Robust a Posteriori Error Estimates for HDG method for Convection-Diffusion Equations
We propose a robust a posteriori error estimator for the hybridizable discontinuous Galerkin (HDG) method for convection-diffusion equations with dominant convection. The reliability and efficiency of the estimator are established for the error measured in an energy norm. The energy norm is uniformly bounded even when the diffusion coefficient tends to zero. The estimators are robust in the sense that the upper and lower bounds of error are uniformly bounded with respect to the diffusion coefficient. A weighted test function technique and the Oswald interpolation are key ingredients in the analysis. Numerical results verify the robustness of the proposed a posteriori error estimator.
Key words and phrases:hybridizable discontinuous Galerkin method, a posteriori error estimates, convection-diffusion equations
2000 Mathematics Subject Classification:65N30, 65L12
Given a bounded, polyhedral domain , we consider the convection-diffusion equations
The data and the right-hand sides in (1.1) satisfy the following assumptions:
, , and .
There is a function and a positive constant such that .
Assumption (A1) includes the case of the convection-dominated regime. According to , Assumption (A4) is satisfied if has no closed curves and .
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 or outflow layer of width . Standard numerical methods, e.g., standard finite element method or central finite difference method, are not robust when the quantity 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 , residual free bubble methods [7, 8, 10], local projection schemes , 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  and the first order least squares method . 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 near the layers should be small enough, where 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 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 , using regularization and duality techniques. Verfürth  proposed semi-robust estimators in the energy norm for the standard Galerkin approximation and the streamline upwind Petrov-Galerkin (SUPG) discretization. In  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  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  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  for mixed finite element methods,  for cell-centered finite volume scheme,  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 , the HDG method was proposed and analyzed for the problem (1.1) on shape-regular mesh. The stabilization parameter of the HDG method in  can be determined clearly, meanwhile the penalty parameters of the DG schemes  need to be chosen empirically. Moreover, the condition number of stiffness matrix of a new way of implementing HDG method in  was proven to be bounded by and independent of the diffusion coefficient . 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  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 .
We notice that the a posteriori error estimators for nonconforming finite element method  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 . 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 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 instead. One can refer to (2.7) for the definition of paparemeter . 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 -error of the scalar solution when . The second one is to address this issue and to employ a weighted function to derive the estimates for the error which contains -error of the scalar solution. This idea goes back to Nävert’s work  for convection-diffusion problems and was used to obtain the -stability of the original DG method for pure hyperbolic equation  and extended to convection-diffusion equations using the IP-DG method [30, 4], the HDG method  and the first order least squares method .
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 , where is the number of elements, depends on the polynomial order .
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.
2. Notation, HDG method, error estimator, and main results
In this section, we begin with some basic notation and hypotheses of meshes. Secondly, we introduce the HDG method for (1.1) in . Then, we define the corresponding a posteriori error estimator. Finally, we give the main results of reliability and efficiency.
2.1. Notation and the mesh
Let be a conforming, shape-regular simplicial triangulation of . For any element , denotes the set of its edges in the two dimensional case and of its faces in the three dimensional case. Elements of will be generally referred to as faces, regardless of dimension, and denoted by . We define . We denote by the set of all faces in the triangulation (the skeleton), while the set of all interior (boundary) faces of the triangulation will be denoted (). Correspondingly, we refer to the set of vertices and to the set of interior vertices. For any , let be the diameter of element . Similarly, for any , we define . Throughout this paper, we use the standard notations and definitions for Sobolev spaces (see, e.g., Adams). We also use the notation and to denote the -norm on the elements and faces , respectively.
2.2. The HDG method
The HDG method is based on a first order formulation of the convection-diffusion equation (1.1), which can be rewritten in a mixed form as finding such that
For any element and any face , we define
where is the space of polynomials of total degree not larger than on . The finite element spaces are given by
where and .
The HDG method seeks finite element approximations satisfying
for all , where the normal component of numerical flux is given by
and the stabilization function is a piecewise, nonnegative constant defined on . Here, we define and . One of the advantages of the HDG method is the elimination of both and from the system (2.2) to obtain a formulation in terms of numerical trace only, one can refer to [17, 31, 36, 38] for the implementation.
The stabilization function in (2.3) is chosen as
Here, . We emphasize that the choice of in (2.4) is the second type of stabilization function in . According to , the HDG method (2.2) with stabilization function (2.4) has a unique solution. Compared with a recent work , the intrinsic idea of choosing in the above HDG method is similar to the strategy of setting the ultra-weakly imposed boundary condition in the first order least squares method for (1.1).
2.3. A posteriori error estimator
We define the elementwise residual function as
where can be any element or any face . In addition, for any , we introduce
Now, we are ready to introduce the a posteriori error estimator in the following.
(A posteriori error estimator) The a posteriori error estimator is defined as
Here, for any interior face in , we define the jump of scalar function and the jump of normal component of vector field by
2.4. Reliability and efficiency of a posteriori error estimator
From now on, we use to denote generic constants, which are independent of the diffusion coefficient and the mesh size.
For any , we define the energy norm
We outline main results by showing the reliability and efficiency of the a posteriori error estimator, respectively, in the following theorems. Using the techniques developed in this paper, we would like to emphasize that all the a posteriori error estimates in this paper will also hold for the mixed hybrid method in .
(Reliability) If the Dirichlet data is contained in , then
(Efficiency) We have
Here, the data oscillation term , and is the orthogonal projection onto .
In addition, for any with or any with , we have the following efficiency results with explicit dependence on .
(Efficiency on refined element) For any , if , then
For any , if , then
Here, is the union of elements sharing the common face , and is the data oscillation term introduced in Theorem 2.3.
3. Auxiliary results
We collect some auxiliary results in this section for the proof of reliability and efficiency.
For every element , we denote by the union of all elements that share at least one point with . For any face , the set is defined analogously, meanwhile, is defined to be the union of elements sharing the common face . The following Clément-type interpolation is crucial for the proof of reliability.
(cf. ) One can define a linear mapping via
where is nodal bases function for every vertex , is the support of a nodal bases function which consists of all elements that share the vertex , and is the corresponding conforming finite element space defined by
The interpolation in Definition 3.1 has the following approximation properties.
The proof of Lemma 3.2 can be obtained by the Lemmas 3.1 and 3.2 in . For the particular case , the constant is set to be for any or in , and the associated energy error excludes the -error . In this paper, a weighted function technique used in  shall be employed, such that we can also obtain the estimates for the energy error including the -error even when . Hence, is always set as (2.6).
For the approximation of function in and the Dirichlet boundary data by continuous finite element space, we need to introduce Oswald interpolation. If is contained in , then the continuous finite element space is not empty. Hence, we can introduce the Oswald interpolation . Given a function , the operator is prescribed at the Lagrangian nodes in the interior of by the average of the values of at this node. For the nodes at the boundary , is prescribed at the Lagrangian nodes on by the value of 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 .
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 as , where denotes the linear nodal basis function at th vertex in . Besides the element bubble function, as mention in Lemma 3.3 in  and Lemma 3.6 in , there also exists proper face bubble function with such that the following lemma holds.
4. Proof of reliability
In view of the assumption (A4), we define a weighted function
where is a positive constant to be determined later. Let . We have the following Lemma 4.1.
Let be given in (4.1) with . Then the following estimate holds:
According to the assumptions (A3)-(A4), we have
By the Cauchy-Schwarz and Young’s inequalities, for any , we have
Then, by taking and , we can conclude that the proof is complete. ∎
We are now ready to state a key result of the upper bound estimate of .
If the Dirichlet data is contained in , then
According to Lemma 4.1, we have
Let . By the definition of , clearly, we have . Adding and subtracting into , we have
In the last step of (4.5), we have used the equation (2.1b) and the fact that (since jumps of vanish on all interior faces and on ). Inserting (4.5) into (4.4) and subtracting and adding into in the first term of the right-hand side of (4.5), we have
Note that the equation (2.2d) indicates . Hence, we have