Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems
Introduction
Let Ω be a bounded polygonal or polyhedral domain in ( or 3) with Lipschitz boundary , where . Consider the following stationary singularly perturbed convection-diffusion-reaction problem where is the singular perturbation parameter, , , , , n is the outward unit normal vector to Γ, and equation (1) is scaled such that and . The Dirichlet boundary has a positive -dimensional Lebesgue measure, which includes the inflow boundary . Assume that there are two nonnegative constants β and , independent of ε, satisfying Note that if , then and there is no reaction term in (1).
Adaptive finite element methods (FEMs) are a class of numerical algorithms to approximate partial differential equations (PDEs) arising in scientific and engineering applications. A posteriori error estimation uses numerical solutions and data to assess the quality of the discrete approximation and improve it adaptively. Error estimators in literature can be categorized into three classes: residual based, gradient recovery based, and hierarchical bases based. Each approach has certain advantages.
It is a challenging task to design a robust a posteriori error estimator for singularly perturbed problems. By a robust estimator, we mean the multiplicative constants in the upper and lower bounds for the error are independent of the size of the convection or reaction relative to the diffusion [29].
It is crucial to employ an appropriate norm, since the efficiency of a robust estimator depends fully on the norm. This problem was first investigated by Verfürth [30], in which both upper and lower bounds for error estimator in an ε-weighted energy norm was proposed. It was shown that the estimator was robust when the local Péclet number is not very large. Generalization of this approach can be found in, e.g., [8], [21], [24], [27]. He considered also robust estimators in an ad hoc norm in [31], which is an ε-weighted energy norm plus a dual norm of the derivative of the convection term. In [28], Sangalli pointed out that the ad hoc norm may not be appropriate for problem (1). To resolve the problem, he proposed a residual-type a posteriori estimator for convection-diffusion problems in one-dimension, which is robust up to a logarithmic factor in the global Péclet number. Recently, John and Novo [20] used the natural SUPG norm in the a priori analysis to develop a robust a posteriori error estimator. In [2], fully computable and guaranteed upper bounds are established for discretization errors in the energy norm. Very recently, Tobiska and Verfürth [29] presented robust residual a priori error estimates for a wide range of stabilized FEMs. Du and Zhang [17] proposed a dual norm, which is induced by an ε-weighted energy norm and a related -norm. A uniformly robust a posteriori estimator for numerical errors was provided, though the dual norm couldn't be directly computed. We shall remark that this dual norm is equivalent to the natural SUPG norm in [20] when the convection term is dominated and the high order terms in the natural SUPG norm are negligible; cf. Section 2.2 for details. We further refer the readers to a recent review article on adaptive methods for problems with layers [25].
It is well known that a posteriori error estimators of the recovery type possess many appealing properties, such as simplicity, universality, and asymptotical exactness, which have resulted in their popular adoptions, especially in the engineering community (cf., e.g., [3], [4], [7], [11], [33], [34], [35], [36]). However, when applied to problems with practical challenges, such as interface singularities, shock-like fronts/discontinuities, and boundary or interior layers, they lose not only asymptotical exactness but also efficiency on relatively coarse meshes. They may lead to overrefinement and fail to reduce global errors (see [6], [22], [23]). To overcome this difficulty, Cai and Zhang [9] developed a global recovery approach for the interface problem. The flux is recovered in conforming finite element (FE) spaces, such as the Raviart-Thomas (RT) or the Brezzi-Douglas-Marini (BDM) spaces, by global weighted -projection or local averaging. The resulting recovery-based (implicit and explicit) estimators are measured in the standard energy norm, which turned out to be robust if the diffusion coefficient is monotonically distributed. This approach was further extended for solving general second-order elliptic PDEs [10]. The implicit estimators based on the -projection and recovery procedures were proposed to be the sum of the error in the standard energy norm and the error of the recovered flux in a weighted norm. The global reliability and the local efficiency bounds for these estimators were established. For singularly perturbed problems, the estimators developed in [9], [10] are not robust with respect to ε. To the authors' knowledge, no robust recovery-type estimators have been proposed for such problems in the literature.
Motivated by aforementioned works, we extend the approach in [17] and develop robust recovery-based a posteriori error estimators for the SUPG method for singularly perturbed problems. Three procedures will be applied, which are the explicit recovery through local averaging in spaces, the implicit recovery based on the global weighted -projection in and spaces, and the implicit recovery procedure. Numerical errors will be measured in a dual norm presented in [17]. Note that these estimators are different from those in [17], since the jump in the normal component of the flux consists of a recovery indicator in addition to an incidental term (see Remark 4.1). Our recovery procedures are also different from those in [9], [10] (e.g., the flux recovery based on the local averaging provides an appropriate choice of weight factor, the recovery procedure develops a stabilization technique, the recovery procedures treat Neumann boundary conditions properly, etc.). Moreover, the estimators developed here are uniformly robust with respect to ε and β.
The rest of this paper is organized as follows. In Section 2, we introduce the variational formulation and some preliminary results. In Section 3, we define an implicit flux recovery procedure based on the -projection onto the lowest-order RT or BDM spaces, and an explicit recovery procedure through local averaging in the lowest-order RT spaces. In Section 4, for implicit and explicit recovery procedures, we give a reliable upper bound for the numerical error in a dual norm developed in [17]. Section 5 is devoted to the analysis of efficiency of the estimators. Here, the efficiency is in the sense that the converse estimate of upper bound holds up to different higher order terms (usually oscillations of data) and a different multiplicative constant depends only on the shape of the mesh. We show that the estimators are completely robust with respect to ε and β. In Section 6, we define a stabilization recovery procedure, and develop a robust recovery-based estimator by using the main results of Sections 4 and 5. Numerical tests are provided in Section 7 to support the theoretical results.
Section snippets
Variational formulations
For any subdomain ω of Ω with a Lipschitz boundary γ, denote by the inner-product on ω, and by (and ) the inner-product of the duality parings between and (and and for , respectively). Throughout this paper, standard notations for Lebesgue and Sobolev spaces and their norms and seminorms are used [1]. In particular, for and , the norm of the fractional Sobolev space is defined as
Flux recovery
Introducing the flux variable , the variational form of the flux reads: find such that In this paper, we use standard or elements to recover the flux, which are and respectively.
Let be the solution to (5) and be or . We recover the flux by solving the following problem: find such that
Remark 3.1 For -projection recovery with , we
A posteriori error estimates
For and , define weights and , and residuals where is the implicit or explicit recovered flux. Let We have the following error estimates.
Theorem 3 Let u and be the solutions to (4) and (5), respectively. Let Φ be defined in (17). If is the recovered flux obtained by the explicit approximation (14), then
Analysis of efficiency
Let . For each , define the edge/face residual along e by where is defined in (25). Let be an oscillation of data, where for every , and for each . Here is an -projection operator into the space of polynomials of total degree not greater than k in d or variables, whichever is appropriate. The following efficient estimate is found in [17].
Lemma 4 Let u and be the
A stabilization recovery
Let be the approximation of the solution u to (1). A stabilization recovery procedure is to find such that where is a stabilization parameter to be determined in below. Recalling the exact flux , define the approximation error of the flux recovery by
Theorem 10 Suppose that the exact flux . The following a priori error bound for the
Numerical experiments
In this section, we demonstrate the performance of our a posteriori error estimators in two example problems. Linear conforming finite element is used to approximate the exact solutions. The lowest order element is used for the three proposed recovery approaches. Similar numerical results will be obtained if the element is used for recovering.
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Cited by (0)
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This research was partially supported by the Natural Science Foundation of Chongqing (cstc2018jcyjAX490) and the Education Science Foundation of Chongqing (KJZD-K201900701).
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This research was partially supported by the US National Science Foundation grant DMS 1217268, the National Natural Science Foundation of China under grant 11428103, and a University Research Grant of Texas A&M International University.