Research Article | Open Access
Affine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in
We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in and the right-hand side belongs to ; 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 in for every with ( or ) 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 in when the right-hand side belongs to verifying for every , for some
In this work we consider, in dimension or , the discontinuous Galerkin (dG) method approximation of the Dirichlet problemwhere is an open bounded set of , is a coercive matrix with coefficients in , and belongs to .
The solution of (1) does not belong to for a general right-hand side in . 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 well-posed in the terminology of Hadamard.
For this problem the standard -nonconforming finite elements approximation, related to a triangulation of , namely,
with the discrete bilinear form yet to be designed, has a unique solution, since the right-hand side (2) is correctly defined for and the bilinear form is consistent.
Using the ideas which are at the root of the SWIP (Symmetric Weighted Interior Penalty) method, in the case , D. A. Di Pietro and A. Ern have proved, in , that the unique solution of (2) converges to the unique solution of (1) in the following sense:
with the broken gradient and the jump seminorm yet to be designed.
The framework in this paper is the same as in . The unique difference here is that is considered instead of ; and we ourselves focus on the two cases and The same convergence results are proved.
Notations. In the present work, denotes an open bounded subset of with or . A particular case is the case where is an open bounded polyhedron. We use the notation for the scalar product of the vector by the vector (which is often denoted by ). For a measurable set , we denote by the measure of and by the complement of .
For and , we have
We define also the following function spaces:
For , we define the broken polynomial space with polynomial degree k 1.
In that case, , which leads us to define the broken gradient such that, ,
and the broken divergence operator such that, ,
Moreover, for any mesh element , we denote
And for a scalar-valued function v defined on (which can admit two possible traces) the average of v is defined as
and the jump of v as
For any face F and for any integer , we define the (local) lifting operator as follows. For all ,
and for any function , 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 such that as
the weighted average operator for all and for a.e. as
on boundary faces F such that F , we setand the skew-weighted average operator for all and for a.e. , as
The SWIP bilinear form is defined by (see Lemma 4.47 in )
where the quantity 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 , we denote by the Marcinkiewicz space whose norm is defined by
For every real number we define the truncation by
For every in , we adopt the following notations:(i), , denote the vertices of .(ii), , denote the centers of the faces .(iii), , designate the barycentric coordinates with respect to the ’s.(iv)for every we put where are the shape functions related to ; it is known that with .(v)If N designate the number of all interior centers of faces F in we define the interpolation operator and the truncated interpolation operator by with .(vi)Finally, we define the stiffness matrix ; namely, As in , the main assumption of the present paper is that is a diagonally dominant matrix; namely,
2. Statement of the Main Result
We consider a matrix such thatfor some , and a right-hand side such that
A function is the renormalized solution of the problem (1) if satisfies
Theorem 1. Assume that and satisfy (33), (34), and (35). Then there exists a renormalized solution of (1). This solution is unique. Moreover this unique solution belongs to for every with . It depends continuously on the right-hand side in the following sense: if is a sequence which satisfieswhen tends to zero, then the sequence of the renormalized solutions of (1) for the right-hand sides satisfies for every and for every with when tends to zero, where is the renormalized solution of (1) for the right-hand side . Finally, if and belong to , and if and are the renormalized solutions of (1) for the right-hand sides and , then, for every , the function belongs to and for every with one hasWhere the constant only depends on , , and .
Remark 2. Throughout all this paper, we denote by any real constant which only depends on the parameters , , and …. We can use the same notation for different constants.
Now we consider a family of triangulations satisfying for each the following assumption:
Note that because of (iv) the triangulations are conforming. A particular case is the case where is a polyhedron of , and where coincides with for every .
In practice, the diffusion coefficient (i.e., matrix A) has more regularity than just belonging to . Henceforth, we make the following assumption (assumption 4.43 ):
An important assumption on the mesh sequence is its compatibility with the partition in the following sense (assumption 4.45 ).
Assumption 3 (mesh compatibility). We suppose that the admissible mesh sequence is such that, for each , each is a subset of only one set of the partition . In this situation, the meshes are said to be compatible with the partition .
For every , we denote by the diameter of and by the diameter of the ball inscribed in We setand we assume that tends to zero.
We also assume that the family of triangulations is regular in the sense of P. G. Ciarlet ; namely, there exists a constant such that
For every triangulation , we consider the discrete problem:
Note that the right-hand side of (48) makes sense since and . The discrete bilinear forme is consistent and coercive (see (128)) on , so a straightforward consequence of the Lax-Milgram Lemma is that the discrete problem (48) is well-posed. The solution of (48) exists and is unique.
As in , the main result of this paper is the following.
Theorem 4. Assume that , , and satisfy (33), (34), (35), (44), (46), (47), and (32). Then the unique solution of (48) satisfies for every and for every with when tends to zero, where is the unique renormalized solution of (1).
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  when is a bounded Radon measure remain valid in our case. In Section 6 we also show that if we assume in addition that for every , we obtain for smooth solutions an error estimate in -norm (Section 6.1), and for Low-Regularity solutions an 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 is acute.
Theorem 5. Assume that satisfiesfor some . Then, for every with ,where the constant only depends on , , and
Lemma 6. Under assumption (47), for all T and all , one has
Lemma 7. Under assumption (47), for every q such that 1 q, the following bound holds for any :
Proof. For every , we denote by the (constant) gradient of the restriction of to . With this notation, using the continuity of across any in at the mass center of any internal , the fact that vanishes at the mass center of any external , and the known inequalityand using (52) we getwhich is (56) with
Lemma 8. Let and let If for some there exists with , then there exists a -simplex with such thatand the strictly positive constant only depends on .
Proof. Let be a simplex from the triangulation , , and , such that Consider (i)If , thus so , and such that .(ii)If , thus , such that so , and .In both cases, there exists an element in such that .
But , so In other words and since one obtainsso and as soon as For this purpose we define the -simplex such that soand to estimate the measure of , it is clear to verify first that . Let be the reference unit -simplex with vertices , , where is the canonical basis of . Let be the invertible affine mapping that maps onto and set .
Since is affine, it is easy to check that for are the barycentric coordinates with respect to the ’s and thatwhere