundary Γ=ΓDΓN with ΓDΓN=. The coefficient εR, ε>0, is the diffusion coefficient, b is the convection field, c the reaction coefficient, and f models sources. The prescribed boundary conditions on the Dirichlet boundary ΓD are denoted by g and n is the outward pointing unit normal vector on the boundary of Ω. From the physical point of view, one has to prescribe Dirichlet boundary conditions at the inflow boundary, i.e., Γ={xΓ:b(x)n(x)<0}ΓD. Mathematically, the convection-dominated regime is described by εLbL(Ω), where L is a characteristic length scale. Note that this inequality is correct with respect to the physical units.

Many discretizations are based on an underlying mesh. In the convection-dominated regime, the size of the layers is much smaller than the affordable mesh width. It is well known that standard discretizations, like the central finite difference method or the Galerkin finite element method cannot cope with this situation. They try to compute all important scales of the solution, which is not possible since the layers cannot even be represented on affordable grids. Numerical solutions computed with these schemes are globally polluted by spurious oscillations, i.e., unphysical values. One has to introduce some stabilizing component in the discretization, leading to so-called stabilized discretizations. A survey on stabilized methods, in particular in the framework of finite element methods, can be found in the monograph [1]. Since the publication of this monograph, several methods and their numerical analysis have been developed further, e.g., the analysis of algebraic flux correction (AFC) schemes, e.g., see [2].

Screening the literature on finite element methods for steady-state convection–diffusion–reaction equations, one finds that by far most publications are for conforming finite elements with Lagrangian basis functions, often of first order. But even for stabilized methods using such basis functions, there are a number of important unresolved questions, which are formulated in [3]. The most popular stabilization for conforming finite elements is the SUGP (streamline-upwind Petrov–Galerkin) or streamline-diffusion method from [4], [5], [6].

For discontinuous Galerkin (DG) methods, one can find, in comparison with conforming finite element methods, only rather few contributions for convection–diffusion equations. The first proposal of using discontinuous finite element functions, for first order hyperbolic problems, dates back to [7]. During the last decades, DG approaches gained also popularity for discretizing second order elliptic equations, e.g., see the monographs [8], [9], [10]. A big advantage of DG methods, in comparison with conforming finite element methods, is that they allow comparatively easily to use hp-adaptivity, e.g., see [11], even for polygonal or polyhedral meshes, [12]. Concerning convection–diffusion–reaction equations, error analysis can be found in [9], [13], [14], [15], [16], [17], which will be discussed in some detail at the end of Section 2. In the competitive numerical study [18], a DG method was included. On the one hand, this method computed numerical solutions with very sharp layers. But on the other hand, the solutions possessed very large over- and undershoots in a vicinity of layers.

The goal of this paper consists in studying approaches for reducing these spurious oscillations. Of course, the ideal situation in practice is a numerical solution without such oscillations, but often small spurious oscillations can be tolerated. For conforming finite elements, there are many proposals of methods for reducing spurious oscillations. A large number of these methods takes the SUPG method as basic stabilized discretization and then adds an additional term to reduce the spurious oscillations of the SUPG method, e.g., see [19] for a survey of these so-called SOLD (spurious oscillations at layers diminishing) methods and [20] for a more recent proposal. Usually, SOLD methods are nonlinear and they are used for lowest order finite elements. Another idea consists in optimizing the stabilization parameter of the SUPG method in order to reduce spurious oscillations, e.g., see [21], [22]. For DG methods, the numerical analysis in [9], [14], [15], [16], [17] shows that there is a control of the error of the streamline derivative without introducing a stabilization term of streamline-diffusion type. This situation is of advantage in practice since there is no need to choose a stabilization parameter. Because of this advantage, we will not consider the SUPG stabilization for DG methods, although it is possible to utilize it, e.g., see [17, Remark 3.1]. Thus, the optimization of a stabilization parameter is not possible. There would be still the way of adding terms like in SOLD methods, as it was done in [23]. In this paper, a suitable formulation of a discrete maximum principle for DG methods is provided, a nonlinear discretization is proposed, and the satisfaction of the discrete maximum principle is proved for one-dimensional problems. Another nonlinear scheme in combination with a piecewise linear DG method was studied in [24]. However, we decided not to pursue such approaches for the following reasons. First, we liked to consider only linear methods. It has been observed for many SOLD methods that the solution of the nonlinear problems requires often many iterations and it is time-consuming, e.g., see [19]. And second, we liked to study also methods with higher polynomial degree. A high polynomial degree is a good choice away from layers, but in a vicinity of layers a low polynomial degree is more appropriate, since Sobolev norms of the analytic solution at layers, which have an impact on the error, scale with inverse powers of the diffusion parameter and the power increases with the order. For these reasons, post-processing approaches will be considered that replace in a vicinity of layers the higher order polynomial by a low order one, whose definition utilizes a slope limiter. Such methods were proposed for constant replacements in [25], [26] and for (at most) linear replacements in [8], [27]. Note that such an easy local change of the polynomial degree is not possible for conforming finite elements. A main contribution of this paper consists in presenting some generalizations and modifications of the post-processing methods. A second main contribution is the first step of a systematic numerical investigation of these methods for steady-state convection–diffusion equations.

The paper is organized as follows. Section 2 introduces the DG method that is studied. The approaches for reducing spurious oscillations are described in Section 3. Numerical studies of these approaches at two standard problems are presented in Section 4. Finally, a summary and an outlook are provided.

Section snippets

DG methods for convection–diffusion–reaction equations

Throughout the paper, standard notations will be used for Lebesgue and Sobolev spaces and their norms. A norm of a space X is denoted by X, a seminorm by ||X, and the inner product in L2(Ω) is denoted by (,).

Starting point of a DG method is the weak formulation of (1): Find uHD,g1(Ω) such that (εu,v)+(bu+cu,v)=(f,v)vHD,01(Ω),where HD,g=vH1(Ω):v|ΓD=g,HD,0=vH1(Ω):v|ΓD=0.The Lax–Milgram theorem shows that under the conditions (μ(x))2=c12bμ0>0,ΓD,bn0 on ΓN,problem (2)

Approaches for reducing spurious oscillations in numerical solutions of convection–diffusion–reaction equations

This paper considers post-processing techniques for reducing spurious oscillations. After having computed the discrete solution with the DG method (3)–(6), the idea consists in identifying those subregions where unphysical oscillations might occur and then to reduce or clip the degree of the polynomial approximation in these subregions, thereby utilizing slope limiters. Thus, the post-processing techniques consist of the following two steps:

  • 1.

    Identify and mark cells where the numerical solution

Numerical studies

The goal of the numerical studies consists in investigating to which extent the methods presented in Section 3 reduce spurious oscillations that are introduced by the DG method (3)–(6). To this end, two standard benchmark problems for convection–diffusion equations in two dimensions are considered.

All simulations were performed with the code ParMooN, cf. [30], [31]. The implementation of the DG method was validated by first considering a smooth solution and comparing the orders of convergence

Summary and outlook

In this paper, a discretization of steady-state convection–diffusion equations by a DG finite element method was considered. Post-processing methods for reducing the size of the spurious oscillations in the obtained numerical solutions were studied: three methods from the literature and several new modifications and extensions. All these methods are computationally very efficient since they do not require to solve any linear or nonlinear system of equations.

The first step of a systematic

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