Elsevier

Applied Numerical Mathematics

Volume 168, October 2021, Pages 23-40
Applied Numerical Mathematics

Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems

https://doi.org/10.1016/j.apnum.2021.05.020Get rights and content

Abstract

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for second order convection-diffusion-reaction equations with singular perturbation in a new dual norm presented in [17]. The flux can be recovered in two different manners: local averaging in conforming H(div) spaces, and weighted global L2 projection onto conforming H(div) spaces. We further propose a recovery stabilization procedure, and provide completely robust a posteriori error estimators with respect to the singular perturbation parameter ε. Numerical experiments are provided to confirm theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in the diffusion parameter ε.

Introduction

Let Ω be a bounded polygonal or polyhedral domain in Rd (d=2 or 3) with Lipschitz boundary Γ=ΓDΓN, where ΓDΓN=. Consider the following stationary singularly perturbed convection-diffusion-reaction problem{Lu:=εΔu+au+bu=finΩ,u=0onΓD,εun=gonΓN, where 0<ε1 is the singular perturbation parameter, a(W1,(Ω))d, bL(Ω), fL2(Ω), gL2(ΓN), n is the outward unit normal vector to Γ, and equation (1) is scaled such that ||a||L=O(1) and ||b||L=O(1). The Dirichlet boundary ΓD has a positive (d1)-dimensional Lebesgue measure, which includes the inflow boundary {xΩ:a(x)n<0}. Assume that there are two nonnegative constants β and cb, independent of ε, satisfyingb12aβand||b||L(Ω)cbβ. Note that if β=0, then b0 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 H1/2(Ω)-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 H(div) conforming finite element (FE) spaces, such as the Raviart-Thomas (RT) or the Brezzi-Douglas-Marini (BDM) spaces, by global weighted L2-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 L2-projection and H(div) 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 H(div) 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 RT0 spaces, the implicit recovery based on the global weighted L2-projection in RT0 and BDM1 spaces, and the implicit H(div) 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 H(div) 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 L2-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 H(div) 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 (,)ω the L2 inner-product on ω, and by <,>ω (and <,>e) the L2 inner-product of the duality parings between H1(ω) and H1(ω) (and H1/2(e) and H1/2(e) for eγ, respectively). Throughout this paper, standard notations for Lebesgue and Sobolev spaces and their norms and seminorms are used [1]. In particular, for 1p< and 0<s<1, the norm of the fractional Sobolev space Ws,p(ω) is defined as||v||Ws,p(ω):={||v||Lp(ω)p+ωω|v(x)v(y

Flux recovery

Introducing the flux variable σ=εu, the variational form of the flux reads: find σH(div;Ω) such that(ε1σ,τ)=(u,τ)τH(div;Ω). In this paper, we use standard RT0 or BDM1 elements to recover the flux, which areRT0:={τH(div;Ω):τ|KP0(K)d+xP0(K)KTh} andBDM1:={τH(div;Ω):τ|KP1(K)dKTh}, respectively.

Let uh be the solution to (5) and V be RT0 or BDM1. We recover the flux by solving the following problem: find σνV such that(ε1σν,τ)=(uh,τ)τV.

Remark 3.1

For L2-projection recovery with V=BDM1, we

A posteriori error estimates

For KTh and eE, define weights αK:=min{hKε1/2,β1/2,hK1/2} and αe:=min{he1/2ε1/2,ε1/4β1/4,1}, and residualsRK:=f+εΔuhauhbuhandR˜K:=fσhauhbuh, where σh is the implicit or explicit recovered flux. LetΦ=(KThαK2(||RK||K2+||R˜K||K2)+||ε1/2uh+ε1/2σh||2)1/2. We have the following error estimates.

Theorem 3

Let u and uh be the solutions to (4) and (5), respectively. Let Φ be defined in (17). If σh=σˆRT0(uh) is the recovered flux obtained by the explicit approximation (14), then|||uuh|||Φ+

Analysis of efficiency

Let τ=εuh. For each eEΩ, define the edge/face residual along e byRe:={Je(τ)ifeΓ,g+τneifeΓN,0ifeΓD, where Je(τ) is defined in (25). Letosch:=(KThαK2||DK||K2+eΓNαe2||De||e2)1/2, be an oscillation of data, where DK=RKΠ1RK for every KTh, and De=ReΠ0Re for each eΓN. Here Πk is an L2-projection operator into the space of polynomials of total degree not greater than k in d or d1 variables, whichever is appropriate. The following efficient estimate is found in [17].

Lemma 4

Let u and uh be the

A stabilization H(div) recovery

Let uhVh be the approximation of the solution u to (1). A stabilization H(div) recovery procedure is to find σTV such that(ε1σT,τν)+KThγK(σT,τν)K=(uh,τν)+KThγK(fauhbuh,τν)KτνV, where γK is a stabilization parameter to be determined in below. Recalling the exact flux σ=εu, define the approximation error of the flux recovery by||σσT||B,Ω2:=(ε1(σσT),σσT)+KThγK((σσT),(σσT))K.

Theorem 10

Suppose that the exact flux σH(div;Ω). 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 RT0 element is used for the three proposed recovery approaches. Similar numerical results will be obtained if the BDM1 element is used for recovering.

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  • Cited by (0)

    1

    This research was partially supported by the Natural Science Foundation of Chongqing (cstc2018jcyjAX490) and the Education Science Foundation of Chongqing (KJZD-K201900701).

    2

    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.

    3

    This research was partially supported by the National Natural Science Foundation of China under grants 11871092 and NSAF U1930402.

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