Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in L 1

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence formwith coefficients in L∞(Ω) and the right-hand side belongs to L1(Ω); we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges inW1,q 0 (Ω) for every q with 1 ≤ q < d/(d − 1) (d = 2 or d = 3) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate inW1,q 0 (Ω) when the right-hand side f belongs to Lr(Ω) verifying Tk(f) ∈ H1(Ω) for every k > 0, for some r > 1.


Introduction
In this work we consider, in dimension  = 2 or 3, the P 1 discontinuous Galerkin (dG) method approximation of the Dirichlet problem −div (∇) =  in Ω,  = 0 on Ω, (1) where Ω is an open bounded set of R  ,  is a coercive matrix with coefficients in  ∞ (Ω), and  belongs to  1 (Ω).
The solution of (1) does not belong to  1 0 (Ω) for a general right-hand side in  1 (Ω).Actually, in order to correctly define the solution of (1), one has to consider a specific framework, the concept of renormalized (or equivalently entropy) solution (see for example [1,2]).These definitions allow one to prove that in this new sense problem (1) is wellposed in the terminology of Hadamard.
For this problem the standard P 1 -nonconforming finite elements approximation, related to a triangulation T ℎ of Ω, namely, where with the discrete bilinear form   ℎ yet to be designed, has a unique solution, since the right-hand side (2) ∫ Ω V ℎ d is 2 International Journal of Differential Equations correctly defined for  ∈  1 (Ω) and the bilinear form   ℎ is consistent.
Using the ideas which are at the root of the SWIP (Symmetric Weighted Interior Penalty) method, in the case  ∈  2 (Ω), D. A. Di Pietro and A. Ern have proved, in [3], that the unique solution  ℎ of (2) converges to the unique solution  of (1) in the following sense: ,      ℎ     → 0, ℎ ℎ → 0, with the broken gradient ∇ ℎ and the jump seminorm | ⋅ |  yet to be designed.
To do that, the authors in [3] assume that the family of triangulations T ℎ belong to an admissible mesh sequence in the sense of 17 and is compatible with the partition P Ω (see Assumption 3).
The framework in this paper is the same as in [3].The unique difference here is that  ∈  1 (Ω) is considered instead of  ∈  2 (Ω); and we ourselves focus on the two cases  = 2 and  = 3.The same convergence results are proved.
Notations.In the present work, Ω denotes an open bounded subset of R  with  = 2 or  = 3.A particular case is the case where Ω is an open bounded polyhedron.We use the notation V for the scalar product of the vector V by the vector  (which is often denoted by  ⋅V).For a measurable set  ⊂ Ω, we denote by || the measure of  and by   the complement Ω \  of .
In that case, P 1  (T ℎ ) ⊂  1 (T ℎ ), which leads us to define the broken gradient ∇ ℎ :  1 and the broken divergence operator ∇ ℎ : (div, T ℎ ) → [ 2 (Ω)]  such that, ∀V ∈ (div, T ℎ ), Moreover, for any mesh element  ∈ T ℎ , we denote And for a scalar-valued function v defined on Ω (which can admit two possible traces) the average of v is defined as and the jump of v as For any face F ∈ F ℎ and for any integer  ≥ 0, we define the (local) lifting operator   , :  2 () → P   (T ℎ ) as follows.For all  ∈  2 (), and for any function V ∈  1 (T ℎ ), we define the (global) lifting of its interface and boundary jumps as We also introduce the normal diffusion coefficient to one face F as the diffusion-dependent penalty parameter (harmonic average of normal diffusion) as the weighted average operator for all  ∈ F  ℎ such that  ⊂  1 ∩  2 as the weighted average operator for all  ∈ F  ℎ and for a.e. ∈  as on boundary faces F ∈ F  ℎ such that F ⊂  ∩ Ω, we set and the skew-weighted average operator for all  ∈ F  ℎ and for a.e. ∈ , as The SWIP bilinear form is defined by (see Lemma 4.47 in where the quantity  > 0 denotes a user-dependent penalty parameter which is independent of the diffusion coefficient.
And the SWIP norms are defined by with the diffusion-dependent jump seminorms The discrete Galerkin norm is defined by with the jump seminorm For every  with 1 <  < +∞, we denote by  ,∞ (Ω) the Marcinkiewicz space whose norm is defined by For every real number  > 0 we define the truncation   : R → R by For every  − simplex  in R  , we adopt the following notations: (i)  , ,  = 0, . . ., , denote the vertices of .
(iv) for every  ∈ R  we put  , () fl 1 −  , ()   = 0, . . ., , where ( , ) 0≤ ≤  are the P 1 shape functions related to ; it is known that with International Journal of Differential Equations (v) If N designate the number of all interior centers   of faces F in T ℎ we define the interpolation operator Π ℎ and the truncated interpolation operator   ℎ by with (vi) Finally, we define the × stiffness matrix  = (  ); namely, As in [4], the main assumption of the present paper is that  is a diagonally dominant matrix; namely,

Statement of the Main Result
We consider a matrix  such that a.e  ∈ Ω : ∀ ∈ R  :  ()  ≥           2 , (34) for some  > 0, and a right-hand side  such that A function  is the renormalized solution of the problem (1) if  satisfies It is known (see [1,5]) that when  belongs to  1 (Ω) ∩  −1 (Ω), the usual weak solution of (1), namely, is a renormalized solution of (1) and conversely the main interest of definition of renormalized solution is the following existence, uniqueness, and continuity theorem (see [1,4]).
Then there exists a renormalized solution of ( 1).This solution is unique.Moreover this unique solution belongs to  1, 0 (Ω) for every  with 1 ≤  < /( − 1).It depends continuously on the right-hand side  in the following sense: if   is a sequence which satisfies when  tends to zero, then the sequence   of the renormalized solutions of (1) for the right-hand sides   satisfies for every  > 0 and for every  with 1 ≤  < /( − 1) (  ) →   ()    Now we consider a family of triangulations T ℎ satisfying for each ℎ > 0 the following assumption: the triangulation T ℎ is made of a finite number of closed -simplices (namely triangles when  = 2 and tetrahedra when  = 3) such that : (ii) for every compact set  with  ⊂ Ω, there exists ℎ 0 () > 0 such that  ⊂ Ω ℎ for every ℎ with ℎ < ℎ 0 () , (iv) every face of every  of T ℎ is either a subset of Ω ℎ , or a face of another   of T ℎ . (44) Note that because of (iv) the triangulations are conforming.A particular case is the case where Ω is a polyhedron of R  , and where Ω ℎ coincides with Ω for every ℎ.
In practice, the diffusion coefficient (i.e., matrix A) has more regularity than just belonging to  ∞ (Ω).Henceforth, we make the following assumption (assumption 4.43 [3]): An important assumption on the mesh sequence T ℎ fl (T H ) ℎ∈H is its compatibility with the partition  Ω in the following sense (assumption 4.45 [3]).
Assumption 3 (mesh compatibility).We suppose that the admissible mesh sequence T H is such that, for each ℎ ∈ H, each  ∈ T ℎ is a subset of only one set Ω  of the partition  Ω .In this situation, the meshes are said to be compatible with the partition  Ω .
For every  ∈ T ℎ , we denote by ℎ  the diameter of  and by   the diameter of the ball inscribed in .We set and we assume that ℎ tends to zero.We also assume that the family of triangulations T ℎ is regular in the sense of P. G. Ciarlet [6]; namely, there exists a constant  such that For every triangulation T ℎ , we consider the discrete problem: Note that the right-hand side of (48) makes sense since  ∈  1 (Ω) and  ℎ ⊂  ∞ (Ω).The discrete bilinear forme   ℎ is consistent and coercive (see ( 128)) on  ℎ , so a straightforward consequence of the Lax-Milgram Lemma is that the discrete problem (48) is well-posed.The solution  ℎ of (48) exists and is unique.
As in [4], the main result of this paper is the following.
This theorem will be proved in Section 4, using the tools that we will prepare in Section 3. In Section 5, we will explain why the results of [4] when  is a bounded Radon measure remain valid in our case.In Section 6 we also show that if we assume in addition that   () ∈  1 (Ω) for every  > 0, we obtain for smooth solutions an (ℎ 4(1−1/) ) error estimate in ‖ ‖ , -norm (Section 6.1), and for Low-Regularity solutions an (ℎ 4  (1−1/) ) error estimate in ‖ ‖ , -norm (Section 6.2).Finally, in Section 7 we show that in the case where A is the identity matrix, condition (32) remains satisfied when every inner angle of every d-simplex of T ℎ is acute.

Tools
We are going to prove Theorem 4 in several steps.We begin by proving the following result which is a piecewise P 1 variant of a result of L. Boccardo & T. Gallouët [2,5].
As in [4], to prove Theorem 5, we use the following lemmas.

Lemma 6.
Under assumption ( 47), for all T ∈ T ℎ and all  ∈ F  , one has Proof.Indeed, let  ∈ T ℎ and  ∈ F  , so and by (47) one has which combined with the fact that where  = Π/4 in 2D, and  = Π/6 in 3D, implies (52).

Lemma 7.
Under assumption (47), for every q such that 1 ≤ q, the following bound holds for any V ℎ ∈  ℎ : Proof.For every  ∈ T ℎ , we denote by (∇ ℎ V)  the (constant) gradient of the restriction of V to .With this notation, using the continuity of V across any  in F  at the mass center   of any internal , the fact that V vanishes at the mass center   of any external , and the known inequality and using (52) we get and the strictly positive constant () only depends on .
Proof.Let  be a −simplex from the triangulation T ℎ , V ℎ ∈  ℎ , and  > 0, such that sup In both cases, there exists an element  in  such that | ℎ ()| ≥ .
But  ℎ ∈ P 1 (), so In other words and since one obtains and as soon as For this purpose we define the -simplex  = ( −  0 ) +  0 such that Since   is affine, it is easy to check that λ =  −1  ∘  , for  = 0, 1, . . .,  are the barycentric coordinates with respect to the â 's and that where () = | Ŝ|/| T| is a constant that depends only on .This proves the result.
Proof.Fix  > 0 and  > 0 such that  <  and consider Let  ∈   and  ∈ T ℎ with  ∈ .It is easily checked that which implies that max  |V ℎ | > .So there are four possibilities.
(i) V ℎ changes sign in ; then, by continuity, There are three possibilities: We can then conclude that Convergence ( 106) is then consequence of ( 98) and (99).
The result and the proof of the Proposition (2.7) in [4] can be conserved without changes.

Proof of the Main Theorem
We first show an a priori estimate (compared with (50)) on the solution  ℎ of (48).
the inequality (123) is then proved.
Proposition 18.Under the assumption of Theorem 4, the solution  ℎ of (48) satisfies for every  > 0 and every ℎ > 0 and, in particular,  ℎ satisfies where the constant   only depends on , , and d.
The proof makes use of results appearing in [3] that we reproduce as follows.
Under assumption (47), one has for all V ℎ ∈ P 1  (T ℎ ), all T ∈ T ℎ , and all F ∈ F where   only depends on  and d.
Theorem 21.Under the assumptions of Theorem 4, the solution  ℎ of (48) satisfies for every  with 1 ≤  < /( − 1) when ℎ tends to zero, where  is the unique renormalized solution of (1).
To complete the proof of Theorem 4, it remains to prove the following proposition.
Proposition 22.Under the assumptions of Theorem 4, the solution  ℎ of (48) satisfies Consequently By the fact that and since  is the renormalized solution of (1), it is known that (see [4]) Finally, from (152) and (154), we deduce that lim sup which combined with the weak convergence (148) implies Owing to Proposition 4.36 in [3], for all V ℎ ∈  ℎ and all  > ( + 1) concluding the proof of (145).

The Case Where f Is a Bounded Radon Measure
The materials used in [4], to handle the case where f belongs to M  (Ω), are not specific to the case of P 1 finite elements approximation; only the weak convergence (148) requires clarification; in our approach it is based on the result of Proposition 15 whose proof involves only properties of V ℎ not .So we can also state the following convergence result.

Error Estimates for Smooth Solutions
Assumption 24 (regularity of exact solution and space  * ).As in [3], we assume that T ℎ is compatible with the partition  Ω International Journal of Differential Equations in the sense of Assumption 3, and the unique solution V is such that And we set The convergence analysis is performed in the spirit of (Theorem 1.35 [3]) by establishing discrete coercivity, consistency, and boundedness for   ℎ . The discrete bilinear form   ℎ is extended to  * ℎ ×  ℎ .Without further knowledge on the exact solution v apart from the domain Ω and the datum  ∈  2 (Ω), Assumption 24 can be asserted for instance if the domain Ω is convex; see Grisvard [7].
A straightforward consequence of the Lax-Milgram Lemma is that the discrete problem (48) is well-posed.

Error Estimates for Low-Regularity Solutions
Assumption 27 (regularity of exact solution and space  * ).As in [3], we assume that the mesh T ℎ is compatible with the partition  Ω in the sense of Assumption 3,  ≥ 2, and that there is  such that 2/( + 2) <  ≤ 2; the unique solution V is such that where  2, ( Ω ) =  2, (T) designate that the mesh T is compatible with the partition  Ω , and we set We also assume  < 2 since, in the case  = 2, Assumption 27 amounts to Assumption 24.
7. The Case Where A Is the Identity MatrixΩ ∇ ℎ Ω ∇ ℎ