Two-scale method for the Monge-Amp\`ere Equation: Pointwise Error Estimates

In this paper we continue the analysis of the two-scale method for the Monge-Amp\`ere equation for dimension $d \geq 2$ introduced in [10]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with H\"older and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.


Introduction
We consider the Monge-Ampère equation with Dirichlet boundary condition where f ≥ 0 is uniformly continuous, Ω is a uniformly convex domain and g is a continuous function. We seek a convex solution u of (1.1), which is critical for (1.1) to be elliptic and have a unique viscosity solution [8].
The Monge-Ampère equation has a wide spectrum of applications, which has led to an increasing interest in the investigation of efficient numerical methods. There are several existing methods for the Monge-Ampère equation, as described in [12]. Error estimates in H 1 (Ω) are established in [3,4] for solutions with H 3 (Ω) regularity or more. Awanou [1] also proved a linear rate of convergence for classical solutions for the wide-stencil method, when applied to a perturbed Monge-Ampère equation with an extra lower order term δu; the parameter δ > 0 is independent of the mesh and appears in reciprocal form in the rate.
On the other hand, Nochetto and Zhang followed an approach based on the discrete Alexandroff estimate developed in [13] and established pointwise error estimates in [14] for the method of Oliker and Prussner [15]. In this paper we follow a similar approach and derive pointwise rates of convergence for classical solutions of (1.1) that have Hölder or Sobolev regularity and for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.
It is worth mentioning a rather strong connection between the semi-Lagrangian method of Feng and Jensen [5] and our two-scale approach introduced in [12]. In fact, for an appropriate choice of discretization of symmetric positive semidefinite matrices with trace one, discussed in [5] along with the implementation, one can show that the discrete solutions of both methods coincide. Therefore, the error estimates in this paper extend to the fully discrete method of [5]. This rather surprising equivalence property is fully derived in a forthcoming paper, along with optimal error estimates in special cases via enhanced techniques for pointwise error analysis.
1.1. Our contribution. The two-scale method was introduced in [12] and hinges on the following formula for the determinant of the semi-positive Hessian D 2 w of a smooth function w, first suggested by Froese and Oberman [6]: where S ⊥ is the set of all d−orthonormal bases in R d . To discretize this expression, we impose our discrete solutions to lie on a space of continuous piecewise linear functions over an unstructured quasi-uniform mesh T h of size h; this defines the fine scale. The mesh also defines the computational domain Ω h , which we describe in more detail in Section 2. The coarser scale δ corresponds to the length of directions used to approximate the directional derivatives that appear in (1.2), namely for any w ∈ C 0 (Ω); To render the method practical, we introduce a discretization S ⊥ θ of the set S ⊥ governed by the parameter θ and denote our discrete solution by u ε , where ε = (h, δ, θ) represents the scales of the method and the parameter θ. We define the discrete Monge-Ampère operator to be where ∇ 2,± δ are the positive and negative parts of ∇ 2 δ . In Section 2 we review briefly the role of each term in the operator T ε and recall some key properties of T ε .
The merit of this definition of T ε is that it leads to a clear separation of scales, which is a key theoretical advantage over the original wide stencil method of [6]. This also yields continuous dependence of discrete solutions on data, namely Proposition 4.6, which allows us to prove rates of convergence in L ∞ (Ω) for our method depending on the regularity of u; this is not clear for the wide stencil method of [6]. Moreover, the two-scale method is formulated over unstructured meshes T h , which adds flexibility to partition arbitrary uniformly convex domains Ω. This is achieved at the expense of points x i ± δv j no longer being nodes of T h , which is responsible for an additional interpolation error in the consistency estimate of T ε . To locate such points and evaluate ∇ 2 δ u ε (x i ; v j ), we resort to fast search techniques within [16,17] and thus render the two-scale method practical. Compared with the error analysis of the Oliker-Prussner method [13], we do not require T h to be cartesian.
In [12] we prove existence and uniqueness of a discrete solution for our method, and convergence to the viscosity solution of (1.1), without regularity beyond uniform continuity of f and g. This entails dealing with the L ∞ −norm and using the discrete comparison principle for piecewise linear functions (monotonicity). Within this L ∞ framework and under the regularity requirement u ∈ W 2 ∞ (Ω), we now prove rates of convergence for classical solutions with either Hölder or Sobolev regularity and for a special class of viscosity solutions. Therefore, our two-scale method [12] and the Oliker-Prussner method [15,14] are the only schemes known to us to converge to the viscosity solution and have provable rates of convergence.
The first important tool for proving pointwise rates of convergence is the discrete Alexandroff estimate introduced in [13]: if w h is an arbitrary continuous piecewise linear function, w h ≥ 0 on ∂Ω h , and Γw h stands for its convex envelope, then where ∂Γw h is the subdifferential of Γw h and C − (w h ) represents the lower contact set of w h , i.e. the set of interior nodes To control the measure of the subdifferential at each node, we show the following estimate such that the ball centered at x i and of radius δ is contained in Ω h . Combining both estimates, we derive the following continuous dependence estimate h . This result is instrumental and, combined with operator consistency and a discrete barrier argument close to the boundary, eventually leads to the following pointwise error estimates provided u ∈ C 2+k,α (Ω) with 0 < α ≤ 1 and k = 0, 1, as well as provided u ∈ W s p (Ω) with 2 + d/p < s ≤ 4 and p > d, and δ is suitably chosen in terms of h; see Theorems 5.3 and 5.4. We also consider a special case of viscosity solutions with bounded but discontinuous Hessians, and manage to prove a rate of convergence (see Theorem 5.5). Since these theorems are proven under the nondegeneracy assumption f > 0, we examine in Theorem 5.6 the effect of degeneracy f ≥ 0. In [12] we explore numerically both classical and W 2 ∞ viscosity solutions and observe linear rates with respect to h for both cases, which are better than predicted by this theory.
1.2. Outline. We start by briefly presenting the operator T ε in Section 2 and recalling some important results from [12]. In Section 3 we mention the discrete Alexandroff estimate and combine it in Section 4 with some geometric estimates to obtain the continuous dependence of the discrete solution on data. This is much stronger than stability, and is critical to prove rates of convergence for fully nonlinear PDEs. Lastly, in Section 5 we combine this result with operator consistency and a discrete barrier argument close to the boundary to derive rates of convergence upon making judicious choices of δ and θ in terms of h.

Key Properties of the Discrete Operator
We recall briefly some of the key properties of operator T ε , as proven in [12].
2.1. Definition of T ε . Let T h be a shape-regular and quasi-uniform triangulation with meshsize h. The computational domain Ω h is the union of elements of T h and Ω h = Ω. If N h denotes the nodes of T h , then N b h := {x i ∈ N h : x i ∈ ∂Ω h } are the boundary nodes and N 0 h := N h \ N b h are the interior nodes. We require that N b h ⊂ ∂Ω, which in view of the convexity of Ω implies that Ω h is also convex and Ω h ⊂ Ω. We denote by V h the space of continuous piecewise linear functions over T h . We let S ⊥ be the collection of all d-tuples of orthonormal bases and v := (v 1 , . . . , v d ) ∈ S ⊥ be a generic element, whence each component v i ∈ S, the unit sphere S of R d . We next introduce a finite subset S θ of S governed by the angular parameter θ > 0: Likewise, we let S ⊥ θ ⊂ S ⊥ be a finite approximation of S ⊥ : for any For x i ∈ N 0 h , we use centered second differences with a coarse scale δ whereδ := ρδ with 0 < ρ ≤ 1 the biggest number such that the ball centered at x i of radiusδ is contained in Ω h ; we stress that ρ need not be computed exactly. This is well defined for any w ∈ C 0 (Ω), in particular for w ∈ V h . We define ε := (h, δ, θ) and we seek u ε ∈ V h such that u ε ( where we use the notation , 0) to indicate positive and negative parts of the centered second differences.

2.2.
Key Properties of T ε . One of the critical properties of the Monge-Ampère equation is the convexity of the solution u. The following notion mimics this at the discrete level.
Definition 2.1 (discrete convexity). We say that w h ∈ V h is discretely convex if The following lemma guarantees the discrete convexity of subsolutions of (2.2) [12, Lemma 2.2].
then w h is discretely convex and as a consequence

Conversely, if w h is discretely convex, then (2.3) is valid.
Another important property of operator T ε that relies on its monotonicity is the following discrete comparison principle [12,Lemma 2.4].
We now state a consistency estimate, proved in [12,Lemma 4.1], that leads to pointwise rates of convergence. To this end, given a node x i ∈ N 0 h , we denote by whereδ is defined in (2.1). We also define the δ-interior region and the δ-boundary region: .

Discrete Alexandroff Estimate
In this section, we review several concepts related to convexity as well as the discrete Alexandroff estimate of [13]. We first recall several definitions. (i) The subdifferential of a function w at a point x 0 ∈ Ω h is the set (ii) The discrete lower contact set C − (w h ) of a function w h ∈ V h is the set of nodes where the function coincides with its convex envelope, i.e.
Remark 3.4 (minima of w h and Γw h ). A consequence of Definition 3.2 (convex envelope and discrete lower contact set) is that the minima of w h ∈ V h and Γw h are attained at the same contact nodes and are equal.
We can now present the discrete Alexandroff estimate from [13], which states that the minimum of a discrete function is controlled by the measure of the subdifferential of its convex envelope in the discrete contact set. Proposition 3.5 (discrete Alexandroff estimate [13]). Let v h be a continuous piecewise linear function that satisfies v h ≥ 0 on ∂Ω h . Then,

Continuous Dependence on Data
We derive the continuous dependence of the discrete solution on data in Section 4.3, which is essential to prove rates of convergence. To this end, we first prove a stability estimate in the max norm in Section 4.1 and the concavity of the discrete operator in Section 4.2.
4.1. Stability of the Two-Scale Method. We start with some geometric estimates. The first and second lemmas connect the discrete Alexandroff estimate with the 2-scale method. They allow us to estimate the measure of the subdifferential of a discrete function w h in terms of our discrete operator T ε [w h ], defined in (2.2). Lemma 4.1 (subdifferential vs hyper-rectangle). Let w ∈ C 0 (Ω h ) be convex and Proof. Take p ∈ ∂w(x i ) and write Consequently, for any 1 ≤ j ≤ d we infer that This implies that p belongs to the desired set.
is the canonical basis in R d . We now seek a more convenient representation of K because K is an orthogonal hyper-rectangle.
Combining Lemmas 4.1 and 4.2 we get the following corollary.
Remark 4.3 (artificial factor δ h ). The above estimate is critical in deriving Proposition 4.6 (continuous dependence on data) and subsequently rates of convergence in Section 5. We thus wish to provide here intuition about the reduced rates of convergence of Theorem 5.3 (rates of convergence for classical solutions) relative to the numerical experiments in [12]. To this end, we let w ∈ C 2 (Ω) be a convex function. Then, Corollary 4.1 implies that However, it was shown by Nochetto and Zhang in [14,Proposition 5.4] that We can now see that using this estimate introduces an extra factor δ h ≫ 1, which could possibly explain the suboptimal rate proved in Theorem 5.3 (rates of convergence for classical solutions).
Applying Corollary 4.1 (subdifferential vs discrete operator) to the convex function Γw h (x i ) at a contact point x i ∈ C − (w h ) and recalling Remark 3.3, we have where the last equality follows from Lemma 2.2 (discrete convexity).

4.2.
Concavity of the Discrete Operator. We recall concavity properties of (det A) 1/d for symmetric positive semi-definite matrices A and extend them to T ε . The results can be traced back to [9,11], but we present them here for completeness. Proof. We proceed in three steps.
Let P be an orthogonal matrix that converts B 1/2 AB 1/2 into a diagonal matrix D, namely D = P B 1/2 AB 1/2 P T . Applying the geometric mean inequality yields where we have used the invariance of the trace under cyclic permutations of the factor to write the last two equalities. This shows that This inequality is actually equality provided A is invertible. In fact, we can take B = (det A) 1/d A −1 , which is SPD and det B = 1. This proves (i) for A nonsingular.
Step 2: Proof of (i) for A singular. Given the singular value decomposition of A , we can assume that k > 0 for otherwise A = 0 and the assertion is trivial. Given a parameter σ > 0, let B be defined by which proves (i) for A singular since B is SPD.
Step 3: Proof of (ii). Let A and B be SPSD matrices and 0 ≤ λ ≤ 1. Then λA + (1 − λ)B is also SPSD and we can apply (i) to This completes the proof.
Upon relabeling A = λA and B = (1 − λ)B, which are still SPSD, we can write Lemma 4.5 (ii) as follows: We now show that our discrete operator T ε [·] possesses a similar property.

Corollary 4.2 (concavity of discrete operator). Given two functions
Proof. We argue in two steps. Step Step 2. We now apply this formula to the discrete operator. Since both u h , w h are discretely convex at x i ∈ N 0 h , so is u h + w h , and we can apply Lemma 2.2 (discrete convexity) to write Making use again of (2.4), this time for u h and w h and for the specific set of directions v just found, we obtain where the second inequality is given by . This is the asserted estimate.

Continuous Dependence of the Two-Scale Method on Data.
We are now ready to prove the continuous dependence of discrete solutions on data. This will be instrumental later for deriving rates of convergence for the two-scale method.
Proposition 4.6 (continuous dependence on data). Given two functions u h , w h ∈ V h such that u h ≥ w h on ∂Ω h and where we have made use of Lemma 2.2 (discrete convexity). Invoking Corollary 4.2 (concavity of discrete operator) for u h − w h and w h , we deduce This completes the proof.

Rates of Convergence
We now combine the preceding estimates to prove pointwise convergence rates for solutions with varying degree of regularity. We first present in Theorem 5.3 the case of a classical solution with Hölder regularity. This allows us to introduce the main techniques employed for deriving the rates of convergence. We then build on these techniques and prove error estimates for three more cases of increasing generality. In Theorem 5.4 we assume a classical solution with Sobolev regularity, which requires the use of embedding estimates and accumulating the truncation error in l d , rather than l ∞ . We next deal with a non-classical solution that is globally in W 2 ∞ (Ω) but its Hessian is discontinuous across a d − 1 dimensional Lipschitz surface. To prove rates for this case we need to take advantage of the small volume affected by this discontinuity and combine it with the techniques used in Theorem 5.3 and Theorem 5.4. Lastly, we remove the non-degeneracy assumption f ≥ f 0 > 0 used in the previous three cases to obtain rates of convergence for a piecewise smooth viscosity solution with degenerate right hand side f . This corresponds to one of the numerical experiments performed in [12]. Our estimates do not require h small and are stated over the computational domain Ω h ⊂ Ω.

Barrier Function.
We recall here the two discrete barrier functions introduced in [12, Lemmas 5.1, 5.2]. The first one is critical in order to control the behavior of u ε close to the boundary of Ω h and prove the convergence to the unique viscosity solution u of (1.1). We now use the same barrier function to control the pointwise error of u ε and u close to the boundary. The second barrier allows us to treat the degenerate case f ≥ 0, using techniques similar to the case f > 0.
Lemma 5.1 (discrete boundary barrier). Let Ω be uniformly convex and E > 0 be arbitrary. For each node z ∈ N 0 Error Estimates for Solutions with Hölder Regularity. We now deal with classical solutions u of (1.1) of class C 2+k,α (Ω), with k = 0, 1 and 0 < α ≤ 1, and derive pointwise error estimates. We proceed as follows. We first use Lemma 5.1 (discrete boundary barrier) to control u ε − I h u in the δ-neighborhood ω h,δ of ∂Ω h , where the consistency error of T ε [I h u] is of order one according to Lemma 2.4 (consistency of T ε [I h u]). In the δ-interior region Ω h,δ we combine the interior consistency error of T ε [I h u] from Lemma 2.4 and Proposition 4.6 (continuous dependence on data). Judicious choices of δ and θ in terms of h conclude the argument.  (2.2). If u ∈ C 2,α (Ω) for 0 < α ≤ 1 and Otherwise, if u ∈ C 3,α (Ω) for 0 < α ≤ 1 and with a hidden constant depending on Ω, d, f 0 . We proceed in three steps. The estimates for max Ω h (I h u − u ε ) are similar and thus omitted. Adding the interpola- [2] readily gives the asserted estimates because k+α 2+k+α ≤ 1 2 for k = 0, 1 and 0 < α ≤ 1.
Step 1: Boundary estimate. We show that for z ∈ N 0 Given the function p h of Lemma 5.1 (discrete boundary barrier), for z fixed, we examine the behavior of u ε + p h . For any interior node ∞ (Ω) . Since I h u = u ε and p h ≤ 0 on ∂Ω h , we deduce from Lemma 2.3 (discrete comparison principle) that Step 2: Interior estimate. We show that for all with k = 0, 1 and Step 1 guarantees that (Ω) δ and note that d ε ≥ 0 on ∂Ω h,δ . We then apply Proposition 4.6 (continuous dependence on data) to d ε in Ω h,δ , in conjunction with Lemma 2.4 (consistency of T ε [I h u]), to obtain Since the cardinality of C − (d ε ) is bounded by that of N h , which in turn is bounded by Ch −d with C depending on shape regularity, we end up with Step 3: Choice of δ and θ. To find an optimal choice of δ and θ in terms of h, we minimize the right-hand side of the preceding estimate. We first set θ 2 = h 2 δ 2 and equate the last two terms .
We observe that according to Theorem 5.3 the rate of convergence is of order h 1/2 whenever u ∈ C 3,1 (Ω). However, our numerical experiments in [12] indicate linear rates of convergence, which correspond to Lemma 2.4 (consistency of T ε [I h u h ]). This mismatch may be attributed to the factor δ h ≫ 1 in (5.1), which relates to Remark 4.3 (artificial factor δ h ). This issue will be tackled in a forthcoming paper.

5.3.
Error Estimates for Solutions with Sobolev Regularity. We now derive error estimates for solutions u ∈ W s p (Ω) with s > 2 + d p so that W s p (Ω) ⊂ C 2 (Ω). We exploit the structure of the estimate of Proposition 4.6 (continuous dependence on data) which shows that its right-hand side accumulates in l d rather than l ∞ . Theorem 5.4 (convergence rate for W s p solutions). Let f ≥ f 0 > 0 in Ω and let the viscosity solution u of (1.1) be of class W s p (Ω) with d p < s − 2 − k ≤ 1, k = 0, 1. If u ε is the discrete solution of (2.2) and Proof. We proceed as in Theorem 5.3 to show an upper bound for u ε − I h u. The boundary estimate of Step 1 remains intact, namely On the other hand, Step 2 yields where C 1 (u) and C 2 (u) are defined in Lemma 2.4 (consistency of T ε [I h u]) and 0 < α = s− 2 − k − d p ≤ 1 corresponds to the Sobolev embedding W s p (B i ) ⊂ C 2+k,α (B i ). In the following calculations we resort to the Sobolev inequality [7, Theorem 2.9] |u| C 2+k,α (Bi) ≤ C|u| W s p (Bi) , involving only semi-norms. We stress that C > 0 depends on the Lipschitz constant of B i but not on its size. The latter is due to the fact that the Sobolev numbers of W s−2−k p (B i ) and C 0,α (B i ) coincide: 0 < s − k − 2 − d/p = α ≤ 1. We refer to [7, Theorem 2.9] for a proof for 0 < s < 1. We now use the Hölder inequality with Since the cardinality of the set of balls B i containing an arbitrarily given x ∈ Ω is proportional to δ h d , while the cardinality of N 0 h is proportional to h −d , we get  Exploiting that α + k + d p + 1 = s − 1, we readily arrive at In addition, we have  Collecting the previous estimates, we end up with To find an optimal relation among h, δ and θ, we first choose θ 2 = h 2 δ 2 and next equate the two terms in the second summand to obtain Adding the interpolation error estimate u − I h u L ∞ (Ω) h 2 |u| W 2 ∞ (Ω) , and using that 2 > 2 s ≥ 1 − 2 s for 2 < s ≤ 4, leads to the asserted estimate.
The error estimate of Theorem 5.4 (convergence rate for W s p -solutions) is of order 1 2 for s = 4 and u ∈ W 4 p (Ω) with p > d. This rate requires much weaker regularity than the corresponding error estimate in Theorem 5.3, namely u ∈ C 3,1 (Ω) = W 4 ∞ (Ω). In both cases, the relation between δ and h is δ ≈ h 1 2 .

Error Estimates for Piecewise Smooth Solutions.
We now derive pointwise rates of convergence for a larger class of solutions than in Section 5.3. These are viscosity solutions which are piecewise W s p but have discontinuous Hessians across a Lipschitz (d − 1)-dimensional manifold S; we refer to the second numerical example in [12]. Since T ε [I h u] has a consistency error of order one in a δ-region around S, due to the discontinuity of D 2 u, we exploit the fact that the measure of this region is proportional to δ|S|. We are thus able to adapt the argument of Theorem 5.4 (convergence rate for W s p solutions), and accumulate such consistency error in l d , at the expense of an extra additive term of order h −1 δ 1+ 1 d . This term is responsible for a reduced convergence rate when u ∈ W s p (Ω \ S), s > 2 + 1 d . Theorem 5.5 (convergence rate for piecewise smooth solutions). Let S denote a (d − 1)-dimensional Lipschitz manifold that divides Ω into two disjoint subdomains , for i = 1, 2 and d p < s − 2 − k ≤ 1, k = 0, 1, be the viscosity solution of (1.1). If u ε denotes the discrete solution of (2.2), then for β = min{s, 2 Proof. We proceed as in Theorems 5.3 and 5.4. The boundary layer estimate relies on the regularity u ∈ W 2 ∞ (Ω) which is still valid, whence for all x ∈ Ω h such that dist(x, ∂Ω h ) ≤ δ we obtain ∩ Ω h,δ as follows:

Consider now the internal layer
An argument similar to Step 2 (interior estimate) of Theorem 5.3, based on combining Proposition 4.6 (continuous dependence on data) and Lemma 2.4 (consistency because the consistency error in S δ h,1 is bounded by C 2 (u) = C|u| d W 2 ∞ (Bi) . As in Theorem 5.4 (convergence rate for W s p solutions), C 1 (u) satisfies . Since the number of nodes x i ∈ S δ h,1 is bounded by C|S|δh −d , we deduce For I 2 we distinguish whether x i belongs to Ω 1 or Ω 2 and accumulate C 1 (u) in ℓ p , exactly as in Theorem 5.4, to obtain Collecting the previous estimates and using the definition of β yields We finally realize that this estimate is similar to that in the proof of Theorem 5.4 except for the middle term on the right-hand side. Therefore, we proceed as in Theorem 5.4 to find the relation between δ, θ and h, add the estimate u − I h u L ∞ (Ω) h 2 |u| W 2 ∞ (Ω) , and eventually derive the asserted error estimate.

Error Estimates for Piecewise Smooth Solutions with Degenerate f .
We observe that in all three preceding theorems we assume that f ≥ f 0 > 0. This is an important assumption in the proofs, since it allows us to use the concavity of t → t 1/d and Proposition 4.6 (continuous dependence on data) to obtain where e is related to the consistency of the operator in Lemma 2.4 (consistency of T ε [I h u]). We see that this is only possible if f 0 > 0. If we allow f to touch zero, then (5.2) reduces to with equality for f (x i ) = 0. This leads to a rate of order δ To circumvent this obstruction, we use Lemma 5.2 (interior barrier function) which allows us to introduce an extra parameter σ > 0 that compensates for the lack of lower bound f 0 > 0 and yields pointwise error estimates of reduced order.
Proof. We employ the interior barrier function q h of Lemma 5.2 scaled by a parameter σ > 0 to control u ε − I h u and I h u − u ε in two steps. The parameter σ allows us to mimic the calculation in (5.2). In the third step we choose σ optimally with respect to the scales of our scheme.
Step 1: Upper bound for u ε − I h u. We let w h := u ε + σq h and v h : and proceed as in Step 1 of Theorem 5.3 to show w h (z) ≤ v h (z) for all z ∈ N 0 h such that dist(z, ∂Ω h ) ≤ δ. We now focus on Ω h,δ and define the auxiliary function d ε := v h −w h and contact set C δ − (d ε ) := C − (d ε ) ∩ Ω h,δ . Since the previous argument guarantees that d ε ≥ 0 on ∂Ω h,δ , Proposition 4.6 (continuous dependence on data) gives If e i is the local consistency error given in Lemma 2.4, we further note that We now minimize the right-hand side upon choosing δ, θ and σ suitably with respect to h. We first recall the definition of β and choose δ and θ as in Theorem 5.5. At this stage it only remains to find σ upon solving which leads to Since β > 2 we get h 2 + δ ≤ 1 + R(u) − 1 β h 2 β and This yields the asserted estimate and finishes the proof.
Theorem 5.6 is an extension of Theorem 5.5 to the degenerate case f ≥ 0, but the same techniques and estimates extend as well to Theorems 5.3 and 5.4. We stress that Theorems 5.5 and 5.6 correspond to non-classical viscosity solutions that are of class W 2 ∞ (Ω). In order to deal with discontinuous Hessians and degenerate right hand sides, we rely on techniques that give rise to reduced rates. For Theorem 5.5 we obtain rates that depend on the space dimension, whereas for Theorem 5.6 we resort to a regularization procedure that leads to further reduction of the rates. Although the derived estimates are suboptimal with respect to the computational rates observed in [12], we wish to emphasize that Theorem 5.6 is, to our knowledge, the only error estimate available in the literature that deals with degenerate right hand sides.

Conclusions
In this paper we extend the analysis of the two-scale method introduced in [12]. We derive continuous dependence of discrete solutions on data and use it to prove rates of convergence in the L ∞ norm in the computational domain Ω h for four different cases. We first prove rates of order up to h 1/2 for smooth classical solutions with Hölder regularity. We then exploit the structure of the continuous dependence estimate of discrete solutions on data to derive error estimates for classical solutions with Sobolev regularity, thereby achieving the same rates under weaker regularity assumptions. In a more general scenario, we derive error estimates for viscosity solutions with discontinuous Hessian across a surface with appropriate smoothness, but otherwise possessing piecewise Sobolev regularity. Lastly, we use an interior barrier function that allows us to remove the nondegeneracy assumption f > 0 at the cost of a reduced rate that depends on dimension. Our theoretical predictions are sub-optimal with respect to the linear rates observed experimentally in [12] for a smooth classical solution and a piecewise smooth viscosity solution with degenerate right-hand side f ≥ 0. This can be attributed to the fact that the continuous dependence estimate of discrete solutions on data introduces a factor δ h ≫ 1 in the error estimates. This feature is similar to the discrete ABP estimate developed in [10] and is the result of using sets of measure ≈ δ d instead of ≈ h d to approximate subdifferentials. In a forthcoming paper we will tackle this issue and connect our two-scale method with that of Feng and Jensen [5].