A HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS WITH DIRAC DELTA SOURCE

. In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in 𝐿 2 -norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in 𝐿 2 -norm and 𝑊 1 ,𝑝 -seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.


Introduction
In this article, we consider the following problem = 0 on Ω. (1.1b) where Ω ⊂ R  ( = 2, 3) is an open, bounded, polygonal or polyhedral domain with Lipschitz boundary Ω, and  0 is a Dirac measure concentrated at the interior point  0 ∈ Ω.An instance of system (1.1) can be found in the electric field generated by a point charge.Another instance appears in the PDE-constrained optimal control problem [11,12] and in the acoustic monopoles or pollutant transport and degradation in an aquatic media [3].
Since the Dirac function  0 does not belong to  −1 (Ω), the solution of problem (1.1) is not in  1 (Ω).In spite of the fact that the solution of problem (1.1) has a very low regularity, it still can be numerically approximated by standard finite element methods.For  = 2, Babuška [8] obtained the convergence rate (ℎ 1− ) ( > 0) for the error in  2 -norm.In [33], Scott removed  and yielded the convergence rate (ℎ 2−/2 ).For an arbitrary Borel measure, Casas derived in [11] a similar result by using different techniques.As for the interior maximum norm error estimates, it has been proved in [34] by Schatz and Wahlbin.It is worth noting that Eriksson [19] proved a priori error estimates for  1 -and  1,1 -errors on adequately refined meshes.
The singular nature of problem (1.1) suggests that the meshes adequately refined around the delta support should be used to boost the accuracy of approximation.In [6], Apel et al. proved the  2 -error estimates of almost optimal order by using graded meshes on a convex and polygonal domain.In [2], adaptive finite element methods based on a posteriori error estimators were considered for  = 2.The efficient and reliable a posteriori error estimators for   -norm ( ∈ (1, ∞)) and  1, -seminorm ( ∈ ( 0 , 2)) were obtained by using duality argument, where  0 ∈ [1, 2) is a real number depending on the largest inner angle of the domain Ω.For the error in fractional Sobolev space   (Ω) ( ∈ ( 1 2 , 1)), the residual type a posteriori error estimators with specifically tailored oscillation were derived by Gaspoz et al. [25].In [7], Agnelli et al. developed a reliable and efficient a posteriori error estimator for the weighted  1   -norm ( ∈ I ⊂ (  2 − 1,  2 )).In 2012, Houston and Wihler [26] studied the discontinuous Galerkin (DG) methods for problem (1.1) with  = 2.The convergence rate (ℎ) for  2 -error was proved under a constraint that  0 lies in the interior of an element.In addition, a posteriori error estimator, which is efficient and reliable for an extended  2 -norm, was also shown.It is well known that the DG method is very flexible when it is applied to solve the partial differential equations, however too many globally coupled degrees of freedom are always used and the discrete system is large in particular when the high order polynomial is utilized.Relatively, the HDG methods, proposed by Cockburn et al. [15], not only can keep the advantage of DG methods, but also can get a system with significantly reduced degrees of freedom by introducing the Lagrange multipliers.Currently, it has been used for many problems such as elliptic [13,14], convection diffusion [21], fluid flow [16,28,32], and optimal control [18,23,29], etc.. To the best of our knowledge, there still has no work on error analysis of HDG methods for problems with Dirac measures.
Therefore, in this paper, we investigate error analysis of HDG methods for problem (1.1).In particular, a priori error estimate with convergence rate (ℎ 2−/2 ) is obtained for the error in  2 -norm.On the other hand, a posteriori error estimator, which provides an upper and a lower bounds for the error in  2 -norm, is proved in a convex domain.Moreover, a posteriori error estimator that is efficient and reliable for the error in  1, -seminorm is also derived in a non-convex Lipschitz polygon, where  ∈ ( Ω , 2) and  Ω > 0 is a real number depending on the largest inner angle of the domain Ω.Finally, some numerical examples are presented to validate the numerical analysis.
Compared with [2], we need introducing the Oswald interpolation to prove a posteriori error estimator for  1, -seminorm, moreover the a posteriori error estimators obtained in this paper incorporate the term ‖ ℎ − ̂︀  ℎ ‖ 0,, that does not appear in [2], see Lemma 4.4 and Section 4 for more detail, where ( ℎ , ̂︀  ℎ ) is the HDG solution of problem (1.1).It is worth noting that Oswald interpolation is a very important tool in a posteriori error analysis of HDG methods [4,5,14,17], because it provides a continuous approximation for a discontinuous piecewise polynomial function.Compared with [26], we not only prove a priori and a posteriori error estimates for  2 -norm in two-and three-dimensional cases, but also derive a posteriori error estimate for  1, -seminorm in two dimensional case.
The rest of this article is arranged as follows: In Section 2, notation and definition corresponding to Sobolev spaces and meshes are provided.In addition, the HDG scheme and some known results are also presented in this section.In Section 3, a priori error estimates are proved.In Sections 4 and 5, the reliability and efficiency of a posteriori error estimators are shown respectively.In Section 6, some numerical examples are presented to validate the numerical analysis.Finally, we end this paper by some conclusions in Section 7.
Throughout this paper, let  with or without subscript be a generic positive constant independent of the mesh size, which may be different in different places.For ease of exposition, we denote  ≤  by   and  ≈  by   .

Weak formulation and HDG discretization
, then the dual space to  , 0 (Ω) is denoted by  −, ′ (Ω), where  , 0 (Ω) is the closure of  ∞ 0 (Ω) in the space  , (Ω) [1].If no confusion induced, we also use ⟨•, •⟩ to denote the duality pairing between spaces  −, ′ (Ω) and  , 0 (Ω).Finally, we define (div, The weak formulation of problem (1.1) is to find From the embedding theorem, we know that where  + and  − are the traces of  on  =  + ∩  − , and n + and n − denote the unit vector normal to the face  .As for  ∈ ℰ  ℎ , we formally set Next, we define the mesh-dependent inner products and norms: For  ≥ 1, the discontinuous finite element spaces corresponding to the partition  ℎ are given as follows: where   () denotes the set of polynomials of degree no larger than  on the domain .Then the HDG discretization of problem (2.1), that approximates the exact solution (, | ℰ ℎ ), reads as follows: where  0 = { ∈  ℎ :  0 ∈ }, ♯ 0 denotes the number of elements in  0 , and  ℎ is defined by Here  is the stabilization parameter.According to the inverse estimate and trace inequality, we have for any ( ℎ ,  ℎ ) ∈   ℎ ×   ℎ , where  is a constant depending on the polynomial degree and shape-regularity of the mesh.Therefore the bilinear form  ℎ is coercive with respect to the norm (•, •) 1,ℎ for  = 0 ℎ  on each face  ∈ ℰ ℎ with  0 sufficiently large.So the HDG formulation (2.2) has an unique solution ( ℎ , ̂︀ for  0 sufficiently large.Notice that the effect of  0 on the error will be discussed by numerical experiments in Section 6.Now we end this section by introducing some known results that will play an important role in the subsequent proofs.
Lemma 2.2.For each element  ∈  ℎ and any given nonnegative integer , the following estimates hold Proof.The approximation results (2.4) and (2.5) can be found in Lemma 4.5.3,Theorem 1.6.6 from [9].
Proof.The result can be found in Theorem 4.6.12from [9].
The next result shows that the polynomial  ℎ ∈   ℎ can be approximated by a continuous function ̃︀ It is well-known that this is the so-called Oswald interpolation.Lemma 2.5.For any  ℎ ∈   ℎ , there exists a function ̃︀ ) Proof.Here we only prove the error estimate (2.12).As for the error estimate (2.11), it can be proved similarly.
For the case of  = 2, the approximation (2.12) has been proved in Theorem 2.2 from [27].As for other cases, the method of proof is similar.

A PRIORI error analysis
This section mainly focuses on a priori error analysis for the error ‖ −  ℎ ‖ 0,Ω .To this end, we introduce the following auxiliary problem: ) The problem (3.1) has an unique weak solution which holds the following regularities from the different cases: and the domain Ω is convex, we have and the domain Ω is a non-convex Lipschtiz polygon, we have where (2 −   ) ′ < 2 and  >  is the largest inner angle of the domain Ω.Proof.The result of case (i) is well-known [10].The result of case (ii) can be found in Section 4 from [2] and Theorem 4.4.4.13 from [24].
Following the approach in [26,33], we define  ℎ ∈   ℎ such that  ℎ = 0 on  ℎ ∖ 0 , and Then with ( [26], Subsect.3.1) and inverse estimate (2.4), we can obtain Furthermore, we define   ℎ as the solution of problem (3.1) with  = 1 ♯ 0  ℎ , and the weak formulation is to From now on, we assume in the rest of this section that the domain Ω is convex and the partition  ℎ is quasi-uniform.
By a simple calculation, we can find  −  ℎ =  −   ℎ +   ℎ −  ℎ , hence a priori error analysis for the error ‖ −  ℎ ‖ 0,Ω can be divided into two parts.Firstly, we are going to deal with the error ‖ −   ℎ ‖ 0,Ω .
) and   ℎ ∈  1 0 (Ω) be the solutions of problems (2.1) and (3.6).If the domain Ω is convex and the mesh is quasi-uniform, it holds that Proof.Let   be the solution of problem (3.1) with  ∈  2 (Ω).Let  ,ℎ ∈  1 ℎ and   ℎ ,ℎ ∈  1 ℎ be the standard finite element approximations of   and   ℎ .Then we have the following standard error estimates (see, e.g., [10]): ) Moreover, integration by parts can yield which, together with (3.5) and Lemma 3.2, yields the following a priori error estimate.
Theorem 3.3.Let  and ( ℎ , ̂︀  ℎ ) be the solutions of problems (2.1) and (2.2).If the domain Ω is convex and the mesh is quasi-uniform, we have

Reliability of A POSTERIORI error estimators
In this section, we consider a posteriori error analysis for the HDG scheme (2.2).Specifically, we will prove a posteriori error estimators for the errors ‖ −  ℎ ‖ 0,Ω and ‖∇( −  ℎ )‖ ,Ω .
If  0 is a node of the partition  ℎ , we define the local error estimators   and  , by Before proving a posteriori error estimates, we first show the following properties for the HDG scheme (2.2).
Proof.By setting  ℎ = 0 in (2.2), we arrive at where  = 0 ℎ  on each face  ∈ ℰ ℎ .Hence the result (4.1) can be obtained by the above equality.Moreover the equality (4.2) can be derived directly by using (4.1), (2.2) and integration by parts.Now we are going to prove a posteriori error estimate for the error ‖ −  ℎ ‖ 0,Ω by the duality argument.Lemma 4.2.Let  and ( ℎ , ̂︀  ℎ ) be the solutions of problems (2.1) and (2.2).If the domain Ω is convex, we have Proof.Let   be the solution of problem (3.1) with  ∈  2 (Ω).From the case (i) of Theorem 3.1, we know that the solution   satisfies the regularity (3.2).Hence by integration by parts, we arrive at Insert (4.2) in the above equality to yield for any  ℎ ∈   ℎ .Note that we have used (4.1) and the fact that ∇  ∈ (div, Ω) in the derivation of (4.4).Let  ℎ =   in (4.4), we can get by (2.5) and (2.6) and As for the term  1 , if  0 is a node of the partition  ℎ , we have  1 = 0, otherwise, the Lagrange interpolation error estimate (2.8) can get Hence the result (4.3) can be achieved by using (3.2) and (4.4)-(4.7).
Proof.Since  ℎ does not belong to the space  1, (Ω), we can not apply the duality argument directly for − ℎ by (4.8).Here, let ̃︀  ℎ be the Oswald interpolation of  ℎ , then using duality argument for  − ̃︀  ℎ we have Insert (4.2) in the above equality to infer that for any  ℎ ∈   ℎ .Let  ℎ =  ∈  1 ℎ in (4.10), the trace inequality (2.5) and Lagrange interpolation error estimate (2.7) can get and )︁ (4.12) If  0 is a node of the partition  ℎ ,  1 = 0, otherwise, we have by the Lagrange interpolation error estimate (2.9).As for the term  3 , the Oswald interpolation error estimate (2.12) can yield Hence combining (4.9) and (4.10)-(4.14),we have which, together with the Oswald interpolation error estimate (2.12) and the triangle inequality, concludes the proof.

Efficiency of A POSTERIORI error estimators
In this section, we mainly prove the efficiency of a posteriori error estimators.Before this, we first introduce the element and face bubble functions   and   as that in [35].Following the approach in [2,20], we define for any  ∈  ℎ , and for  ∈ ℰ  ℎ , where   := ⋃︀ { :  ⊂ ,  ∈  ℎ }.
Proof.The result (5.1) can be obtained directly by the definitions of   and   , and the approximation result (5.2) can be derived by Lemma 3 from [20].
For the case of  0 is not a node of the partition  ℎ , we know that the error estimators   and  , include an additional term.In order to control this term, we define a cutoff function  0 for the element  ∈  0 such that 0 ≤  0 ≤ 1 ∀ ∈ Ω, (5.7) ) ) where   := ⋃︀ { ′ ∈  ℎ :  ∩  ′ ̸ = ∅}, and  denotes the distance of  0 to the boundary of   .Note that the cutoff function has been used in [2].Then using the fact that ℎ   and (5.10), we have (5.11) Lemma 5.3.For any  ∈  0 , let  0 and   be defined as above.Then we have and for  ∈ ( Ω , 2).
Obviously, the remaining task is to bound the error estimator ‖ ℎ − ̂︀  ℎ ‖ 0,, .But, seemingly, it is not an easy task.First of all, we show the following relationships.
Lemma 5.4.Let ( ℎ , ̂︀  ℎ ) be the solution of problem (2.2).Then the following relationships hold and ) Proof.To prove (5.16) and (5.17), we only need to prove the following results: By using (4.1) and the definition of the jump [ ℎ ], we yield Hence we can get (5.18) by the above inequality.
On the other hand, by using the triangle inequality, we yield Therefore the estimate (5.19) can be derived by the above two inequalities.Now we are going to bound the error estimator ‖[∇ ℎ • n]‖  0,, for  ∈ ℰ  ℎ .
(i) If the domain Ω is convex, we have (ii) If d=2 and the domain Ω is a Lipschitz polygon, we have for  ∈ ( Ω , 2), where  Ω = max{1, 2/(1 +   )} and  is the largest inner angle of the domain Ω.
Finally the approximation result (5.26) can be obtained by Lemma 4.4 and the fact that )︁ 1/ .

Numerical experiments
In this section, some numerical experiments are provided to validate the theoretical analysis.For the adaptive HDG algorithm designed by the obtained a posteriori error estimators, we use the following marking strategy   3.6253e-2 9.1561e-3 6.9934e-3 )︁ 1/ .
Example 6.1.Based on the domain Ω = (0, 1) 2 , we consider the problem (1.1) with  0 = (0.5, 0.5).In this example, the Dirichlet boundary conditions are imposed so that the exact solution is given by We test this example under the constraint that  0 be a vertex of all partitions.
In Tables 1 and 2, the convergence history and convergence order of the error ‖  ‖ 0,Ω for different  0 and  are provided on uniform meshes.We find that the convergence rate (ℎ) can be achieved.From the perspective of error, we find from Figure 1 that the best choice of  0 may be 25 and 150 for  = 1 and  = 2. Furthermore, in Figure 2 and Table 3, the convergence histories of ‖ −  ℎ ‖ 0,Ω for finite element methods (FEM) and HDG methods are presented for  = 1.Obviously, involving a suitable choice of  0 , we can expect that the HDG methods of this paper are slightly better than the finite element methods.
From now on, we set  0 = 25 and we test this example with adaptive HDG algorithm.The meshes and the corresponding surfaces of  ℎ , generated by  2 and  1.5 , are provided in Figures 3 and 5 for  = 1 and  = 0.2.Obviously, the mesh nodes are concentrated around  0 .In Figure 4, we present the convergence histories of  2 and ‖  ‖ 0,Ω for different  and .We observe from the left graph of Figure 4 that the convergence rate ( −(+1)/2 ) can be obtained.From the right graph of Figure 4, we can see that for the same numerical accuracy the number of required vertices will be increased as  becomes large.
The initial mesh consists of 12 triangles.According to the definition of  Ω , we know that  Ω = 1.2.Throughout this section, let  0 = 15 and  0 be a vertex of all partitions.
In Figure 7, the convergence histories of  2 and ‖  ‖ 0,Ω and the efficiency index  2 / are given.The convergence rate ( −(+1)/2 ) can be obtained.In Figure 8, the adaptive meshes, generated by  1.3 and  1.5 , after    Finally, we make a comparison between [2] and this paper for  = 1 and  = 0.3.Note that the estimators and errors obtained in [2] are labeled by  Ref  and  Ref  .In Figure 10, the convergence histories are performed, and the convergence rate ( −1/2 ) can be obtained.Obviously, the error estimators and the errors derived in this paper are smaller than that achieved by [2].Therefore, from the perspective of error, the HDG result of this paper is better than the FEM result of [2] for the suitable choice of  0 .

Conclusions
In this paper, we investigate HDG methods for elliptic problems with Dirac measures.Firstly, a priori error estimate with convergence rate (ℎ) is proved for the error in  2 -norm.Then, by duality argument and Oswald interpolation, the efficient and reliable a posteriori error estimators for the errors in  2 -norm and  1, -seminorm are obtained.
Finally the obtained a posteriori error estimators are used to design adaptive HDG algorithm, and some numerical examples are provided to verify the theoretical analysis and show the performance of the obtained a posteriori error estimators.By the numerical results, we find that the HDG scheme and error estimators of this paper are slightly better than the finite element discretization and error estimators of [2] based on a suitable choice of  0 , see Figures 2 and 10.
2 2 or    , and   is the restriction of  on the element  ∈  ℎ .   .We refine the marking set ℳ by bisections to generate a new mesh.Note here that the figure of convergence history is plotted in log-log coordinates in this section.Moreover we set   =  −  ℎ and to select the marking set ℳ, where  ∈ (0, 1],  =