We consider the integral fractional Laplacian on a bounded domain ,
and corresponding homogeneous Dirichlet problem
where We assume that is a bounded Lipschitz domain satisfying an exterior ball condition and a bounded function. We are interested in analyzing the performance of a greedy algorithm for approximating solutions to (1.2) by continuous piecewise linear functions over graded bisection meshes.
that limits the smoothness of solutions. Heuristically, let us consider thatis the half-line . Thus, we can interpret the behavior (1.3) as , and wonder under what conditions this function belongs to a Sobolev space with differentiability index and integrability index . For that purpose, let us compute (Riemann-Liouville) derivatives of order of the function , :
We observe that is -integrable near if and only if , namely, if . This heuristic discussion illustrates the natural interplay between the differentiability order and integrability index for membership of solutions to (1.2) in the class , at least for dimension . It turns out that the restriction is needed irrespective of dimension (cf Theorem 3.7 below).
For the sake of approximation, one can find the optimal choice of the indexes by inspecting a DeVore diagram; see Figure 1.1 for an illustration in the two-dimensional setting. Recall the definition of Sobolev number and the Sobolev line corresponding to the nonlinear approximation scale of ,
In order to have a compact embedding , we require or equivalently to lie above this line. In addition, the regularity restriction , derived heuristically earlier, is depicted in red and intersects the Sobolev line at .
Letting and with arbitrarily small, we have
while the condition is also satisfied. This yields an optimal choice of parameters in dimension .
One can perform an analogous argument for arbitrary dimension : the optimal approximation space can be found on a DeVore diagram by intersecting the Sobolev line corresponding to with the regularity line . For these two lines are parallel, while for these lines meet at the point with
This indicates that the optimal regularity one can expect corresponds to the differentiability order for .
In this paper we justify this heuristic argument rigorously and exploit it to construct suitable graded bisection meshes via a greedy algorithm that delivers quasi-optimal convergence rates for continuous piecewise linear approximation. In Section 2, we introduce some notation regarding fractional-order Sobolev spaces and the weak formulation of (1.2). Section 3 is devoted to providing a rigorous proof of the regularity estimates discussed above. In Section 4 we study the performance of the greedy algorithm. Finally, Section 5 includes some numerical experiments for illustrating our theoretical findings: we observe optimal convergence rates and that the singular boundary layer (1.3) dominates reentrant corner singularities.
2. Fractional Sobolev Spaces
In this section we set the notation and review some properties of the spaces involved in the rest of the paper. We start by recalling some function spaces.
Given and , we consider the seminorm
We see that these constants vanish linearly in as and as . For we set the constant as in the definition (1.1) of the fractional Laplacian , which is consistent with these requirements.
We adopt the convention that zero-order derivatives correspond to the identity, and write , and . Given , let be the largest integer number smaller or equal than , , and we define
with the norm
The Sobolev number of is defined to be .
For our purposes, we need to consider zero-extension spaces as well. For , we denote by its extension by zero on . If and , we define to be the space of functions whose trivial extensions are globally in ,
These spaces characterize the regularity of functions across . It is important to realize that if , , then the spaces and are identical; equivalently, any function in can be extended by zero without changing its regularity (cf. [15, Corollary 188.8.131.52]). In contrast, if , then the notion of trace is well defined in and its subspace of functions with vanishing trace coincides with . Finally, the case is exceptional and corresponds to the so-called Lions-Magenes space [18, Theorem 1.11.7].
From now on, for any given function we will drop the tilde to denote its zero extension, and assume that the domain of is . An important feature of these zero-extension spaces is the following Poincaré inequality: if for and , then . Therefore,
defines a norm equivalent to in .
As usual, we denote Sobolev spaces with integrability index by using the letter instead of . Hence , and we define and its duality pairing. For , the weak formulation of (1.2) reads: find such that
for all . Existence and uniqueness of weak solutions in , as well as the stability of the solution map , are straightforward consequences of the Lax-Milgram theorem. We point out that, in the left-hand side of (2.3), the integration region is effectively .
3. Regularity of solutions
The purpose of this section is to provide regularity estimates for solutions of (1.2) in terms of fractional Sobolev norms with arbitrary integrability index . In a similar fashion to , our starting point shall be the precise weighted Hölder estimates derived by X. Ros-Oton and J. Serra . In  these estimates were employed to obtain regularity estimates in weighted Sobolev spaces with differentiability , where the weight is a power of the distance to the boundary of . As we show below, such regularity estimates are optimally suited for the case . Here, we derive optimal regularity estimates for any . Our technique consists in recasting the estimates from  in unweighted Sobolev spaces with differentiability index but at the expense of an integrability index .
3.1. Hölder regularity.
We start with two important assumptions. We first assume satisfies an exterior ball condition. In  this assumption allows the construction of suitable barriers that lead to -regularity of solutions up to the boundary of , or equivalently, exploiting the zero extension [24, Proposition 1.1]
Secondly, we shall assume possesses certain Hölder continuity. This assumption allows us to derive higher-order regularity estimates on the solution.
We will employ the letter to denote either the distance functions
For , we denote by the -seminorm. If , let us set with integer and . We consider the seminorm
and the associated norm in the following way: for ,
while for ,
Theorem 3.1 (weighted Hölder regularity).
Let be a bounded Lipschitz domain and be such that neither nor is an integer. Let be such that , and be a solution of (1.2). Then, and
We next recast this estimate depending , with the exceptional case . According to (3.1) and the definition of , we have
Corollary 3.2 (pointwise weighted bounds).
Let be a bounded Lipschitz domain that satisfies the exterior ball condition and . If , then
Case : we have and
Case and : we have
Case : we have
The following interior Hölder estimate [24, Lemma 2.9] will also be used later.
Lemma 3.3 (interior regularity).
If and , then verifies
where R = and the constant C depends only on and , and blows up only when .
3.2. Sobolev regularity.
Our goal for the remainder of this section is to use the Hölder estimates we have reviewed to derive bounds on Sobolev norms of . We first show that under suitable assumptions on the right-hand side , the first-order derivatives of the solution are -integrable. For such a purpose, we resort to the following result , which utilizes the asymptotic behavior as of the scaling factor in the definition (2.1) of the seminorm .
Proposition 3.4 (limits of fractional seminorms).
Assume , . Then, it holds that
Remark 3.5 (integrability of powers of the distance function to the boundary).
On Lipschitz domains, powers of the distance function to the boundary have the following integrability property: for every it holds that (cf. for example [9, Lemma 2.14])
Theorem 3.6 (-regularity).
Let be a bounded Lipschitz domain satisfying an exterior ball condition, , such that , and satisfy the following regularity assumptions:
Then, the solution to (1.2) satisfies , which coincides with .
We shall prove that, for every sufficiently small, and
From (3.6) and the fact that is compactly supported, (3.8) implies that . Since on it follows that and has a well defined and vanishing trace because . To exploit symmetry of the integrand in the definition of we decompose the domain of integration into
and its complement within and realize that
Similarly, for the rest of the domain of integration we have
We further split the effective domain of integration into the sets
and rewrite the seminorm defined in (2.1) as
where as according to (2.2). We finally fix if and if , and estimate the contributions on and .
For and distinguish two cases. In the case , we resort to (3.5)
and to write
provided . This implies
and simply setting yields
For , we resort to (3.2) with , namely
because . We next integrate in polar coordinates to get
Thus, letting we obtain
We now aim to prove higher-order Sobolev regularity estimates.
Theorem 3.7 (Sobolev regularity).
Let be a bounded Lipschitz domain satisfying an exterior ball condition, , for some . Furthermore, let and . Then, the solution of (2.3) belongs to , with
where the constant is robust with respect to and .
Our hypotheses imply that . Therefore, we distinguish between two cases: either or . We shall focus on the case because the case can be dealt with the same arguments, but performed over the function instead of its gradient.
Since , it turns out that , whence . Moreover, we have and consequently we can apply Theorem 3.6 (-regularity) to deduce . This concludes the proof in the case ; if we next aim to bound . Similarly to the proof of Theorem 3.6, we split the domain of integration into the sets
because the assumption ensures the convergence of the integral on . In view of (3.7) with , the integral in the right hand side above is convergent and is of order , whence
On the set , we utilize the pointwise bound given in (3.4) to write
because either , whence , or and . Consequently, since ,
For the purposes of approximation, we aim to take as large as possible. On the one hand, we have the limitation from the hypotheses of Theorem 3.7 (Sobolev regularity). On the other hand, one requires in order to have . These two straight lines meet at , whence we deduce the extreme differentiability and integrability indices
This is in agreement with the estimates in weighted spaces from [2, 6]. Let us now specify admissible choices of differentiability parameter and integrability parameter so that is as close to and as close to as possible.
Corollary 3.8 (optimal regularity for ).
Let and satisfy . If , then any yields