Abstract
We prove the infsup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) meshdependent norms for quasiuniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasioptimal error control.
Similar content being viewed by others
1 Introduction
Discontinuous Galerkin (dG) methods are a popular family of nonconforming finite elementtype 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 [5, 21]; 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 \({\text {C}} ^{1}\)conforming elements whose derivation and implementation can become very complicated [5, 13, 23].
Infsup 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 \({\text {L}} _{2}\) and \({\text {H}} ^{2}\)like meshdependent norms. This is in keeping with the spirit of [3, 4], 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 infsup stability for both \({\text {L}} _{2}\) and \({\text {H}} ^{2}\)like meshdependent norms under the assumption that the underlying mesh is quasiuniform.
The analysis presented utilises a new \({\text {H}} ^{2}\)conforming reconstruction operator, based on Hsieh–Clough–Tochertype \({\text {C}} ^{1}\) reconstructions. Such reconstructions, based on nodal averaging, are used for the proof of a posteriori bounds for nonconforming methods for elliptic [7, 13, 18, 24] and hyperbolic problems [12, 16]. 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 \({\text {H}} ^{2}\)conforming reconstruction from [13].
The argument is quite general and allows the derivation of infsup stability results whenever the numerical scheme has a well posed discrete adjoint (dual) problem over an appropriately constructed nonconforming finite element space. This is contrary to the Aubin–Nitsche \({\text {L}} _{2}\) 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 \({\text {H}} ^{2}(\Omega )\) [17]. 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 quasioptimal \({\text {L}} _{2}\) 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 \({\text {H}} ^{1}\) in both 1 and 2 spatial dimensions. 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 Sect. 2 we introduce the problem and present the analysis cumulating in infsup stability for problems with smooth data. In Sect. 3 we examine a particular problem with rough data and prove quasioptimal convergence in this case. In addition we give some numerical validation of the method.
2 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 \(\Omega \subset {\mathbb R} ^d\) be an open convex domain and consider the problem: given \(f\in {\text {L}} _{2}(\Omega )\) find \(u\in {\text {H}} ^{2}(\Omega )\cap {\text {H}} ^{1}_0(\Omega )\), such that
We consider \({\mathscr {T}} ^{}\) to be a conforming triangulation of \(\Omega \), namely, \({\mathscr {T}} ^{}\) is a finite family of sets such that

(1)
\(K\in {\mathscr {T}} ^{}\) implies K is an open simplex (segment for \(d=1\), triangle for \(d=2\), tetrahedron for \(d=3\)),

(2)
for any \(K,J\in {\mathscr {T}} ^{}\) we have that \(\overline{K}\cap \overline{J}\) is a full lowerdimensional simplex (i.e., it is either \(\emptyset \), a vertex, an edge, a face, or the whole of \(\overline{K}\) and \(\overline{J}\)) of both \(\overline{K}\) and \(\overline{J}\) and

(3)
\(\bigcup _{K\in {\mathscr {T}} ^{}}\overline{K}=\overline{\Omega }\).
The shape regularity constant of \({\mathscr {T}} ^{}\) is defined as the number
where \(\rho _K\) is the radius of the largest ball contained inside K and \(h_K\) is the diameter of K. An indexed family of triangulations \(\left\{ {{\mathscr {T}} ^{n}}\right\} _n\) is called shape regular if
Further, we define \(h:\Omega \rightarrow {\mathbb R} \) to be the piecewise constant meshsize function of \({\mathscr {T}} ^{}\) given by
In addition if
we call \({\mathscr {T}} ^{}\) quasiuniform. If an entire indexed family of triangulations satisfy (4), we call it a quasiuniform family. In what follows we shall assume that all triangulations are shaperegular and quasiuniform.
We let \({\mathscr {E}} {}\) be the skeleton (set of common interfaces) of the triangulation \({\mathscr {T}} ^{}\) and say \(e\in {\mathscr {E}} \) if e is on the interior of \(\Omega \) and \(e\in \partial \Omega \) if e lies on the boundary \(\partial \Omega \) and set \(h_e\) to be the diameter of e. We also define the “broken” gradient \(\nabla _h\), Laplacian \(\Delta _h\) and Hessian \(\mathrm {D}^2_h\) to be defined elementwise by \(\nabla _h w_K = \nabla w\), \(\Delta _h w_K = \Delta w\), \(\mathrm {D}^2_h w_K = \mathrm {D}^2w\) for all \(K\in {\mathscr {T}} ^{}\), respectively, for respectively smooth functions on the interior of K,
We let \({\mathbb P} ^{k}({\mathscr {T}} ^{})\) denote the space of piecewise polynomials of degree k over the triangulation \({\mathscr {T}} ^{}\), and introduce the finite element space \(\mathbb {V}:= {\mathbb P} ^{k}({\mathscr {T}} ^{})\) to be the usual space of discontinuous piecewise polynomial functions of degree k. We define average operators for arbitrary scalar functions v and vectors \(\mathbf {v}\) over an edge e shared by elements \(K_1\) and \(K_2\) as \(\{\!\{ v \}\!\} = {\tfrac{1}{2}\!\left( {v_{K_1} + v_{K_2}}\right) }\), \(\{\!\{ \mathbf {v} \}\!\} = {\tfrac{1}{2}\!\left( {\mathbf {v}_{K_1} + \mathbf {v}_{K_2}}\right) }\) and jump operators as , . Note that on the boundary of the domain \(\partial \Omega \) the jump and average operators are defined as \(\{\!\{ v \}\!\} \Big \vert _{\partial \Omega } := v\), \(\{\!\{ \varvec{v} \}\!\} \Big \vert _{\partial \Omega } := \varvec{v}\), , ,
Definition 2.1
(Mesh dependent norms) We introduce the mesh dependent \({\text {L}} _{2}\), \({\text {H}} ^{1}\)and \({\text {H}} ^{2}\)norms to be
Note for \(w_h \in \mathbb {V}\) in view of scaling each mesh dependent norm is equaivalent to the continuous counterpart, that is \(\Vert {{w_h}} \Vert _{0,h} \sim \Vert {w_h}\Vert _{{\text {L}} _{2}(\Omega )}\) for example.
Consider the interior penalty (IP) discretisation of (1), to find \(u_h \in \mathbb {V}\) such that
where
where \(\sigma _0 >0, \sigma _1 \ge 0\) represent penalty parameters. Note that a standard choice is to take \(\sigma _1 = 0\). The choice \(\sigma _1 \ne 0\) results in a class of stabilised dG methods [8].
Proposition 2.2
(Continuity and coercivity of \({\mathscr {A}} _h\!\left( {\cdot ,\cdot }\right) \) [1, 11, c.f.]) For \(\sigma _1 \ge 0\) and \(\sigma _0\) large enough and any \(u_h,v_h\in \mathbb {V}\) the bilinear form \({\mathscr {A}} _h\!\left( {\cdot ,\cdot }\right) \) satisfies
LaxMilgram Theorem guarantees a unique solution to the problem (7). Also, since \(u\in {\text {H}} ^{2}(\Omega )\), the bilinear form is consistent, hence, Strang’s Lemma yields quasioptimal convergence of the method in the \(\Vert {{\cdot }} \Vert _{1,h}\) norm:
Conforming reconstruction operators The key tool in the proof of the infsup 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 \({\text {H}} ^{2}\)conforming finite element space. For example a \({\text {C}} ^{1}\) Hsieh–Clough–Tocher (HCT) macroelement conforming space for \({\text {H}} ^{2}\) conformity [6, 13, 25, c.f.]. Another option is the use of Argyristype reconstructions [6].
Example 2.3
(Construction of the HCT(4) space for \(d=2\)) Since the HCT spaces are an integral part of our analysis, we will illustrate the construction of the HCT(4) space for \(d=2\), over triangles, noting that for \(d=1\), we can simply consider families of cubic splines, while for \(d=3\) corresponding constructions are possible [19, Ch. 18]. Consider a triangle K that is partitioned in 3 subtriangles, \(\{K_i \}_{i=1}^3\) by connecting each of the vertices to the barycentre as illustrated in Fig. 1. We then take
The dimension of \({\mathbb P} ^{4}(K_i)\) is 15. It can then be verified that, upon enforcing continuity of functions and their derivatives across the subtriangulation edges for the degrees of freedom depicted in Fig. 1, the dimension of the HCT(4) space is 21 and those degrees of freedom and unisolvent.
For the general construction of the macroelement HCT(r) space for simplicial and boxtype elements, we refer to [10, 19, 22].
Definition 2.4
(\({\text {H}} ^{2}(\Omega )\) reconstructions) An example of \({\text {H}} ^{2}(\Omega )\) reconstruction operator \(E^2(u_h)\) of kth order Lagrange elements is defined as follows. Let \(\mathbf {x}\) be a degree of freedom of the \(H^2\)conforming space \(\text {HCT}(k+2)\) consisting of HCTtype macroelements of degree \(k+2\), and let \(\widehat{K_{\mathbf {x}}}\) be the set of all elements sharing the degree of freedom \(\mathbf {x}\). 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 Fig. 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 \({\text {H}} ^{2}(\Omega )\) subspace in \(\mathbb {V}\) for low k; for instance, the existence of an \({\text {H}} ^{2}(\Omega )\)conforming space requires \(k\ge 5\) in two dimensions (Argyris space). Corresponding reconstructions for higher polynomial degrees have been considered in [6, 13], for instance.
Even though the \(\text {HCT}(k+2)\) space contains functions that are not polynomial, it does include \({\mathbb P} ^{k+2}(K)\) and hence the \(\text {HCT}(k+2)\) interpolant preserves \({\mathbb P} ^{k+2}(K)\) functions, hence by BrambleHilbert the \(\text {HCT}(k+2)\) space has quasioptimal approximability.
We refer also to the discussion in [13], regarding reconstructions of boxtype kth order Lagrange elements into \({ HCT}(k+2)\).
Lemma 2.5
(Reconstruction bounds [13, Lem 3.1]) For \(d=1,2\), the \(\text {HCT}(k+2)\) reconstruction operator \(E^2 : \mathbb {V}\rightarrow {\text {H}} ^{2}(\Omega )\) satisifies the following bound for all \(u_h\in \mathbb {V}\):
with the constant \(C>0\) independent of \(u_h\) and of h.
Remark 2.6
Lemma 2.5 is proven in [13] for \(d=2\). For \(d=1\), we can recover into cubic or quartic splines and the proof is completely analogous. The proof for \(d=3\) using one of the trivariate \(C^1\)elements with nodal and normal derivative degrees of freedom presented in [19] is conjectured to follow along the same lines to the proof of [13, Lem 3.1].
Using this \(\text {HCT}(k+2)\)reconstruction, we can construct a further \(\text {HCT}(k+2)\)reconstruction admitting the same bounds, but also satisfying an adjoint orthogonality property.
Definition 2.7
(\(HCT(k+2)\) Ritz reconstruction) We define the Hsieh–Clough–Tocher \({\text {H}} ^{2}(\Omega )\)conforming Ritz reconstruction operator \(E^{{\mathscr {R}}}: \mathbb {V}\rightarrow \text {HCT}(k+2)\) such that
Lemma 2.8
(Properties of \(E^{{\mathscr {R}}}\)) The \(\text {HCT}(k+2)\)Ritz reconstruction is welldefined and satisfies the orthogonality condition:
In addition, for \(\alpha =0,1,2\), we have
for \(C>0\) constants, independent of \(u_h\) and of h.
Proof
Fixing \(u_h\in \mathbb {V}\), \(E^{{\mathscr {R}}}(u_h)\) is welldefined. Indeed, setting \(\widetilde{v}=E^{{\mathscr {R}}}(u_h)\) in (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
Notice that in order to invoke coercivity of \({\mathscr {A}} _h\) over W(h) we must choose \(\sigma _0\) larger than if we merely required coercivity over \(\mathbb {V}\) since, as already mentioned in Example 2.4, W(h) contains piecewise polynomials two degrees higher than \(\mathbb {V}\). Using the properties of \(E^2(u_h)\) from Lemma 2.5 shows the claim for \(\alpha =1\). The result for \(\alpha = 2\) follows by an inverse inequality.
For \(\alpha = 0\) we use a duality argument. Take \(z\in {\text {H}} ^{2}(\Omega )\cap {\text {H}} ^{1}_0(\Omega )\) as the solution of the dual problem
then
in view of the orthogonality property (15). Integrating by parts we see
The result follows using the approximability of the \(\text {HCT}(k+2)\) space [9] that can be inferred through the dimensional analysis in Definition 2.3 and the regularity of the dual problem, specifically
thereby concluding the proof. \(\square \)
Theorem 2.9
(inf–sup stability over W(h)) For polynomial degree \(k \ge 2\) there exists a \(\gamma _h>0\), depending on the quasiuniformity constant \(C_{qu}\), such that, when \(\sigma _0, \sigma _1\) are chosen large enough, we have for all \(w_h\in \mathbb {V}\)
where \(W(h):= \mathbb {V}+\text {HCT}(k+2)\).
Proof
The proof consists of two steps. We first show for a given \(w_h\in \mathbb {V}\) that there exists a \(\widetilde{v}\in W(h)\) such that
and that
which, along with the quasiuniformity assumption on the mesh, yields the infsup condition (22).
Firstly note that, after an integration by parts, the IP method (7) can be written as
Upon setting \(\widetilde{v}= w_h  E^{{\mathscr {R}}}(w_h)  \alpha h^2 \Delta _h w_h\), for some parameter \(\alpha \in {\mathbb R} \) to be chosen below, we compute
The orthogonality property of the \(\text {HCT}(k+2)\)Ritz reconstruction (15) yields
Repeated use of the Cauchy–Schwarz inequality, therefore, gives
We proceed to bound each of the terms \({\mathscr {I}} _i\) individually. Note that in view of scaling and inverse inequalities we have for any \(w_h\in \mathbb {V}\):
for any edge/face \(e:=\bar{K}_1\cap \bar{K}_2\in {\mathscr {E}} \), and elements \(K_1, K_2\in {\mathscr {T}} ^{}\), with \(C_1,C_2\) depending only on the meshregularity and shaperegularity constants.
For \({\mathscr {I}} _1\), in view of Lemma 2.8, we have
with constant \(C_3>0\) being the maximum of all constants in (16) for all \(\alpha \).
For \({\mathscr {I}} _2\), (29) and Lemma 2.8 yield
For \({\mathscr {I}} _3\), (30) and Lemma 2.8 yield
For \({\mathscr {I}} _4\), we have
for any \(\epsilon _4>0\), while for \({\mathscr {I}} _5\), we get
for any \(\epsilon _5>0\); similarly for \({\mathscr {I}} _6\) and for any \(\epsilon _6>0\), we have
Finally, the last term \({\mathscr {I}} _7\) can be bounded as follows:
for any \(\epsilon _7>0\).
Collecting the results (31)–(37) and substituting this into (28) we deduce
To arrive to (23), we can choose \(\epsilon _4 = \epsilon _5 =\epsilon _6 = \epsilon _7 = \frac{1}{5}\), \(\alpha =(\max \!\left( {\sigma _0^2,\sigma _1^2}\right) )^{1}\) and \(\sigma _0\) and \(\sigma _1\) large enough.
For (24), we use Lemma 2.8 to see that
which, together with the quasiuniformity of the meshes, completes the proof. \(\square \)
Lemma 2.10
(Stability of the Ritz projection) Let \(R:W(h) \rightarrow \mathbb {V}\) denote the \({\mathscr {A}} _h\!\left( {\cdot ,\cdot }\right) \) orthogonal projector into \(\mathbb {V}\). Then, for \(\widetilde{w}\in W(h)\), there exists a \(C>0\), independent of h but possibly dependent on the quasiuniformity constant, \(C_{qu}\), such that
Proof
Let \(\widetilde{g}\in W(h)\) be the solution to the discrete dual problem such that
Note that this is well posed owing to coercivity as long as the penalty parameters are tuned to account for the fact that W(h) contains piecewise polynomials over the subpartition two degrees higher than \(\mathbb {V}\) itself. Then, we have
Let \(\Pi : {\text {H}} ^{1}(\Omega )\rightarrow \mathbb {V}\cap {\text {H}} ^{1}_0(\Omega )\) a suitable projection with optimal approximation properties. Then
through the continuity of \({\mathscr {A}} _h\!\left( {\cdot ,\cdot }\right) \). From the optimal approximation properties of the projection/interpolant \(\Pi \), we have
and, using the discrete regularity of \(\widetilde{g}\) induced by the infsup condition in Theorem 2.9
Hence we see that
in view of the quasibest approximation in \(\Vert {{\cdot }} \Vert _{1,h}\) from (10). The conclusion follows from standard inverse inequalities. \(\square \)
Theorem 2.11
(inf–sup stability over \(\mathbb {V}\)) For polynomial degree \(k \ge 2\) there exists a \(\gamma _h>0\), independent of h, but dependent on \(C_{qu}\) such that, when \(\sigma _0, \sigma _1 \ge 1\) are chosen as in Theorem 2.9, we have for all \(w_h \in \mathbb {V}\)
Proof
To show (47) we fix \(w_h\) and let \(\Phi \in \mathbb {V}\) be the solution of the dual problem
Following the same arguments as in the proof of Theorem 2.9, it is clear that there exists a \(C>0\) such that
where \(\widetilde{v}:= \Phi  E^{{\mathscr {R}}}(\Phi )  \alpha h^2 \Delta _h \Phi \). Now it is clear that
and hence in view of Lemma 2.10 we have, with R denoting the \({\mathscr {A}} _h\) orthogonal projector into \(\mathbb {V}\), that
through inverse inequalities. Hence
Now arguing as in the Proof of Theorem 2.9, and noting the constant will now depend on \(C_{qu}\), we may show that
Combining the previous two inequalities yields
concluding the proof. \(\square \)
Corollary 2.12
(Convergence) Let u solve (1) and \(u_h \in \mathbb {V}\) be the interior penalty approximation from (7), then, under the assumptions of Theorem 2.11,
where \(\gamma _h\) is the discrete infsup constant and \(C_B\) is the continuity constant. If \(u\in {\text {H}} ^{k+1}(\Omega )\) for \(k \ge 2\) the following a priori bound holds:
Proof
Using the infsup condition from Theorem 2.11 we see for all
Now using the natural continuity bound
we see
Hence, in view of the triangle inequality
The bound (55) follows since \(w_h\) was arbitrary and (56) follows from the best approximation of \(\mathbb {V}\). \(\square \)
3 Applications to problems with rough data
In this section we examine some problems of the form
where f may be as rough as \({\text {H}} ^{2}(\Omega ) \backslash {\text {H}} ^{1}(\Omega )\) and so, \(u\in {\text {L}} _{2}(\Omega ) \backslash {\text {H}} ^{1}(\Omega )\). This means that the problem (75) cannot be characterised through a weak formulation, rather an ultra weak formulation, whereby we seek \(u\in {\text {L}} _{2}(\Omega )\) such that
and the right hand side of (62) is understood as a duality pairing. In this setting standard tools pertaining to the analysis of Galerkin methods may not apply, for example the Aubin–Nitsche duality argument.
However, the stabilised IP method is still well defined and the infsup condition still holds. Indeed, we define the modified IP method: seek \(u_h\in \mathbb {V}\) such that
where \(E^2_0:\mathbb {V}\rightarrow H^2_0(\Omega )\) is the modification of \(E^2\) recovering onto \(H^2_0(\Omega )\) which is constructed by setting to zero all the degrees of freedom on \(\partial \Omega \). We note that Lemma 2.5 still holds verbatim when \(E^2\) is replaced by \(E^2_0\). The nonstandard definition of the righthand side allows us to make sense of extremely rough source terms [15] by interpreting the righthand side a a duality pairing. Correspondingly, we also denote by \(E^{{\mathscr {R}}}_0\) the recovery given by Definition 2.7 when \(E^2\) is replaced by \(E^2_0\). Also, we note that Lemma 2.8 holds for \(E^{{\mathscr {R}}}_0\) also.
Since the infsup condition given in Theorem 2.11 is a condition only on the operator itself, the best approximation result given in Corollary 2.12 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 (63).
Theorem 3.1
(quasioptimal error control for problems with rough data) Let \(u\in {\text {L}} _{2}(\Omega )\) solve (62) and \(u_h\in \mathbb {V}\) be the approximation defined through (63), then
Proof
The proof takes some inspiration from that of [17], where inconsistency terms arise from the fact that the solution of an elliptic problem may only lie in \({\text {H}} ^{1}(\Omega )\), for which the operator \({\mathscr {A}} _h\!\left( {u,v_h}\right) \) may not be well defined. Here, the situation is more involved, since the solution \(u\in {\text {L}} _{2}(\Omega )\backslash {\text {H}} ^{1}(\Omega )\).
Using the infsup condition from Theorem 2.11 we have
Now, by adding and subtracting appropriate terms and using (62) and (63), we see
by the orthogonality properties of \(E^{{\mathscr {R}}}_0(w_h)\) given in (14). Now we may use that
through the stability of \(E^{{\mathscr {R}}}(v_h)\).
Finally, using the approximation properties of \(E^{{\mathscr {R}}}_0(\cdot )\) and \(E^2_0(\cdot )\) we see
Substituting (67) and (68) into (66) we have
and hence
as required. \(\square \)
3.1 Numerical experiments
The implementation of all the numerical experiments was performed in \(\textsf {Matlab}^{\textregistered }\) on a laptop computer with a 2.3GHz Intel i7 processor and 16 GB of RAM. All computations took less than 2 min on this machine.
3.1.1 Test 1: an onedimensional example
We begin by assessing the method (63) for \(d=1\). We set \(\Omega = (0,1)\) and consider the problem of finding u such that
where \(\delta _{\bar{x}}'\) denotes the distributional derivative of the Dirac distribution at a point \(\bar{x} \in \Omega \). This onedimensional problem was a motivating example in the classical work of Babuška and Osborn [3]; this computation is included here as a tribute to that inspiring work.
For this problem we can even characterise a distributional solution, indeed we have that
solves (75). If we assume that \(\bar{x}\) does not lie on the skeleton of the triangulation we can define our approximation as seeking \(u_h\in \mathbb {V}\) such that
Using Theorem 3.1 we are able to show this approximation satisfies the a priori bound
since \(\delta _{\bar{x}} \in {\text {H}} ^{s}(\Omega )\) for all \(s > 1/2\).
We fix \(k=2\) and solve (73) over a sequence of uniform meshes in 1d with \(h = 1/2, 1/4, \dots , 1/1024\). We take \(\bar{x} = 1/2 + \sqrt{2}/100000\) as to not align it with the nodes of the mesh. In Fig. 2 we show the numerical approximation over the finest mesh along with the experimental order of convergence.
3.1.2 Test 2: \(d=2\) with a Dirac source term at a point
We now take \(\Omega = B(0,1)\), the open ball of radius 1 centred at the origin, and consider the problem of finding u such that
The exact solution u is the fundamental solution of Laplace’s problem
for which we have \(u\in {\text {L}} _{2}(\Omega ) \backslash {\text {H}} ^{1}(\Omega )\).
We define our approximation as seeking \(u_h \in \mathbb {V}\) such that
and, under the assumptions of Theorem 3.1, we are able to show that this approximation satifies the a priori bound
This is in agreement with the results of [2] for conforming finite elements applied to this problem.
We fix \(k=2\) and solve (80) over a sequence of unstructured, quasiuniform triangulations of B(0, 1). For the coarsest mesh we have \(h \approx 0.13\) and for the most fine \(h \approx 0.0019\). In Fig. 3, we show the numerical approximation over the finest mesh along with the error measured in the \({\text {L}} _{2}\)norm and its convergence history. We remark, that the boundary approximation by straightfaced elements is not detrimental to the convergence rate in this case as the error is measured in the \({\text {L}} _{2}\)norm.
3.2 Test 3: \(d=2\), rough source terms defined over an onedimensional manifold
We now test the proposed method on a more complicated problem, whereby, we set \(\Omega = (0,1)^2\) and consider the problem of finding u such that
where \(\alpha \in \{0,1\}\) and \({\mathscr {M}}:= \{ (x,y) : \left x1/2\right< 1/4 \text { and } y = 1/2 \text { or } x = 1/2 \text { and } \left y1/2\right < 1/4\}\) is an onedimensional manifold. When \(\alpha = 1\), we have \(u\in {\text {H}} ^{1}(\Omega )\), whereas when \(\alpha = 0\), we have \(u\in {\text {L}} _{2}(\Omega )\backslash {\text {H}} ^{1}(\Omega )\).
We seek \(u_h \in \mathbb {V}\) such that
We fix \(k=2\) and solve (80) over a uniform, crisscross triangulation of \(\Omega \) with \(h \approx 0.015\). In Fig. 4 we show the numerical approximation over this mesh for both values of \(\alpha \).
We conclude this exposition by noticing that, since the singularity in all tests were isolated, adaptive approximations should be able to recover best approximation. This motivates the extension of this analysis from the quite restrictive quasiuniform meshes to those that allow for some grading. Note the recent work [14] where, using a posteriori localisation techniques, analagous infsup results for the classical interior penalty dG scheme have been proven over meshes satisfying a mesh variation condition.
References
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02)
Babuška, I.: Errorbounds for finite element method. Numer. Math. 16(1), 322–333 (1971)
Babuška, I., Osborn, J.: Analysis of finite element methods for second order boundary value problems using mesh dependent norms. Numer. Math. 34(1), 41–62 (1980)
Babuška, I., Osborn, J., Pitkäranta, J.: Analysis of mixed methods using mesh dependent norms. Math. Comput. 35(152), 1039–1062 (1980)
Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31(137), 45–59 (1977)
Brenner, S.C., Gudi, T., Sung, L.Y.: A weakly overpenalized symmetric interior penalty method for the biharmonic problem. Electron. Trans. Numer. Anal. 37, 214–238 (2010)
Brenner, S.C., Gudi, T., Sung, L.Y.: An a posteriori error estimator for a quadratic \(C^0\)interior penalty method for the biharmonic problem. IMA J. Numer. Anal. 30(3), 777–798 (2010)
Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43(5), 2012–2033 (2005). (electronic)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. NorthHolland Publishing Co., Amsterdam (1978)
Douglas Jr., J., Dupont, T., Percell, P., Scott, R.: A family of \(C^{1}\) finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér. 13(3), 227–255 (1979)
Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, Volume of 159 Applied Mathematical Sciences. Springer, New York (2004)
Georgoulis, E.H., Hall, E., Makridakis, C.: A posteriori error control for discontinuous Galerkin methods for first order hyperbolic problems. In: Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations: 2012 John H Barrett Memorial Lectures (2014)
Georgoulis, E.H., Houston, P., Virtanen, J.: An a posteriori error indicator for discontinuous Galerkin approximations of fourthorder elliptic problems. IMA J. Numer. Anal. 31(1), 281–298 (2011)
Georgoulis, E.H., Makridakis, C.G., Pryer, T.: BabuškaOsborn techniques in discontinuous Galerkin methods: \(L^2\) error estimates for unstructured meshes. Submitted—Technical Report (2017). arXiV arXiv:1704.05238
Georgoulis, E.H., Pryer, T.: Recovered finite element methods. Submitted—Technical Report (2017). ArXiV arXiv:1705.03649
Giesselmann, J., Makridakis, C., Pryer, T.: A posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws. SIAM J. Numer. Anal. 53(3), 1280–1303 (2015)
Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79(272), 2169–2189 (2010)
Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of secondorder elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003). (electronic)
Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations, Volume of 110 Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2007)
Makridakis, C.G.: On the BabuškaOsborn approach to finite element analysis: \(L^2\) estimates for unstructured meshes. Preprint (2016)
Nitsche, J.: Über ein Variationsprinzip zur Lösung von DirichletProblemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971). (Collection of articles dedicated to Lothar Collatz on his sixtieth birthday)
Percell, P.: On cubic and quartic Clough–Tocher finite elements. SIAM J. Numer. Anal. 13(1), 100–103 (1976)
Pryer, T.: Discontinuous Galerkin methods for the \(p\)biharmonic equation from a discrete variational perspective. Electron. Trans. Numer. Anal. 41, 328–349 (2014)
Pryer, T.: An a posteriori analysis of some inconsistent, nonconforming Galerkin methods approximating elliptic problems. Submitted—Technical Report (2015). ArXiV arXiv:1505.04318
Smears, I.: Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton–Jacobi–Bellman equations. J. Sci. Comput. 1–30 (2015)
Acknowledgements
TP acknowledges funding by the EPSRC grant EP/P000835/1.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Georgoulis, E.H., Pryer, T. Analysis of discontinuous Galerkin methods using meshdependent norms and applications to problems with rough data. Calcolo 54, 1533–1551 (2017). https://doi.org/10.1007/s1009201702405
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1009201702405