Robust a Posteriori Error Estimates for HDG method for Convection-Diffusion Equations

It is well known that solutions of (1.1) may develop layers (cf. [24, 29]). In particular, the solutions may have singular interior layer of width $O\left(\sqrt{ϵ}\right)$ or outflow layer of width $O\left(ϵ\right)$. Standard numerical methods, e.g., standard finite element method or central finite difference method, are not robust when the quantity $ϵ/\parallel \beta {\parallel }_{L^{\infty }\left(\Omega \right)}$ is small compared to the mesh size. In order to stabilize the numerical method, several remedies are proposed for addressing the issue, for instance, streamline diffusion method [6], residual free bubble methods [7, 8, 10], local projection schemes [37], subgrid scale method [16, 5], continuous interior penalty (CIP) methods [11, 12], discontinuous Galerkin methods [4, 32, 33], and recently discontinuous Petrov-Galerkin (DPG) methods [9, 13, 23], HDG method [31] and the first order least squares method [15]. One can refer to [40, 44] for more other stabilization techniques. But in order to capture the potential interior or outflow layer of the solutions to the problem (1.1), the local Péclet number $P_e=h\parallel \beta {\parallel }_{L^{\infty }\left(\Omega \right)}/ϵ$ near the layers should be small enough, where $h$ is the mesh size. Hence it would be quite expensive for the stabilized numerical methods used on the quasi-uniform mesh to capture the layers when $ϵ$ is small. If the mesh in the vicinity of the layers can be locally refined, the cost of numerical computations could be reduced. Therefore the adaptive finite element method is a natural choice for the efficient solution of convection-diffusion equations with dominant convection.

The adaptive finite element method based on a posteriori error estimates have been well established for second-order elliptic problems (cf. [2, 47]). In recent years the a posteriori error estimates are also extended to convection-diffusion equation. An early attempt was proposed by Eriksson and Johnson in [26], using regularization and duality techniques. Verfürth [48] proposed semi-robust estimators in the energy norm for the standard Galerkin approximation and the streamline upwind Petrov-Galerkin (SUPG) discretization. In [49] Verfürth improved his results by giving the estimates which are robust with dominant convection in a norm incorporating the standard energy norm and a dual norm of the convective derivative. Very recently, Tobiska and Verfürth [45] derived the same robust a posteriori error estimators for a wide range of stabilized finite element methods such as streamline diffusion methods, local projection schemes, subgrid scale technique and CIP method. However, the energy norm of error used in Verfürth’s estimates is defined through a dual norm which is not easy to compute. Sangalli [41] proposed different norms for the a posteriori error estimates that allow for robust estimators, but the analysis is only valid in the one dimensional case. As to other approaches for the robust error estimations, one can refer to [50] for mixed finite element methods, [51] for cell-centered finite volume scheme, [3] for nonconforming finite element method, [27, 28, 42, 52] for interior penalty discontinuous Galerkin method.

Discontinuous Galerkin (DG) methods have several attractive features compared with conforming finite element methods. For example, DG methods have elementwise conservation of mass, and they work well on arbitrary meshes. However, the dimension of the approximation DG space is much larger than the dimension of the corresponding conforming space. The HDG method [18, 19, 36] was recently introduced to address this issue. HDG methods retain the advantages of standard DG methods and result in significant reduced degrees of freedom. New variables on all interfaces of mesh are introduced such that the numerical solution inside each element can be computed in terms of them, and the resulting algebraic system is only due to the unknowns on the skeleton of the mesh. In [31], the HDG method was proposed and analyzed for the problem (1.1) on shape-regular mesh. The stabilization parameter of the HDG method in [31] can be determined clearly, meanwhile the penalty parameters of the DG schemes [4] need to be chosen empirically. Moreover, the condition number of stiffness matrix of a new way of implementing HDG method in [31] was proven to be bounded by $O\left(h^{-2}\right)$ and independent of the diffusion coefficient $ϵ$. These properties are important for the efficient solution of the problem (1.1) and encourage us to consider the corresponding HDG method on adaptive meshes.

The a posteriori error analysis for the HDG method for second order elliptic problems has been presented in [20, 21], where the error incorporates only the flux and a postprocessed solution used in the estimators. To our best knowledge, no a posteriori error estimates for the HDG discretizations of convection-diffusion problems have been studied in the literature so far. In this work, our objective is to show that the HDG scheme proposed in [31] gives rise to robust a posteriori error estimates for the problem (1.1). In comparison with the postprocessing technique utilized in [50, 20, 21], we establish the estimators without any postprocessed solution since the solution of (1.1) is always not smooth and there is no superconvergence result for the HDG method when $ϵ\ll 1$.

We notice that the a posteriori error estimators for nonconforming finite element method [3] and interior penalty DG method [27, 28] are only semi-robust in the sense that they yield lower and upper bounds of the error which differ by a factor equal at most to $O\left({ϵ}^{-1/2}\right)$. In [42, 52], the a posteriori error estimator is robust in the sense that the ratio of the constants in the upper and lower bounds of error is independent of the diffusion coefficient. However, the energy norm of error in [42, 52] contains the jump term ${\left(h_F{ϵ}^{-1}\parallel \text{llbracket}{u}_h\text{rrbracket}{\parallel }_{0,F}^2\right)}^{1/2}$ on each interior interface of meshes. In contrast, the a posteriori error estimator in this paper is robust based on the energy norm in (2.9), which contains a jump term ${\left({\gamma }_F\parallel \text{llbracket}{u}_h\text{rrbracket}{\parallel }_{0,F}^2\right)}^{1/2}$ instead. One can refer to (2.7) for the definition of paparemeter ${\gamma }_F$. So, our a posteriori error estimator will not enlarge the error estimate too much as the error estimator in [42, 52], when the mesh size is not small compared with the diffusion coefficient.

To derive the reliability and efficiency of the estimators for the error measured in an energy norm which incorporates a scaling flux and the scalar solution of the HDG discretization, two techniques are utilized. The first one is to use the Oswald interpolation operator to approximate a discontinuous polynomial by a continuous and piecewise polynomial function and to control the approximation by the jumps (cf. [34, 35]). For most of a posteriori error estimates mentioned above (e.g. [49, 50, 45, 27, 28, 42, 52]), the analysis only gives the estimates for the energy error without $L^2$-error of the scalar solution when $c-\frac{1}{2}div\beta =0$. The second one is to address this issue and to employ a weighted function to derive the estimates for the error which contains $L^2$-error of the scalar solution. This idea goes back to Nävert’s work [46] for convection-diffusion problems and was used to obtain the $L^2$-stability of the original DG method for pure hyperbolic equation [39] and extended to convection-diffusion equations using the IP-DG method [30, 4], the HDG method [31] and the first order least squares method [15].

In the numerical experiments, the convection-diffusion problems with interior or outflow layers are tested based on the proposed a posteriori error estimator. The robustness of the a posteriori error estimator based on the HDG method is observed for the problems with different diffusion coefficient. We also find that the convergence of the adaptive HDG method is almost optimal, i.e., the convergence rate is almost $O\left(N^{-s}\right)$, where $N$ is the number of elements, $s$ depends on the polynomial order $p$.

The outline of the paper is as follows: We introduce some notations, the HDG method, a posteriori error estimator and main results in the next section. In section 3, we collect some auxiliary results for analysis. Section 4 and section 5 are devoted to the proofs of reliability and efficiency, respectively. In the final section, we give some numerical results to confirm our theoretical analysis.

In this section, we begin with some basic notation and hypotheses of meshes. Secondly, we introduce the HDG method for (1.1) in [31]. Then, we define the corresponding a posteriori error estimator. Finally, we give the main results of reliability and efficiency.

We collect some auxiliary results in this section for the proof of reliability and efficiency.

For every element $T\in T_h$, we denote ${\Omega }_T$ by the union of all elements that share at least one point with $T$. For any face $F\in E_h$, the set ${\Omega }_F$ is defined analogously, meanwhile, ${\omega }_F$ is defined to be the union of elements sharing the common face $F$. The following Clément-type interpolation is crucial for the proof of reliability.

The interpolation ${\pi }_h$ in Definition 3.1 has the following approximation properties.

For the approximation of function in $W_h$ and the Dirichlet boundary data $g$ by continuous finite element space, we need to introduce Oswald interpolation. If $g$ is contained in $C\left(\partial \Omega \right)\cap M_h{|}_{\partial \Omega }$, then the continuous finite element space $W_{h,g}^c:=\{w\in C(\Omega ):w{|}_T\in P_p(T),T\in T_h,w{|}_{\partial \Omega }=g\}$ is not empty. Hence, we can introduce the Oswald interpolation $I_h^{os}:W_h\to W_{h,g}^c$. Given a function $v_h\in W_h$, the operator $I_h^{os}$ is prescribed at the Lagrangian nodes in the interior of $\Omega$ by the average of the values of $v_h$ at this node. For the nodes at the boundary $\partial \Omega$, $I_h^{os}$ is prescribed at the Lagrangian nodes on $\partial \Omega$ by the value of $g$ at this node. The following estimate has been analyzed for nonconforming mesh and conforming mesh in [34, 35] and was extended to variable polynomial degree in [53].

In order to prove the local efficiency of the estimators, proper element and face bubble functions are useful. We set the element bubble function in the element $T$ as $B_T={\prod }_{i=1}^{d+1}{\lambda }_i$, where ${\lambda }_i$ denotes the linear nodal basis function at $i$th vertex in $T$. Besides the element bubble function, as mention in Lemma 3.3 in [48] and Lemma 3.6 in [49], there also exists proper face bubble function $B_F$ with $F\in E_h^0$ such that the following lemma holds.

In this section, we give the proof of Theorem 2.2, which shows reliability of the a posteriori error estimator in Definition 2.1.