On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review

Abstract

An unwelcome feature of the popular streamline upwind/Petrov–Galerkin (SUPG) stabilization of convection-dominated convection–diffusion equations is the presence of spurious oscillations at layers. Since the mid of the 1980s, a number of methods have been proposed to remove or, at least, to diminish these oscillations without leading to excessive smearing of the layers. The paper gives a review and state of the art of these methods, discusses their derivation, proposes some alternative choices of parameters in the methods and categorizes them. Some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods.

Introduction

This paper is devoted to the numerical solution of the scalar convection–diffusion equation

$$-\epsilon \mathrm{\Delta }u+\mathit{b}·\nabla u=f\text{in}\Omega \text{,}u=u_b\text{on}\mathrm{\partial }\Omega \text{,}$$

where $\Omega \subset {\mathbb{R}}^d$, $d=2\text{,}3$, is a bounded domain with a polygonal (resp. polyhedral) boundary $\mathrm{\partial }\Omega$, $\epsilon >0$ is the constant diffusivity, $\mathit{b}\in W^{1\text{,}\infty }(\Omega {)}^d$ is a given convective field satisfying the incompressibility condition div b = 0, $f\in L^2(\Omega )$ is an outer source of u, and $u_b\in H^{1/2}(\mathrm{\partial }\Omega )$ represents the Dirichlet boundary condition. In our numerical tests we shall also consider less regular functions u_b.

Problem (1) describes the stationary distribution of a physical quantity u (e.g., temperature or concentration) determined by two basic physical mechanisms, namely the convection and diffusion. The broad interest in solving problem (1) is caused not only by its physical meaning just explained but also (and perhaps mainly) by the fact that it is a simple model problem for convection–diffusion effects which appear in many more complicated problems arising in applications (e.g. in various fluid flow problems).

Despite the apparent simplicity of problem (1), its numerical solution is still a challenge when convection is strongly dominant (i.e., when $\epsilon \ll |\mathit{b}|$). The basic difficulty is that, in this case, the solution of (1) typically possesses interior and boundary layers, which are small subregions where the derivatives of the solution are very large. The widths of these layers are usually significantly smaller than the mesh size and hence the layers cannot be resolved properly. This leads to unwanted spurious (nonphysical) oscillations in the numerical solution, the attenuation of which has been the subject of extensive research for more than three decades.

In this paper, we concentrate on the solution of (1) using the finite element method which proved to be a very efficient tool for the numerical solution of various boundary value problems in science and engineering. Unfortunately, the classical Galerkin formulation of (1) is inappropriate since, in case of dominant convection, the discrete solution is usually globally polluted by spurious oscillations causing a severe loss of accuracy and stability. This is not surprising since, in simple settings, the standard Galerkin finite element method is equivalent to a central finite difference discretization and it is well known that central difference approximations of the convective term give rise to spurious oscillations in convection dominated regimes (cf. e.g. Roos et al. [58]).

To enhance the stability and accuracy of the Galerkin discretization of (1) in the convection dominated regime, various stabilization strategies have been developed. Initially, these approaches imitated the upwind finite difference techniques. An important contribution to this development was made by Christie et al. [17], who showed that, in the one-dimensional case, a stabilization can be achieved using asymmetric test functions in a weighted residual finite element formulation. Choosing these test functions in a suitable way, they recovered the usual one-sided differences used for the approximation of the convective term in the finite difference method. Two-dimensional upwind finite element discretizations were derived by Heinrich et al. in [32], [33] and by Tabata [62]. Many other finite element discretizations of upwind type have been proposed later.

Like in the finite difference method, the upwind finite element discretizations remove the unwanted oscillations but the accuracy attained is often poor since too much numerical diffusion is introduced. In addition, if the flow field b is directed skew to the mesh, an excessive artificial diffusion perpendicular to the flow (crosswind diffusion) can be observed. A further important drawback is that these methods are not consistent, i.e., the solution of (1) is no longer a solution to the variational problem as it is the case for a Galerkin formulation. Consequently, the accuracy is limited to first order. Moreover, non-consistent formulations are also known to produce inaccurate or wrong solutions when f (or the time derivative in case of transient problems) is significant. It can even happen that the discrete solution is then less accurate than that one produced by the Galerkin method (cf. e.g. Brooks and Hughes [9] for a discussion on shortcomings of upwind methods).

A significant improvement came with the streamline upwind/Petrov–Galerkin (SUPG) method developed by Brooks and Hughes [9] which substantially eliminates almost all the difficulties mentioned above. In contrast with upwind methods proposed earlier, the SUPG method introduces numerical diffusion along streamlines only and hence it possesses no spurious crosswind diffusion. Moreover, the streamline diffusion is added in a consistent manner. Consequently, stability is obtained without compromising accuracy and convergence results may be derived for a wide class of finite elements. In view of its stability properties and higher-order accuracy, the SUPG method is regarded as one of the most efficient procedures for solving convection-dominated equations.

An alternative to the SUPG method is the Galerkin/least-squares method introduced by Hughes et al. [35] who observed that stabilization terms can be obtained by minimizing the square of the equation residual. A variant to this method was proposed by Franca et al. [26] using the idea of Douglas and Wang [23] to change the sign of the Laplacian in the test function. Since the SUPG method is the most popular approach, we shall restrict ourselves to this method in the following.

The SUPG method produces accurate and oscillation-free solutions in regions where no abrupt changes in the solution of (1) occur but it does not preclude spurious oscillations (overshooting and undershooting) localized in narrow regions along sharp layers. It was observed by Almeida and Silva [3] that these oscillations can even be amplified if high-order finite elements are used in these regions. This indicates that using the streamlines as upwind direction is not always sufficient. Although the remaining nonphysical oscillations are usually small in magnitude, they are not permissible in many applications. An example are chemically reacting flows where it is essential to guarantee that the concentrations of all species are nonnegative. Another example are free-convection computations where temperature oscillations create spurious sources and sinks of momentum that effect the computation of the flow field. The small spurious oscillations may also deteriorate the solution of nonlinear problems, e.g., in two-equations turbulence models or in numerical simulations of compressible flow problems, where the solution may develop discontinuities (shocks) whose poor resolution may effect the global stability of the numerical calculations.

The oscillations along sharp layers are caused by the fact that the SUPG method is neither monotone nor monotonicity preserving. Therefore, various, often nonlinear, terms introducing artificial crosswind diffusion in the neighborhood of layers have been proposed to be added to the SUPG formulation in order to obtain a method which is monotone, at least in some model cases, or which at least reduces the local oscillations. This procedure is referred to as discontinuity capturing or shock capturing. However, these names are not really appropriate in our opinion for several reasons. First, the solution of (1) does not possess shocks or discontinuities because of the presence of diffusion. Instead, steep but continuous layers are formed. Second, the position of these layers is in general already captured well by the SUPG formulation. And third, a confusion might arise with shock capturing methods which are used in the numerical simulation of compressible flows. For these reasons, we propose to call the methods spurious oscillations at layers diminishing (SOLD) methods and this name is used throughout the paper.

The literature on SOLD methods is rather extended but the various numerical tests published in the literature do not allow to draw a clear conclusion concerning their advantages and drawbacks. Therefore, the main goal of the present paper is to provide a review of the most published SOLD methods, to discuss the motivations of their derivation, to present some alternative choices of parameters and to classify them. This review is followed by a numerical comparison of these methods at two test problems whose solutions possess characteristic features of solutions of (1). The numerical results will only give a first insight into the behavior of the SOLD methods and they serve as a pre-selection to identify those SOLD methods which deserve further numerical studies. Comprehensive numerical studies will be presented in the second part of the paper. In order to keep the paper in a reasonable length, we do not consider a reaction term in Eq. (1) since special techniques are necessary if this term is dominant.

A basic problem of all SOLD methods is to find the proper amount of artificial diffusion which leads to sufficiently small nonphysical oscillations (requiring that the artificial diffusion is not ‘too small’) and to a sufficiently high accuracy (requiring that the artificial diffusion is not ‘too large’). Since the artificial diffusion is the sum of the contributions coming from the SUPG term and the SOLD term, the definition of both terms will be thoroughly presented and discussed in this paper.

Sometimes, it is claimed that the SUPG method applied on adaptively refined meshes should be preferred to SOLD methods. However, if convection strongly dominates diffusion, the spurious oscillations of the SUPG method disappear only if extremely fine meshes are used along inner and boundary layers. This leads to a high computational cost which further increases if systems of equations or transient problems are considered. The numerical comparison of the SUPG method on adaptively refined grids and several SOLD methods will be a topic of the second part of the paper. Let us also mention that a further reason for using SOLD methods is that they try to preserve the inverse monotonicity property of the continuous problem.

The plan of the paper is as follows. In the next section, we describe the usual Galerkin discretization of (1) and, in Section 3, we introduce the SUPG method. The accuracy of the SUPG method is greatly influenced by the choice of the stabilizing parameter, which is discussed in Section 4. Then, a detailed review of SOLD methods follows in Section 5. Results of our numerical tests with the SOLD methods at two typical examples are reported in Section 6. Finally, the paper is closed by Section 7 containing our conclusions and an outlook.

Throughout the paper, we use the standard notations $L^p(\Omega )$, $W^{k\text{,}p}(\Omega )$, $H^k(\Omega )=W^{k\text{,}2}(\Omega )$, $C(\overline{\Omega })$, etc. for the usual function spaces, see e.g. Ciarlet [18]. The norm and seminorm in the Sobolev space $H^k(\Omega )$ will be denoted by $\parallel ·{\parallel }_{k\text{,}\Omega }$ and $|·{|}_{k\text{,}\Omega }$, respectively. The inner product in the space $L^2(\Omega )$ or $L^2(\Omega {)}^d$ will be denoted by $(·\text{,}·)$. For a vector $\mathit{a}\in {\mathbb{R}}^d$, the symbol $|\mathit{a}|$ stands for its Euclidean norm.