The convection-diffusion-reaction equation in non-Hilbert Sobolev spaces: A direct proof of the inf-sup condition and stability of Galerkin’s method

: While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space 𝐻 10 (Ω) , the Banach Sobolev space 𝑊 1 ,𝑞 0 (Ω) , 1 < 𝑞 < ∞ , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the 𝑊 1 ,𝑞 0 (Ω) - 𝑊 1 ,𝑞 ′ 0 (Ω) functional setting, 1 /𝑞 + 1 /𝑞 ′ = 1 . The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin’s method in this setting, for a diffusion-dominated case and under the assumption of 𝑊 1 ,𝑞 ′ stability of the 𝐻 1 0 -projector.


Introduction
In this paper we prove the inf-sup conditions for the convection-diffusion-reaction problem in a nonsymmetric Sobolev-space setting; in particular, we consider the 1, 0 (Ω)-1, ′ 0 (Ω)-setting, 1 < < ∞, 1/ + 1/ ′ = 1. The main motivation for considering this non-Hilbert setting is to allow more irregular solutions, if, e.g., the right-hand side is not in −1 (Ω) but in −1, ′ (Ω) for some 1 < < 2. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. We furthermore prove an elementary stability result of Galerkin's method in that setting.
In the context of finite element methods (FEMs), well-posedness of the convection-diffusion-reaction variational problem is traditionally established by proving coercivity and continuity of the underlying bilinear form in 1 0 (Ω). Due to the Lax-Milgram theorem this then implies well-posedness of the continuous problem and its discrete counterpart. The concept of coercivity, however, requires that the trial and test spaces should be the same. 1 This is not always possible or desirable, e.g., in the context of Petrov-Galerkin methods or mixed methods, or if the continuous problem itself is stated in a non-symmetric manner such as in the 1, 0 (Ω)-1, ′ 0 (Ω)-setting. A generalization of the Lax-Milgram theorem originally due to Nečas [27] replaces the coercivity requirement by two inf-sup conditions and can be formulated for Banach spaces with distinct test and trial spaces. A version of this theorem for approximations in finite dimensional subspaces is due to Babuška [3]. It is important to note that the inf-sup conditions in the continuous formulation does not imply an inf-sup condition in the discrete setting. In fact, for the convection-diffusionreaction equation in 1, 0 (Ω)-1, ′ 0 (Ω), we will establish discrete well-posedness under much stronger assumptions than in the continuous setting.
In the remainder of the introduction we specify the weak formulation of the underlying problem, stipulate the assumptions and announce our main results.

Problem statement
We consider the following model problem: find such that where > 0, : Ω → R , and : Ω → R are the (positive) diffusion parameter, convection field and reaction coefficient, respectively, and : Ω → R is a given source.
Multiplying (1.2a) by a test function ∈ ∞ (Ω) and integrating by parts yields the bilinear form This allows us to state the following variational problem for some ∈ (1, ∞): find ∈ := 1, 0 (Ω) such that where is allowed to be any element in ′ = −1, (Ω). Furthermore, we endow and with the following norms , (1.5b) with 0 given by Assumption 1.3 below. The proof of well-posedness of this problem with = 2 is well known, see, e.g., [12,Section 3.1]. Furthermore, well-posedness of the problem for ̸ = 2 in smooth domains (e.g., domains with 1 -or 1,1 -boundary) is also wellestablished [16]. For ̸ = 2 and general Lipschitz domains on the other hand, even simply proving well-posedness of the Poisson problem is far more challenging. It is no longer sufficient to require that Ω is a bounded Lipschitz domain. In fact, if ≥ 3, then for any > 3 there exists a Lipschitz domain Ω and a right-hand side ∈ ∞ (Ω) such that the solution to the Poisson problem is not in 1, (Ω),cf., [20]; in two dimensions the same result holds for > 4.
The novelty of our approach is that we provide a direct proof of the inf-sup conditions for the convection-diffusion-reaction equation illustrating that the socalled duality map is invaluable as a replacement for the Riesz isometry in the context of Banach spaces. The duality map is a fundamental operator in Banach spaces and will be recalled in Section 2.2. Furthermore, our proof only relies on standard assumptions on and , and 1, -regularity of the standard 1 0 -Poisson problem for all up to ′ . The latter assumption can be interpreted as an assumption on the domain Ω for a given , or, for a given bounded Lipschitz domain Ω it restricts the values that can be chosen for . Another important technique in our proof (for the second inf-sup condition) is a bootstrapping argument involving repeated use of elliptic regularity and Sobolev embeddings to gain sufficient regularity. To prove the discrete inf-sup condition, as needed for the stability of Galerkin's method, we will further assume a diffusion-dominated case and the 1, ′ -stability of the 1 0 -projector.

Assumptions
We now present the assumptions in detail. One of the main assumptions we rely on is (adjoint-) regularity of the standard 1 0 (Ω)-Poisson problem.
For more details we refer the reader to [20]. For 1 < ≤ 2 it is possible to obtain higher regularity for the solution, i.e., ∈ 2, (Ω), provided the right-hand side is in (Ω).

□
Similar to the analysis of the advection-reaction equation [6,25], we also need some requirements on the advective field and the reaction coefficient.
, and there exists a constant 0 > 0 such that In order to prove well-posedness in the discrete setting, we request that the 1 0 (Ω)projector onto the chosen ℎ-parametrized family of finite-dimensional subspaces is stable in the following sense. (Ω) be a finite-dimensional subspace. The 1 0 (Ω)-projector ℎ : 1 0 (Ω) → ℎ is uniformly stable in 1, ′ (Ω), i.e., there is a stability constant ≥ 1, independent of ℎ, such that for all ∈ 1, ′ 0 (Ω) ⊂ 1 0 (Ω), the unique 1 0 (Ω)-projection ℎ ∈ ℎ that solves the discrete problem satisfies the a priori bound: (1.10) We have not assumed any specific properties of the (finite element) space ℎ or the underlying mesh in order to keep the above assumption as general as possible. Note, however, that the validity of Assumption 1.4 has only been proven in very specific settings and thus it can be expected that this assumption implies certain restrictions. If Ω is a bounded interval in R, a convex polygonal domain in R 2 , or a convex polyhedral domain in R 3 , Assumption 1.4 is known to hold for all ′ > 2 for standard finite element spaces on quasi-uniform meshes, see [28,18] and [4,Chapter 8], and certain graded meshes, see [9,22]. Furthermore, the assumption is valid for all ′ > 2 if ℎ is a spectral space of d-variate polynomials of fixed degree on a (finite union of) star-shaped domain(s), see [10].

Main results
We can now state the main results of this paper, the proofs of which are given in Sections 3 and 4. We prove the inf-sup conditions, both in the continuous and the discrete settings, under the assumptions stated in the previous section. To this end, firstly, we have the well-posedness in the continuous case. Remark 1.6. The proof of (1.12a) given in Section 3 shows the following estimate for : (1.13) Here, , is the constant in the Poincaré-Friedrichs inequality in 1, (Ω), ′ ,Ω is the constant given in Assumption 1.1 for = ′ and )︁ .
For very small the dominating term in the numerator scales like Note that this is equal to 1 for = ′ = 2. The denominator scales like −1 ; the overall scaling for very small is therefore . For very small 0 the lower bound for the inf-sup constant scales like 0 . It can be assumed that this estimate is not optimal, since the scaling in and 0 for = ′ = 2 is less favourable than in the standard proof when = ′ = 2.
□ Secondly, consider the (Bubnov-) Galerkin approximation for a (finite-element) (1.14) In this discrete setting, we have the following result.
and ∈ ∞ (Ω). If Assumption 1.1 and Assumption 1.4 hold, then the following discrete inf-sup condition holds true: Note that it can be guaranteed that the constant̂︀ in (1.16) is strictly positive. Indeed, this is true if the convection-diffusion-reaction problem is sufficiently diffusion-dominated, i.e., the advection and the reaction coefficients are sufficiently small compared to the diffusion parameter . From this perspective, Theorem 1.7 is an elementary result: We strongly believe that (1.16) is a suboptimal result, and we conjecture that the discrete inf-sup condition holds true for any choice of the parameters (although not robustly).
In summary, by standard arguments (see, e.g., [12, Chapter 2]), Theorem 1.5 implies the existence of a unique solution ∈ 1, 0 (Ω) to the convection-diffusionreaction problem, which satisfies the a priori bound: Recall from (1.5a) that ‖ · ‖ is a norm on 1, 0 (Ω) equivalent to, e.g., | · | 1, (Ω) . Theorem 1.7 further implies that in the diffusion-dominated case, Galerkin's method is a stable and convergent method in the 1, 0 (Ω)-1, ′ 0 (Ω)-setting. In particular, the following a priori bound holds as well as the following a priori error estimate This last result can be sharpened, by invoking the error estimate due to Stern [29]: To illustrate the results for Galerkin's method, we verify the bound in (1.17) by performing several numerical experiments in two-and three-dimensional domains on convection-diffusion problems, for which the solution is not in 1 0 (Ω) but in 1, 0 (Ω) for suitable < 2. These numerical experiments show that Galerkin's method is indeed stable for the anticipated values ≤ .

Outline of the paper
The rest of the paper is organized as follows. In Section 2, we present several necessary preliminaries, in particular, we present well-posedness results for the Poisson problem in the 1, 0 -1, ′ 0 (Ω) setting, fundamental properties of duality mappings, and a brief discussion on difficulties in proving the continuous inf-sup conditions. We then present in Section 3 the proof of Theorem 1.5, relying heavily on the preliminaries in the preceding section. Section 3 discusses in the following order: the proof of the continuity of ℬ , the first inf-sup condition, and finally, the second inf-sup condition. In Section 4, we give the proof of Theorem 1.7 (discrete inf-sup condition). Section 5 contains the numerical experiments for Galerkin's method considering the approximation of irregular solutions / ∈ 1 0 (Ω). Finally, the Appendix A contains the proof of Proposition 2.1 (well-posedness of Poisson problem in the 1, In the context of finite element methods the analysis of variational formulations of PDEs is traditionally undertaken in subspaces of 2 (Ω) such as 1 0 (Ω). Even though we are considering a formulation in more general Sobolev spaces that are no longer Hilbert spaces, we are still using many standard techniques that have been developed for the numerical analysis of finite element methods. In this section we focus on illustrating how concepts and techniques that rely on a Hilbert space setting with identical test and trial spaces can be extended to our more general setting. To this end, we first consider the Poisson problem in the 1, 0 (Ω)-1, ′ 0 (Ω)setting as a simplified model problem, which will be crucial later when dealing with the full convection-diffusion-reaction problem.
As part of the proof of the inf-sup-conditions, we will construct a very specific functional in −1, ′ (Ω) that is an example of a duality mapping. We thus continue in Section 2.2 with introducing duality mappings as a general concept. Duality mappings have proven to be a very useful concept for mimicking certain techniques that rely on properties of Hilbert spaces in more general Banach spaces. In some sense duality mappings are a suitable nonlinear replacement for the Riesz map. This becomes particularly evident in the context of residual minimization problems; see, for example, [26].
Finally, we conclude this section by explaining how we intend to include the lower order terms in the convection-diffusion-advection equation. The next sections will then focus on rigorous proofs of Theorem 1.5 and Theorem 1.7.

The Poisson Problem in the
The goal is to extend the well-posedness proof of the Poisson problem to the convection-diffusion-reaction equation provided that Assumption 1.1 holds. We will now see that Assumption 1.1 indeed implies the following well-posedness result for the Poisson problem.
Furthermore, satisfies the a priori estimate Proof. Although the above well-posedness result is known (cf. [20]), we provide in Appendix A an elementary self-contained proof of the inf-sup conditions (2.6a)-(2.6b) using the duality map. We also demonstrate that when = 1, Assumption 1.1 is not needed.
Combining Proposition 2.1 again with Assumption 1.1 immediately implies the following general regularity result; this will be required in the proof of Theorem 1.5 instead of Assumption 1.1 directly.
Proof. This result is a direct consequence of Assumption 1.1 and Proposition 2.1. Suppose ∈ ( , 2), then apply Proposition 2.1 (replacing by and by ) to obtain that ∈ 1, Let us recall that for = 2 well-posedness follows immediately from the Lax-Milgram Theorem since both continuity and coercivity are obvious in this case. The same approach is obviously no longer applicable if ̸ = 2 since it requires the trial and test spaces to be identical. We thus apply a generalized Lax-Milgram Theorem (or BNB Theorem [12,Chapter 2]) that is originally due to Nečas [27]. A version of this theorem for approximations in finite dimensional subspaces was derived by Babuška [3].
Let and be Banach spaces, and assume additionally that is reflexive. Let : × → R be a continuous bilinear form and ℓ ∈ ′ . Furthermore, consider the problem: find ∈ such that Then, (2.5) is well-posed for all ℓ ∈ ′ if and only if Moreover, we have the a priori estimate: For the Poisson problem with = 2, the inf-sup condition (2.6a) is again immediate since in this case The key observation here is that ( , ) is equal to the square of the norm on . For < 2, we can no longer use itself as a test function. But if we suppose that there exists a such that we would again immediately obtain an inf-sup condition.

□
The question now is whether it is possible to construct such a test function ∈ 1, ′ 0 (Ω) for any ∈ 1, 0 (Ω). Let us start with the right-hand side of (2.9a).
As a first step we determine a : Ω → R such that (2.10) If we divide each summand by , we can see that has to be of the form Ideally, we would like to construct ∈ 1, ′ 0 (Ω) such that ∇ = , which, however, seems to be impossible in general. Instead we identify with ℓ ∈ It is easy to check that we have ∈ ′ (Ω) and as a consequence ℓ ∈ −1, ′ (Ω). Furthermore, we can compute ‖ℓ ‖ −1, ′ (Ω) = | | −1 1, (Ω) using the duality of the spaces (Ω) and ′ (Ω). We can now define as the solution to the following Poisson problem: Note that the existence of ∈ 1 0 (Ω) is due to the well-posedness of the Poisson problem in 1 0 (Ω). The higher regularity of then follows from Assumption 1.1 and the a priori estimate in Assumption 1.1 implies (2.9b). Furthermore note that by definition ℓ ( ) = | | 1, (Ω) , which implies (2.9a).
If we instead use˜= we can analogously define a linear functional ℓ˜and a test function˜that satisfies the conditions in Remark 2.4.

Duality Mappings
In the previous section we have heuristically constructed the linear functional ℓt hat allowed us to define a suitable test function. In [26, Section 2] the same idea is undertaken in a rather more abstract setting. The functional ℓ˜can also be defined as the functional ℓ˜∈ −1, ′ (Ω) such that The Hahn-Banach Theorem implies existence of a functional with these properties. In order to obtain uniqueness, we have to require strict convexity of the underlying space, which is true for 1 < < ∞. A linear functional with the above properties is called a duality mapping and we will now give a general definition of duality mappings in Banach spaces.
is called a duality mapping 2 .
□ Due to a corollary of the Hahn-Banach Theorem (see, e.g., [ The following theorem is a special case of Theorem 4.4 in [7, Chapter I] and states that the duality map on can be characterized using the subdifferential of the norm on . This is a key property of the duality map that will allow us to derive the duality map for some specific Banach spaces in the special case that the subdifferential is essentially the Gâteaux or Fréchet derivative of the norm. where ( ) denotes the subdifferential of at .
If ′ is strictly convex and thus the duality mapping is single valued, the subdifferential exists as a Gâteaux derivative. This allows us to explicitly compute the duality mappings for Sobolev spaces with exponent 1 < < ∞. Let us denote the duality mapping on (Ω) by . We compute

Lower Order Terms
We now aim to extend the result we have seen for the Poisson problem to the convection-diffusion-reaction equation. For = 2, well-posedness again follows from the Lax-Milgram Theorem due to the continuity and coercivity of the underlying bilinear form. Let us take a brief look at the well-known proof of coercivity for the bilinear form ℬ under Assumption 1.3 with = 2: (2.20) As for the Poisson problem, coercivity also immediately implies the inf-sup condition (2.6a). For ̸ = 2, we have already seen how the Laplace operator can be treated; thereby, we will now focus on the lower order terms. In (2.20), there are two steps that are applied to the lower order terms: the first one is rearranging everything by integration by parts such that Assumption 1.3 can be applied and the second one is the rather trivial observation that the resulting term can be bounded by the square of the 2 (Ω)-norm of . If ̸ = 2, we again have the problem that we cannot simply test with itself. The idea is to start from the end and mimic the steps in the proof for the Poisson problem by constructing a test function such that Due to the duality of the spaces (Ω) and ′ (Ω) this construction actually becomes easier than for the Poisson problem. Either by using the abstract concept of duality mappings or by explicitly constructing , we can see immediately that must be defined as It is easy to verify that ∈ ′ (Ω). However, to use as a test function we require ∈ 1, ′ 0 (Ω). It turns out, that this is not true for < 2, but it is the case for > 2 (a proof of this will be given in Section 3.3). This suggests to prove the following inf-sup conditions on the adjoint instead: This is equivalent to (1.12) (for an elementary proof, see, e.g., [23,Prop. A.2]). In particular, the inf-sup constant is the same. Another observation is the following. We need to be able to mimic the steps in the proof of coercivity in the -setting: (2.24) The last issue that has to be resolved is that we have two different test functions -one for the Laplace operator and one for the advection operator. In order to obtain an inf-sup condition for the full bilinear form we thus consider a linear combination of both functions and additionally estimate the Laplace term tested with the second test function and the advection-reaction term with the first. The rather technical details of this are presented in the next section.

Proof of Theorem 1.5: Well-posedness of the Convection-Diffusion-Reaction Equation in
1,

(Ω)
In this section we now present a rigorous proof of Theorem 1.5. We start by showing continuity of the bilinear form ℬ in Section 3.1. Sections 3.2 and 3.3 then contain all necessary estimates for the two test functions introduced in Sections 2.1 and 2.3, respectively. In Section 3.4 we then consider a linear combination of the two test functions and combine all of the estimates to conclude the proof of the inf-sup condition (2.23a). We then finish the proof of Theorem 1.5 in Section 3.5 by proving the second inf-sup condition (2.23b). For this last step we employ a so-called bootstrap argument that is more commonly used in the PDE literature, cf., Remark 3.2 below. where we consider ∇ · ( ) ∈ 2 (Ω) as an element in ′ = −1 (Ω) ⊃ 2 (Ω) with its norm on ′ given by A proof of this result is presented in [30,Section 4.4.1]. There are essentially three steps to the proof: firstly, it is easy to prove that the continuity constant with respect to the norms ‖ · ‖ and ‖ · ‖˜is independent of ; secondly, an -independent estimate for the inf-sup constant can be proven using the norm (1.5a) on ; thirdly, it is then possible to bound the term (3.2) from above as well without losing robustness of the inf-sup constant. The first and the third step can easily be extended to ̸ = 2. However, the generalization of the second step to Banach spaces remains an open problem.

Continuity of the bilinear form
In order to prove continuity of the bilinear form (1.11), we apply the Hölder inequality and obtain Here we chose to integrate the advection term by parts and then apply the Hölder inequality. This is a rather arbitrary choice, but note that some -dependence of the continuity constant cannot be avoided even if the advection term is estimated in its current form.

A test function for the Laplace operator
We now start with proving the inf-sup condition (2.23a) by establishing estimates for a test function that is tailored for the diffusion part of the bilinear form. Let ∈ 1, 0 (Ω) be the unique solution to the problem Hence, by definition Next we look at the advection-reaction term; to bound these terms we integrate by parts, and employ Hölder's inequality, the Poincaré-Friedrichs inequality: and the a priori estimate for ∇ (1.7). Thereby, we get (3.7) To simplify the notation we define Using Young's inequality gives

A test function for the advection-reaction operator
We now consider the second test function introduced in Section 2.3 to obtain a lower bound for the advection-reaction term. As we have already mentioned, we require 1, 0 (Ω)-regularity for the test function and hence we first have to prove ′ ( ) ∈ 1, 0 (Ω). By definition ′ ( ) ∈ (Ω); we will now show that where it is essential that ′ > 2. Now we can apply (2.24) to obtain (3.11) Next, we have to bound the diffusion term when testing with ′ ( ). Here, we observe that thereby, we can disregard this term in the inf-sup analysis.

Combining the estimates
We now combine all the above estimates and prove a lower bound for the bilinear form when testing with a linear combination of the two test functions above, i.e., we test with where is a constant to be chosen. Combining the estimates (3.11), (3.12), (3.5) and (3.9), we obtain (3.14) If we now choose = 2 To conclude, we now only need to estimate ‖ ‖ . According to Proposition 2.1 we have Thus, recalling the Poincaré-Friedrichs inequality (3.6), we obtain Similarly, using (3.10), gives Dividing (3.15) by ‖ ‖ and using the above estimates finally yields the inf-sup condition (2.23a). We refer the reader to (1.13) for the final estimate of the inf-sup constant .

The second inf-sup condition
The final step is the proof of the second inf-sup condition (2.23b). To this end, assume there exists 0 ̸ = ∈ 1, 0 (Ω) such that The idea is to use the above equation to gain sufficient regularity in order to test with itself. A priori this is not possible since 1, 0 (Ω) ̸ ⊂ 1, ′ 0 (Ω) for ′ > . We will consider as a solution to a Poisson problem and then use the regularity of the right-hand side in order to employ Corollary 2.2 and the Sobolev embedding theorem to gain higher regularity for . We then iterate this until we have gained sufficient regularity.
Remark 3.2. The idea of exploiting the Sobolev embedding multiple times iteratively is often referred to as a bootstrapping argument. These types of arguments are commonly used to obtain improved integrability or regularity in the context of elliptic partial differential equations. One example is the proof of Lemma 9.16 in [16]. There the same argument is used locally to improve a previously obtained -estimate (cf., [16,Theorem 9.13]) to an ′ -estimate. The fact that we can apply this argument globally, i.e., on the whole domain Ω, in our case is due to the regularity assumption on the domain (Assumption 1.1). For sufficiently smooth boundaries (e.g., 1 or 1,1 ), local estimates can often be turned into global estimates by combining them with boundary estimates (see, e.g., [16,Chapter 6] in the context of Schauder estimates). □ Remark 3.3 (The case > 2). Theorem 1.5 can be analogously proven for > 2. Indeed, instead of proving the inf-sup condition on the adjoint, we can prove the inf-sup condition directly. The steps of the proof in Sections 3.1 and 3.2 are identical with and ′ swapped and the constant 1 has to be adjusted slightly to account for the fact that the advection term is not symmetric. To imitate the argument in Section 3.3, note that this time ( ) ∈ 1, ′ (Ω) for any ∈ 1, (Ω), since > 2 by the same argument as before and we can thus again apply (2.24)

(3.22)
To prove the second inf-sup condition, we observe that for ∈ 1, ′ (Ω) we have − ∇ · ( ) ∈ ′ (Ω) and we can apply the same bootstrap argument as in Section 3.5 after swapping and ′ .

□
In this section, we consider illustrative numerical examples. To this end, we first consider an example with smooth right-hand side on a square domain with a smooth solution to show optimal convergence rates in the 1, (Ω) norm. Secondly, we will illustrate why it is useful to consider a non-Hilbert setting for the convectiondiffusion equation by considering examples in two and three dimensions with right-hand sides with very low regularity. Note that Assumptions 1.1 and 1.4 are satsified since all domains are convex Lipschitz domains.

Convergence Rates for a Simple Example in Two Dimensions
In order to illustrate the quasi-optimality estimates given at the end of Section 1.4, we consider the following simple example (which is essentially the Eriksson-Johnson test case, but with a reaction term): Note that the exact solution of this problem is also given by ( , ). Figure 1 shows the error in the 1, (Ω)-norm for = 1.5, 2, 3, 5, = 1, and with the polynomial degree chosen uniformly as = 1, 2, 3, 4, on a uniform triangluation of the domain. Here, we observe the optimal rate of convergence (ℎ ), as the mesh is uniformly refined for each fixed . This illustrates that in the diffusion-dominated case and sufficiently smooth right-hand side the underlying finite element method performs similarly to the case when = 2.

Examples with Rough Right-hand Sides
To motivate looking at the 1, (Ω)-1, ′ (Ω)-setting instead of the standard 1 0 (Ω)-setting, we consider examples with rough right-hand sides, viz. Dirac delta distributions, both in two and three dimensions.
To this end, consider the problem For the weak formulation to be well-defined, we require that 0 ∈ . In other words, we require 1, ′ (Ω) ⊂ 0 (Ω). According to the Sobolev embedding result (see, e.g., [2]), we have for any bounded Lipschitz domain Ω ⊂ R that Thus we require ′ > 2 for = 2 and ′ > 3 for = 3 or, equivalently, < 2 for = 2 and < 1.5 for = 3. Furthermore, the solution is not in 1, (Ω) for ≥ 2 in two dimensions and ≥ 1.5 in three dimensions.
Indeed, it is well-known, see, e.g., [13], that the fundamental solution of the Poisson problem contains a singularity of the form 1/| | in three dimensions and ln(| |) in two dimensions. Thus the fundamental solution is not contained in 1, (Ω) for ≥ 1.5 in three dimensions and for ≥ 2 in two dimensions. The same applies to the convection-diffusion equation; near the origin the singularity of the fundamental solution behaves like (1/| |) for = 3 and like (ln(| |)) for = 2, cf. [14,11,21]. We can thus only expect convergence of | ℎ | 1, (Ω) to a finite value if < 2 for Ω ⊂ R 2 and if < 1.5 for Ω ⊂ R 3 and should observe divergence otherwise. This is illustrated for the 2D-case in Fig. 2 and for the 3D-case in Fig. 3. These figures plot | ℎ | 1, (Ω) for the finite element method using linear finite elements on a mesh (triangles in 2D, or tetrahedra in 3D) of mesh size ℎ. In two dimensions, we can observe that | ℎ | 1, (Ω) diverges for = 2, 3, 5. For = 2, i.e., the borderline case, divergence is very slow; the values converge for = 1.5. Similarly, in three dimensions, we clearly observe divergence for = 1.5, 1.7, 2, while the values converge for = 1.3.

A Proof of Proposition 2.1
In this section, we give the proof of Proposition 2.1. We establish the inf-sup conditions (2.6a) and (2.6b), employing the standard duality technique that invokes the assumed regularity (i.e., Assumption 1.1). In the 1-D case, the latter assumption is not needed; see below. The proof is brief, since we employ straightforward properties of duality maps.