Analysis of Discontinuous Galerkin Methods using Mesh-Dependent Norms and Applications to Problems with Rough Data

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine a problem with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control.


Introduction
Discontinuous Galerkin (dG) methods are a popular family of non-conforming finite element-type approximation schemes for partial differential equations (PDEs) involving discontinuous approximation spaces. In the context of elliptic problems their inception can be traced back to the 1970s [17,4]; see also [1] for an accessible overview and history of these methods for second order problems. For higher order problems, for example the (nonlinear) biharmonic problem, dG methods are a useful alternative to using C 1 -conforming elements whose derivation and implementation can become very complicated [4,11,18].
Inf-sup conditions form one part of the Banach-Nečas-Babuška condition which guarantees the wellposedness of a given variational problem. In this note, we shall describe an analytical framework to examine the stability of dG approximations for L 2 and H 2 -like mesh-dependent norms. This is in keeping with the spirit of [2,3], where for continuous finite element methods the authors prove equivalent results for second and fourth order problems respectively. The present approach, however, is quite different and results in inf-sup stability for both L 2 -and H 2 -like mesh-dependent norms under the assumption that the underlying mesh is quasi-uniform.
The analysis presented utilises a new H 2 -conforming reconstruction operator, based on Hsieh-Clough-Tocher-type C 1 reconstructions. Such reconstructions, based on nodal averaging, are used for the proof of a posteriori bounds for non-conforming methods for elliptic [15,6,11,19] and hyperbolic problems [10,13]. The new reconstruction operators presented below enjoys certain orthogonality properties; in particular, they are adjoint orthogonal to the underlying Hsieh-Clough-Tocher space and maintain the same stability bounds as the H 2 -conforming reconstruction from [11].
The argument is quite general and allows the derivation of inf-sup stability results whenever the numerical scheme has a well posed discrete adjoint (dual) problem over an appropriately constructed non-conforming finite element space. This is contrary to the Aubin-Nitsche L 2 duality argument whereby it is the underlying partial differential operator itself that requires the well posedness of the adjoint continuous problem.
The use of these recovery operators is not limited to an a posteriori setting, indeed, they have been used to quantify inconsistencies appearing in standard interior penalty methods when the exact solution is not H 2 (Ω) [14]. This allows for quasi optimal a priori bounds for elliptic problems under minimal regularity up to data oscillation. Fundamentally the assumption in this analysis is that the singularity arises from the geometry of the domain rather than through the problem data itself. Our analysis allows us to show quasi-optimal L 2 convergence to problems that have rough problem data. To showcase the result we study the convergence of a method posed for an elliptic problem whose source term is not H −1 . In this case the Aubin-Nitsche and indeed the standard treatment of Galerkin methods are not applicable.
The note is set out as follows: In §2 we introduce the problem and present the analysis cumulating in inf-sup stability for problems with smooth data. In §3 we examine a particular problem with rough data and prove quasi-optimal convergence in this case. In addition we give some numerical validation of the method.

Problem set up and discretisation
To highlight the main steps of the present developments in this area, we consider the Poisson problem with homogeneous Dirichlet boundary conditions. Let Ω ⊂ R d be a convex domain and consider the problem: We consider T to be a conforming triangulation of Ω, namely, T is a finite family of sets such that (1) K ∈ T implies K is an open simplex (segment for d = 1, triangle for d = 2, tetrahedron for d = 3), (2) for any K, J ∈ T we have that K ∩ J is a full lower-dimensional simplex (i.e., it is either ∅, a vertex, an edge, a face, or the whole of K and J) of both K and J and (3) K∈T K = Ω. The shape regularity constant of T is defined as the number where ρ K is the radius of the largest ball contained inside K and h K is the diameter of K. An indexed family of triangulations {T n } n is called shape regular if Further, we define h : Ω → R to be the piecewise constant meshsize function of T given by A mesh is called quasiuniform when there exists a positive constant C such that max x∈Ω h ≤ C min x∈Ω h.
In what follows we shall assume that all triangulations are shape-regular and quasiuniform.
We let E be the skeleton (set of common interfaces) of the triangulation T and say e ∈ E if e is on the interior of Ω and e ∈ ∂Ω if e lies on the boundary ∂Ω and set h e to be the diameter of e. We also define the "broken" gradient ∇ h , Laplacian ∆ h and Hessian D 2 h to be defined element-wise by ∇ h w| K = ∇w, ∆ h w| K = ∆w, D 2 h w| K = D 2 w for all K ∈ T , respectively, for respectively smooth functions on the interior of K, We let P k (T ) denote the space of piecewise polynomials of degree k over the triangulation T , and introduce the finite element space V := P k (T ) to be the usual space of discontinuous piecewise polynomial functions of degree k. We define average operators for arbitrary scalar functions v and vectors v over an edge e shared by elements K 1 and ) and jump operators as v = v| K1 n K1 + v| K2 n K2 , v = v| K1 · n K1 + v| K2 · n K2 . Note that on the boundary of the domain ∂Ω the jump and average operators are defined as Definition 2.1 (mesh dependent norms). We introduce the mesh dependent L 2 −, H 1 -and H 2 -norms to be Note for w h ∈ V in view of scaling each mesh dependent norm is equaivalent to the continuous counterpart, that is w h 0,h ∼ w h L2(Ω) for example.
Consider the interior penalty (IP) discretisation of (1), to find u h ∈ V such that where σ 0 , σ 1 > 0 represent penalty parameters. Note that a standard choice is to take σ 1 = 0. The choice σ 1 = 0 results in a class of stabilised dG methods [7].

Conforming reconstruction operators:
The key tool in the proof of the inf-sup condition is the notion of reconstruction operators. It is commonplace in the a posteriori analysis of nonconforming schemes to make use of such operators. A simple, quite general methodology for the construction of reconstruction operators is to use an averaging interpolation operator into an H 2 -conforming finite element space. For example a C 1 Hsieh-Clough-Tocher (HCT) macro-element conforming space for H 2 conformity [5, 11, c.f.]. Another option is the use of Argyris-type reconstructions [5].
is defined as follows. Let x be a degree of freedom of the H 2 -conforming space HCT(k + 2) consisting of HCT-type macro-elements of degree k + 2, and let " K x be the set of all elements sharing the degree of freedom x. Then, the reconstruction at that specific degree of freedom is given by For the case k = 2, the associated degrees of freedom are illustrated in Figure 1. Notice that the degrees of freedom of the reconstruction are a superset of those of the original finite element. This is due to the lack of existence of a conforming H 2 (Ω) subspace in V for low k; for instance, the existence of an H 2 (Ω)-conforming space requires k ≥ 5 in two dimensions (Argyris space). Corresponding reconstructions for higher polynomial degrees have been considered in [5,11], for instance. HCT-reconstruction with the constant C > 0 independent of u h and of h.
Using this HCT(k + 2)-reconstruction, we can construct a further HCT(k + 2)-reconstruction admitting the same bounds, but also satisfying an adjoint orthogonality property. Definition 2.5 (HCT (k + 2)-Ritz reconstruction). We define the Hsieh-Clough-Tocher H 2 (Ω)-conforming Ritz reconstruction operator E R : V → HCT(k + 2) such that Lemma 2.6 (Properties of E R ). The HCT(k + 2)-Ritz reconstruction is well-defined and satisfies the orthogonality condition: In addition, for α = 0, 1, 2, we have for C > 0 constants, independent of u h and of h. (14), along with a standard inverse estimate, we deduce for C > 0 independent of u h . The orthogonality condition follows from integrating both sides of (14) by parts.
To see (16) we note that Using the properties of E 2 (u h ) from Lemma 2.4 shows the claim for α = 1. The result for α = 2 follows by an inverse inequality.
For α = 0 we use a duality argument. Take z ∈ H 2 (Ω) ∩ H 1 0 (Ω) as the solution of the dual problem in view of the orthogonality property (15). Integrating by parts we see The result follows using the approximability of the HCT(k + 2) space [8] and the regularity of the dual problem.
Proof The proof consists of two steps. We first show there exists a v ∈ W (h) such that and then show that along with the quasi-uniformity assumption on the mesh. Firstly note that, after an integration by parts, the IP method (8) can be written as for some parameter α ∈ R to be chosen below, we compute The orthogonality property of the HCT(k + 2)-Ritz reconstruction (14) yields Repeated use of the Cauchy-Schwarz inequality, therefore, gives We proceed to bound each of the terms I i individually. Note that in view of scaling and inverse inequalities we have for any w h ∈ V: for any edge/face e :=K 1 ∩K 2 ∈ E , and elements K 1 , K 2 ∈ T , with C 1 , C 2 depending only on the mesh-regularity and shape-regularity constants.
For I 1 , in view of Lemma 2.6, we have with constant C 3 > 0 being the maximum of all constants in (16) for all α.
For I 2 , (28) and Lemma 2.6 yield For I 3 , (29) and Lemma 2.6 yield For I 4 , we have for any 4 > 0, while for I 5 , we get for any 5 > 0; similarly for I 6 and for any 6 > 0, we have Finally, the last term I 7 can be bounded as follows: for any 7 > 0.

Lemma 2.8 (Stability of the Ritz projection).
Let R demote the A h (·, ·) orthogonal projector into V, then for w ∈ W (h) we have that Proof Let g ∈ W (h) be the solution to the discrete dual problem such that Let Π : H 1 (Ω) → V ∩ H 1 0 (Ω) a suitable projection with optimal approximation properties. Then through the continuity of A h (·, ·). From the optimal approximation properties of the projection/interpolant Π, we have and, using the discrete regularity of g induced by the inf-sup condition in Theorem 2.7 Hence we see that in view of the quasi-best approximation in · 1,h from (11). The conclusion follows from standard inverse inequalities.
Theorem 2.9 (inf-sup stability over V). For polynomial degree k ≥ 2 there exists a γ h > 0, independent of h, such that when σ 0 , σ 1 1 Proof To show (46) we fix w h and let Φ ∈ V be the solution of the dual problem Following the same arguments as in the proof of Theorem 2.7, it is clear that there exists a C > 0 such that , and hence in view of Lemma 2.8 we have, with R denoting the A h orthogonal projector into V, that through inverse inequalities. Hence arguing as above Corollary 2.10 (Convergence). Let u solve (1) and u h ∈ V be the interior penalty approximation from (8), then If u ∈ H k+1 (Ω) the following a priori bound holds: Proof Using the inf-sup condition from Theorem 2.9 we see Now using the natural continuity bound Hence, in view of the triangle inequality The conclusion follows since w h was arbitrary.

Applications to problems with Rough Data
In this section we examine the problem where δx denotes the Dirac distribution at a pointx ∈ Ω. In this case we have f ∈ H −2 (Ω) H −1 (Ω) and hence the solution u ∈ L 2 (Ω) H 1 (Ω). This means it cannot be characterised through a weak formulation, rather an ultra weak formulation, where we seek u ∈ L 2 (Ω) such that ∀ v ∈ H 2 0 (Ω), and the right hand side of (59) is understood as a duality pairing. In this setting standard methods pertaining to the analysis of Galerkin methods may not apply, for example the Aubin-Nitsche duality arguement. However, if we assume thatx does not lie on the skeleton of the triangulation the stabilised IP method is still well defined and the inf-sup condition still holds. We define our approximation as seeking u h ∈ V such that Since the inf-sup condition given in Theorem 2.9 is a condition only on the operator itself the first statement in Corollary 2.10 holds true. The only uncertainty with the bound is the behaviour of the inconsistency term. The control of this term is the main motivation in the nonstandard definition of the right hand side of (60). Theorem 3.1 (quasi-optimal error control for problems with rough data). Let u ∈ L 2 (Ω) solve (59) and u h ∈ V be the approximation defined through (60), then Proof The proof of this fact takes some inspiration from that of [14] where inconsistency terms arise from the fact that the solution of an elliptic problem may only lie in H 1 (Ω) for which the operator A h (u, v h ) may not be well defined. Here we have even more difficulty since the solution u ∈ L 2 (Ω) H 1 (Ω). Using the inf-sup condition from Theorem 2.9 we have Now by adding and subtracting appropriate terms and using (59) and (60) we see by the orthogonality properties of E R (w h ) given in (14). Now we may use that through the stability of E R (v h ). In addition by the definition of the solution to the PDE Finally, using the approximation properties of E R (·) and E 2 (·) we see Substituting (64), (65) and (66) into (63) we have  In this experiment we test the L 2 convergence of the interior penality method to approximate the distributional solution (69).