NUMERICAL APPROXIMATION OF THE INTEGRAL FRACTIONAL LAPLACIAN

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1 NUMERICAL APPROXIMATION OF THE INTEGRAL FRACTIONAL LAPLACIAN ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Abstract. We propose a new nonconforming finite element algoritm to approximate te solution to te elliptic problem involving te fractional Laplacian. We first derive an integral representation of te bilinear form corresponding to te variational problem. Te numerical approximation of te action of te corresponding stiffness matrix consists of tree steps: (i) apply a sinc quadrature sceme to approximate te integral representation by a finite sum were eac term involves te solution of an elliptic partial differential equation defined on te entire space, (ii) truncate eac elliptic problem to a bounded domain, (iii) use te finite element metod for te space approximation on eac truncated domain. Te consistency error analysis for te tree steps is discussed togeter wit te numerical implementation of te entire algoritm. Te results of computations are given illustrating te error beavior in terms of te mes size of te pysical domain, te domain truncation parameter and te quadrature spacing parameter. 1. Introduction We consider a nonlocal model on a bounded domain involving te Riesz fractional derivative (i.e., te fractional Laplacian). For teory and numerical analysis of general nonlocal models, we refer to te review paper [1] and references terein. Particularly, several applications are modeled by partial differential equations involving te fractional Laplacian: obstacle problems from symmetric α-stable Lévy processes [43, 31, 37]; image denosings [5]; fractional kinetics and anomalous transport [44]; fractal conservation laws [0, 5]; and geopysical fluid dynamics [17, 15, 16, 8]. In tis paper, we consider a class of fractional boundary problems on bounded domains were te fractional derivative comes from te fractional Laplacian defined on all of R d. Te motivation for tese problems is illustrated by an evolution equation considered by Meuller [35] of te form: (1) () u t = Λ s u + f(u), in R + D, u = 0, in D c. Here D is a convex polygonal domain in R d, D c denotes its complement and Λ s u := (( ) s ũ) D wit ũ denoting te extension of u by zero to R d. Tis fractional Laplacian on R d is defined using te Fourier transform F: (3) F(( ) s f)(ζ) = ζ s F(f)(ζ). Date: July 13,

2 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Te formula (3) defines an unbounded operator ( ) s on L (R d ) wit domain of definition D(( ) s ) := {f L (R d ) : ζ s F(f) L (R d )}. It is clear tat te Sobolev space H s (R d ) := { f L (R d ) : (1 + ζ ) s F(f) L (R d ) } is a subset of D(( ) s ) for any s 0. Note tat ( ) s v for s = 1 and v H (R d ) coincides wit te negative Laplacian applied to v. Te term Λ s along wit te boundary condition () represents te generator of a symmetric s-stable Lévy process wic is killed wen it exits D (cf. [35]). Te f(u) term in (1) involves wite noise and will be ignored in tis paper. Te goal of tis paper is to study te numerical approximation of solutions of partial differential equations on bounded domains involving te fractional operator Λ s supplemented wit te boundary conditions (). As finite element approximations to parabolic problems are based on approximations to te elliptic part, we sall restrict out attention to te elliptic case, namely, (4) Λ s u = f, in D, u = 0, in D c. Te above system is sometimes referred to as te integral fractional Laplacian problem. We note tat te variational formulation of (4) can be defined in terms of te classical spaces H s (D) consisting of te functions defined in D wose extension by zero are in H s (R d ). Tis is to find u H s (D) satisfying (5) a(u, φ) = fφ dx, for all φ H s (D), were (6) a(u, φ) = [( ) s/ ũ][( ) s/ φ] dx R d D wit ũ and φ denoting te extensions by 0. We refer to Section 8.1 for te description of model problems. Te bilinear form a(, ) is obviously bounded on H s (D) H s (D) and, as discussed in Section, it is coercive on H s (D). Tus, te Lax-Milgram Teory guarantees existence and uniqueness. We consider finite element approximations of (5). Te use of standard finite element approximation spaces of continuous functions vanising on D is te natural coice. Te convergence analysis is classical once te regularity properties of solutions to problem (5) are understood (regularity results for (5) ave been studied in [1] and [38]). However, te implementation of te resulting discretization suffers from te fact tat, for d > 1, te entries of te stiffness matrix, namely, a(φ i, φ j ), wit {φ k } denoting te finite element basis, cannot be computed exactly. Wen d = 1, s (0, 1/) (1/, 1) and, for example, D = ( 1, 1), te bilinear form can be written in terms of Riemann-Liouville fractional derivatives (cf. [41]), namely, (7) a(φ i, φ j ) = ( s L φ i, R s φ j) D + ( L s φ j, R s φ i) D. cos(sπ)

3 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 3 Here (, ) D denotes te inner product on L (D) and for t (0, 1) and v H0 1 (D), te left-sided and rigt-sided Riemann-Liouville fractional derivatives of order t are defined by (8) Lv(x) t 1 d = Γ(1 t) dt and (9) Rv(x) t 1 d = Γ(1 t) dt x 1 1 x v(y) (x y) t dy v(y) (x y) t dy. Note tat te integrals in (8) and (9) can be easily computed wen v is a piecewise polynomial, i.e, wen v is a finite element basis function. Te computation of te stiffness matrix in tis case reduces to a coding exercise. A representation of te fractional Laplacian for d 1 is given by [4]: (10) (( ) s η)(x) = c d,s P V R d η(x) η(y) dy, for all η S, x y d+s were S denotes te Scwartz space of rapidly decreasing functions on R d, P V denotes te principle value and c d,s is a normalization constant. It follows tat for η, θ S, (11) a(η, θ) = (( ) s η, θ) = c d,s R d R d (η(x) η(y))(θ(x) θ(y)) x y d+s dy dx. A density argument implies tat te stiffness entries are given by (1) a(φ i, φ j ) = c d,s ( φ i (x) φ i (y))( φ j (x) φ j (y)) x y d+s dy dx, R d R d were again φ denotes te extension of φ by zero outside D. It is possible to apply te tecniques developed for te approximation of boundary integral stiffness matrices [39] to deal wit some of te issues associated wit te approximation of te double integral above, namely, te application of special tecniques for andling te singularity and quadratures. However, (1) requires additional truncation tecniques as te non-locality of te kernel implies a non-vanising integrand over R d. Tese tecniques are used to approximate (1) in [3, 1]. In particular, [1] use teir regularity teory to do a priori mes refinement near te boundary to develop iger order convergence under te assumption of exact evaluation of te stiffness matrix. Te metod to be developed in tis paper is based on a representation of te underlying bilinear form given in Section 4, namely, for s (0, 1), 0 r s, η H r (R d ) and θ H s r (R d ), (13) R d [( ) r/ η][( ) (s r)/ θ] dx = c s 0 t s (( )(I t ) 1 η, θ) dt t were (, ) denotes te inner product on L (R d ) (see, also [3]). We note tat for t > 0, (I t ) 1 is a bounded map of L (R d ) into H (R d ) so tat te integrand above is well defined for η, θ L (R d ). In Teorem 4.1, we sow tat for η H r (R d ) and θ H s r (R d ), te formula (13) olds and te rigt and side integral converges absolutely. It follows tat te bilinear form a(, ) is given by (14) a(η, θ) = c s 0 t s (( )(I t ) 1 η, θ) D dt t, for all η, θ H s (D).

4 4 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Tere are tree main issues needed to be addressed in developing numerical metods for (5) based on (14): (a) Te infinite integral wit respect to t must be truncated and approximated by numerical quadrature; (b) At eac quadrature node t j, te inner product term in te integrand involves an elliptic problem on R d. Tis must be replaced by a problem wit vanising boundary condition on a bounded truncated domain Ω M (t j ) (defined below); (c) Using a fixed subdivision of D, we construct subdivisions of te larger domain Ω M (t j ) wic coincide wit tat on D. We ten replace te problems on Ω M (t j ) of (b) above by teir finite element approximations. We address (a) above by first making te cange of variable t = e y wic results in an integral over R. We ten apply a sinc quadrature obtaining te approximate bilinear form (15) a k (η, θ) := c sk N + j= N e syj (( )(e yj I ) 1 η, θ) D, for all θ, η L (D), were k is te quadrature spacing, y j = kj, and N and N + are positive integers. Teorem 5.1 sows tat for θ H s (D) and η H δ (D) wit δ (s, s], we ave a(η, θ) a k (η, θ) C(δ, s) [ e π /(4k) + e (s δ)n + k/ + e skn ] η Hδ (D) θ Hs (D). Balancing of te exponentials gives rise to an O(e π /4k ) convergence rate wit te relation N + + N = O(1/k ). Te size of te truncated domain Ω M (t j ) in (b) is determined by te decay of (e yj I ) 1 f for functions f supported in D. For tecnical reasons, we first extend D to a bounded convex (star-saped wit respect to te origin) domain Ω and set (wit t j = e yj/ ) { Ω M {(1 + tj (1 + M))x : x Ω}, t (t j ) := j 1 {( + M)x : x Ω}, t j < 1. Let t denote te unbounded operator on L (Ω M (t)) corresponding to te Laplacian on Ω M (t) supplemented wit vanising boundary condition. We define te bilinear form a k,m (η, θ) by replacing ( )(e yj I ) 1 in (15) by ( tj )(e yj I tj ) 1. Teorem 6. guarantees tat for sufficiently large M, we ave a k (η, θ) a k,m (η, θ) Ce cm η L (D) θ L (D), for all η, θ L (D). Here c and C are positive constants independent of M and k. Tis addresses (b). Step (c) consists in approximating ( tj )(e yj I tj ) 1 using finite elements. In tis aim, we associate to a subdivision of Ω M (t j ) te finite element space V M (t j) and te restriction a k,m (, ) of a k,m (, ) to V M (t j) V M (t j). As already mentioned, te subdivisions of Ω M (t j ) are constructed to coincide on D. Denoting by V (D) te set of finite element functions restricted to D and vanising on D, our approximation to te solution of (5) is te function u V (D) satisfying (16) a k,m (u, θ) = D fθ dx, for all θ V (D). Lemma 7. guarantees te V (D)-coercivity of te bilinear form a k,m (, ). Consequently, u is well defined again from te Lax-Milgram teory. Moreover, given,

5 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 5 for every t j, a sequence of quasi-uniform subdivisions of Ω M (t j ), we sow (Teorem 7.5) tat for v in H β (D) wit β (s, 3/) and for θ V (D), a k,m (v, θ ) a k,m (v, θ ) C(1 + ln( 1 )) β s v Hβ (D) θ Hs (D). Here C is a constant independent of M, k and, and v V (D) denotes te Scott- Zang interpolation or te L projection of v depending on weter β (1, 3/) or β (s, 1]. Strang s Lemma implies tat te error between u and u in te H s (D)-norm is bounded by te error of te best approximation in H s (D) and te sum of te consistency errors from te above tree steps (see Teorem 7.7). Te online of te paper is as follows. Section introduces notations of Sobolev spaces followed by Section 3 introducing te dotted spaces associated wit elliptic operators. Te alternative integral representation of te bilinear form is given in Section 4. Based on tis integral representation, we discuss te discretization of te bilinear form and te associated consistency error in tree steps (Sections 5, 6 and 7). Te energy error estimate for te discrete problem is given in Section 7. A discussion on te implementation aspects of te metod togeter wit results of numerical experiments illustrating te convergence of te metod are provided in Section 8. We left to Appendix te proof of tecnical result regarding te stability and approximability of te Scott-Zang interpolant in nonstandard norms.. Notations and Preliminaries Notation. We use te notation D R d to denote te polygonal domain wit Lipscitz boundary in problem (5) and ω R d to denote a generic bounded Lipscitz domain. For a function η : ω R, we denote by η its extension by zero outside ω. We do not specify te domain ω in te notation η as it will be always clear from te context. Scalar Products. We denote by (, ) ω te L (ω)-scalar product and by L (ω) := (, ) 1/ ω te associated norm. Te L (R d )-scalar product is denoted (, ) R d. To simplify te notation, we write in sort (, ) := (, ) R d and := L (R ). d Sobolev Spaces. For r > 0, te Sobolev space of order r on R d, H r (R d ), is defined to be te set of functions θ L (R d ) suc tat ( ) 1/ (17) θ H r (R d ) := (1 + ζ ) r/ F(θ)(ζ) dζ <. R d In te case of bounded Lipscitz domains, H r (ω) wit r (0, 1), stands for te Sobolev space of order r on ω. It is equipped wit te norm (18) θ H r (ω) := ( θ L (ω) + θ H r (ω)) 1/, were θ H r (ω) := (θ(x) θ(y)) ω ω x y d+r dx dy. Wen r (1, ) instead, te norm in H r (ω) is given by θ H r (ω) := θ H 1 (ω) + ω ω θ(x) θ(y) x y d+(r 1) dx dy,

6 6 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK were w H1 (ω) := ( w L (ω) + w L (ω) )1/. In addition, H0 1 (ω) denotes te set of functions in H 1 (ω) vanising at ω, te boundary of ω. Its dual space is denoted H 1 (ω). We note tat wen we replace ω wit R d and r [0, ), te norms using te double integral above are equivalent wit tose in (17) (see e.g. [3, 34]). Te spaces H r (D). For r (0, ), te set of functions in D wose extension by zero are in H s (R d ) is denoted H r (D). Te norm of H r (D) is given by Hr (R ). d Note tat for r (0, 1), (10) implies tat for φ in te Scwartz space S, (19) (( ) r φ, φ) = c d,r Tus, we prefer to use (0) φ Hr (D) ( c := d,r R d R d Rd (φ(x) φ(y)) x y d+r dx dy. Rd ( φ(x) φ(y)) ) 1/ x y d+r dx dy as equivalent norm on H r (D) for r (0, 1). Tis is justified upon invoking a variant of te Peetre-Tartar compactness argument on H r (D) H r (D). Coercivity. Since C0 (D) is dense in H s (D) for s (0, 1) [6], (11) and a density argument imply tat for η, θ H s (D), we ave a(η, θ) = c d,s ( η(x) η(y))( θ(x) θ(y)) x y d+s dy dx. R d R d In turn, from te definition (0) of te H s (D) norm, we directly deduce te coercivity of a(, ) on H s (D) (1) a(η, η) = η Hs (D), η H s (D). Diriclet Forms. We define te Diriclet form on H 1 (ω) H 1 (ω) to be d ω (η, φ) := η φ dx. On H 1 (R d ) H 1 (R d ) we write d(η, φ) := d R d(η, φ) := η φ dx. R d ω 3. Scales of interpolation spaces We now introduce anoter set of functions instrumental in te analysis of te finite element metod described in Section 6 and 7. In tis section ω stands for a bounded domain of R d. Given f L (ω), we define θ H 1 0 (ω) to be te unique solution to () (θ, φ) ω + d ω (θ, φ) = (f, φ) ω, for all φ H 1 0 (ω) and define T ω : L (ω) H 1 0 (ω) by (3) T ω f = θ. As discussed in [30], tis defines a densely defined unbounded operator on L (ω), namely L ω f := Tω 1 f for f in D(L ω ) := {T ω φ : φ L (ω)}.

7 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 7 Te operator L ω is self-adjoint and positive so its fractional powers define a Hilbert scale of interpolation spaces, namely, for r 0, Ḣ r (ω) := D(L r/ ω ) wit D(L r ω) denoting te domain of L r ω. Tese are Hilbert spaces wit norms w Ḣr (ω) := Lr/ ω w L (ω). Te space Ḣ1 (ω) coincides wit H 1 0 (ω) wile Ḣ0 (ω) wit L (ω), in bot cases wit equal norms. Hence for r [0, 1], we ave Ḣ r (ω) = (L (ω), H 1 0 (ω)) r,, were (L (ω), H0 1 (ω)) r, denotes te interpolation spaces defined using te real metod. Anoter caracterization of tese spaces stems from Corollary 4.10 in [13], wic states tat for r [0, 1], te spaces H r (ω) are interpolation spaces. Since H 1 (ω) = H0 1 (ω) and H 0 (ω) = L (ω), Hs (ω) coincides wit Ḣr (ω). In particular, we ave (4) C 1 θ Ḣr (ω) θ Hr (ω) C θ Ḣ r (ω), for a constant C only depending on ω. Te intermediate spaces can also be caracterized by expansions in te L (ω) ortonormal system of eigenvectors {ψ i } for T ω, i.e., { } Ḣ r (ω) = φ L (ω) : λ r i (φ, ψ i ) ω <. Here λ i = µ 1 i find tat i=1 were µ i is te eigenvalue of T ω associated wit ψ i. In tis case, we φ Ḣ r (ω) = Lr/ ω and for r (0, 1), (see, e.g., [1]) Here φ Ḣ r (ω) K ω (φ, t) := = sin πr π φ L (ω) = λ r i (φ, ψ i ) ω 0 i=1 t r K ω (φ, t) dt t. inf w H 1 0 (ω) ( φ w L (ω) + t w H 1 (ω) ). Note tat if ω ω ten since te extension of a function φ in H 1 0 (ω ) by zero is in H 1 0 (ω), te K-functional identity implies tat for all r [0, 1], (5) φ Ḣr (ω) φ Ḣ r (ω ), were φ denotes te extension by zero of φ outside ω. Te operator T ω extends naturally to F H 1 (ω) by setting T ω F = u were u H 1 0 (ω) is te solution of () wit (f, φ) ω replaced by F, φ. Here, denotes te functional-function pairing. Identifying f L (ω) wit te functional F, φ := (f, φ) ω, we define te intermediate spaces for r ( 1, 0) by Ḣ r (ω) := (H 1 (ω), L (ω)) 1+r, and set Ḣ 1 := H 1 (ω). Since T ω maps H 1 (ω) isomorpically onto Ḣ1 (ω) and L (ω) isomorpically onto Ḣ (ω), T ω maps Ḣ r (ω) isometrically onto Ḣ r (ω) for r [0, 1].

8 8 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Functionals in H 1 (ω) can also be caracterized in terms of te eigenfunctions of T ω, indeed, H 1 (ω) is te set of linear functionals F for wic te sum λ 1 i F, ψ i is finite. Moreover, F H 1 (ω) = i=1 ( F, θ ) 1/ sup = λ 1 θ H0 1(ω) i F, ψ i θ H 1 (ω) i=1 for all F H 1 (ω). Tis implies tat for r [ 1, 0], Ḣ r (ω) = {F Ḣ 1 : λ r i F, ψ i < } and i=1 ( F Ḣr (ω) = ) 1/ λ r i F, ψ i. i=1 Remark 3.1 (Norm equivalence for Lipscitz domains). For r (1, 3/), it is known tat H r (ω) = H r (ω) H0 1 (ω). On te oter and, we note tat wen ω is Lipscitz, is an isomorpism from H r (ω) H0 1 (ω) to Ḣr (ω); see Teorem 0.5(b) of [9]. We apply tis regularity result into Proposition 4.1 of [7] to obtain H r (ω) H0 1 (ω) = Ḣr (ω). So te norms of Hr (ω) and Ḣr (ω) are equivalent for r [0, 3/) and te equivalence constant may depend on ω. In wat follows, we use H r (D) to describe te smootness of functions defined on D. Wen functions defined on a larger domain (see Section 6 and 7), we will use tese interpolation spaces separately so tat we can investigate te dependency of constants. We end te section wit te following lemma: Lemma 3.1. Let a be in [0, ] and b be in [0, 1] wit a+b. Ten for µ (0, ), we ave (µi + T ω ) 1 φ Ḣ b (ω) µ(a+b)/ 1 φ Ḣa (ω), for all φ Ḣa (ω). Proof. Let φ be in Ḣa (ω) = D(L a/ ω ). Setting θ := L a/ ω φ L (ω), it suffices to prove tat (6) (µi + T ω ) 1 T a/ ω θ Ḣ b (ω) µ(a+b)/ 1 θ L (ω), for all θ L (ω). Te operator T ω and its fractional powers are symmetric in te L (ω) inner product. Terefore, we ave (µi + T ω ) 1 T a/ ω θ Ḣ b (ω) = = i=1 i=1 Inequality (6) follows from te Young s inequality ((µi + T ω ) 1 Tω a/ θ, ψ i ) ω λ b i λ a b i (µ + λ 1 i ) (θ, ψ i) ω. λ (a+b)/ i µ 1 (a+b)/ (µ + λ 1 i ) 1.

9 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 9 4. An Alternative Integral Representation of te Bilinear Form Te goal of tis section is to derive te integral expression (13) and some of its properties. Teorem 4.1 (Equivalent Representation). Let s (0, 1) and 0 r s. For η H s+r (R d ) and θ H s r (R d ), (7) (( ) (s+r)/ η, ( ) (s r)/ θ) = c s were ( y 1 s ) 1 (8) c s := 1 + y dy = sin(πs). π 0 0 t s ( (I t ) 1 η, θ) dt t, Proof. Let I(η, θ) denotes te rigt and side of (7). Parseval s teorem implies tat Rd (9) ( (I t ) 1 ζ η, θ) = 1 + t F(η)(ζ)F(θ)(ζ) dζ. ζ and so (30) I(η, θ) = c s t 1 s 0 ζ R 1 + t F(η)(ζ)F(θ)(ζ) dζ dt. ζ d In order to invoke Fubini s teorem, we now sow tat c s t 1 s ζ 1 + t F(η)(ζ) F(θ)(ζ) dζ dt <. ζ R d 0 Indeed, te cange of variable y = t ζ and te definition (8) of c s implies tat te above integral is equal to c s F(η)(ζ) F(θ)(ζ) t R 1 s ζ 1 + t dt dζ = ζ s F(η)(ζ) F(θ)(ζ) dζ, ζ d R d 0 wic is finite for η H r (R d ) and θ H s r (R d ). We now apply Fubini s teorem and te same cange of variable y = t ζ in (30) to arrive at I(η, θ) = ζ s F(η)(ζ)F(θ)(ζ) dζ = (( ) (s+r)/ η, ( ) (s r)/ θ). R d Tis completes te proof. Teorem 4.1 above implies tat for η, θ in H s (D), (31) a(η, θ) = c s were for ψ L (R d ) 0 t s (w( η, t), θ) D dt t, w(t) := w(ψ, t) := t (I t ) 1 ψ. Examining te Fourier transform of w(ψ, t), we realize tat w(t) := w(ψ, t) := ψ + v(ψ, t) were v(t) := v(ψ, t) H 1 (R d ) solves (3) (v(t), φ) + t d(v(t), φ) = (ψ, φ), for all φ H 1 (R d ). Te integral in (31) is te basis of a numerical metod for (5). Te following lemma, instrumental in our analyze, provides an alternative caracterization for te inner product appearing on te rigt and side of (31).

10 10 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Lemma 4.. Let η be in L (R d ). Ten, (33) (w(η, t), η) = inf { η θ + t d(θ, θ)} =: K(η, t). θ H 1 (R d ) Proof. Let η be in L (R d ). We start by observing tat for any positive t and ζ R d, solves te minimization problem and so ˆφ(ζ) := F(η)(ζ) 1 + t ζ inf { F(η)(ζ) z C z + t ζ z } (34) inf z C { F(η)(ζ) z + t ζ z } = t ζ 1 + t ζ F(η)(ζ). We denote φ to be te inverse Fourier transform of ˆφ. Note tat φ is in H 1 (R d ) (actually, φ is in H (R d )). Applying te Fourier transform, we find tat (35) K(η, t) = inf ( F(η)(ζ) F(θ)(ζ) + t ζ F(θ)(ζ) ) dζ. θ H 1 (R d ) R d Now, φ is te pointwise minimizer of te integrand in (35) and since φ H 1 (R d ), it is also te minimizer of (33). In addition, (34), (35) and (9) imply tat K(η, t) = Rd t ζ Tis completes te proof of te lemma. 1 + t ζ F(η)(ζ) dζ = (w(η, t), η). Remark 4.1 (Relation wit te vanising Diriclet boundary condition case). Te above lemma implies tat for η H s (D), a(η, η) = c s 0 t s K( η, t) dt t. It is observed in te Appendix of [1] tat for any bounded domain ω, and η (L (ω), H0 1 (ω)) s,, te real interpolation space between L (ω) and H0 1 (ω), we ave were η (L (ω),h 1 0 (ω))s, = c s 0 t s K 0 ω(η, t) dt t (36) K 0 ω(η, t) := inf θ H 1 0 (ω) { η θ L (ω) + t d ω (θ, θ)}. Let {ψi 0} H1 0 (ω) denote te L (ω)-ortonormal basis of eigenfunctions satisfying d ω (ψi 0, θ) = λ i (ψi 0, θ) ω, for all θ H0 1 (ω). As te proof in Lemma 4. but using te expansion in te above eigenfunctions, it is not ard to see tat (37) (w ω (η, t), η) ω = K 0 ω(η, t) wit w ω (η, t) = η + v and v H 1 0 (ω) solving (v, θ) ω + t d ω (v, θ) = (u, θ) ω, for all θ H 1 0 (ω).

11 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 11 Tis means tat if η L (ω), K( η, t) K 0 ω(η, t) and ence (w( η, t), η) ω (w ω (η, t), η) ω. 5. Exponentially Convergent Sinc Quadrature In tis section, we analyze a sinc quadrature sceme applied to te integral (31) Te Quadrature Sceme. We first use te cange of variable t = e y so tat (31) becomes a(η, θ) = c s e sy (w( η, t(y)), θ) D dy. Given a quadrature spacing k > 0 and two positive integers N y j := jk so tat and N +, set (38) t j = e yj/ = e jk/ and define te approximation of a(η, θ) by (39) a k (η, θ) := c sk N + j= N e syj (w( η, t j ), θ) D. 5.. Consistency Bound. Te convergence of te sinc quadrature depends on te properties of te integrand ( ) (40) g(y; η, θ) := e sy (w( η, t(y)), θ) D = e sy (e y I ) 1 η, θ. More precisely, te following conditions are required: (a) g( ; η, θ) is an analytic function in te band { B := z = y + iw C : w < π } ; 4 (b) Tere exists a constant C independent of y R suc tat (c) N(B) := π 4 π 4 g(y + iw; η, θ) dw C; ( g(y + i π 4 ; η, θ) + g(y iπ 4 ; η, θ) ) dy <. In tat case, tere olds (see Teorem.0 of [33]) (41) g(y; η, θ) dy k g(kj; η, θ) N(B) sin(π /(4k)) e π /(4k). j= In our context, tis leads to te following estimates for te sinc quadrature error.

12 1 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Teorem 5.1 (Sinc quadrature). Suppose θ H s (D) and η H δ (D) wit δ (s, s]. Let a(, ) and a k (, ) be defined by (5) and (39), respectively. Ten we ave (4) a(η, θ) a k (η, θ) N(B) sin(π /(4k)) e π /(4k) + + δ s e(s δ)n k/ η Hδ (D) θ Hs (D) + k s e sn η L (D) θ L (D). Proof. We start by sowing tat te conditions (a), (b) and (c) old. For (a), we note tat g( ; η, θ) in analytic on B if and only if te operator mapping z (e z I ) 1 is analytic on B. To see te latter, we fix z 0 B and set p 0 := e z0. Clearly, p 0 I is invertible from L (R d ) to L (R d ). Let M 0 := (p 0 I ) 1 L (R d ) L (R d ). For p C, we write pi = (p p 0 )I + (p 0 I ) = (p 0 I ) ( (p p 0 )(p 0 I ) 1 + I ), so tat te Neumann series representation (pi ) 1 = ( 1) j (p p 0 ) j (p 0 I ) j (p 0 I ) 1 j=0 is uniformly convergent provided (p p 0 )(p 0 I ) 1 L (R d ) L (R d ) < 1 or p p 0 < 1/M 0. Hence (pi ) 1 is analytic in an open neigborood of p 0 = e z0 for all p 0 B and (a) follows. To prove (b) and (c), we first bound g(z; η, θ) for z in te band B. Assume η H β (D) and θ H s (D) wit β > s. For z B, we use te Fourier transform and estimate g as follows g(z; η, θ) = ζ esz R e z + ζ F( η)f( θ) dζ d e srez ζ R e Rez + ζ F( η) F( θ) dζ. d Here we used te fact tat e z + ζ Ree z + ζ (e Rez + ζ )/. If Rez < 0, we deduce tat (43) g(z; η, θ) e srez η L (D) θ L (D). Instead, wen Rez 0, we write g(z; η, θ) e (s δ)rez/ ( ζ ) 1 (δ+s)/ (e Rez ) (δ+s)/ e Rez + ζ ζ δ+s F( η) F( θ) dζ. Wence, Young s inequality guarantees tat R d (44) g(z; η, θ) e (s δ)rez/ η Hδ (D) θ Hs (D).

13 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 13 (45) Gatering te above two estimates (43) and (44) gives d g(y + iw; η, θ) dw d π η L (D) θ L (D), y < 0, π η Hδ (D) θ Hs (D), y 0, and (46) N(B) 4 δ s η Hδ (D) θ Hs (D) + s η L (D) θ L (D). Estimates (45) and (46) prove (b) and (c) respectively. Having establised (a), (b), and (c), we can use te sinc quadrature estimate (41). In addition, from (43) and (44) we also deduce tat k g(kj; η, θ) k s e sn η L (D) θ L (D) and (47) j N 1 k j N + +1 g(kj; η, θ) + δ s e(s δ)n k/ η Hδ (D) θ Hs (D). Combining (41) and (47) sows (4) and completes te proof. Remark 5.1 (Coice of N and N + ). Balancing te tree exponentials in (4) leads to te following coice π /(4k) (δ s)n + k/ sn k. Hence, for given te quadrature spacing k > 0, we set (48) N + π π := k and N := (δ s) 4sk. Wit tis coice, (4) becomes (49) a(η, θ) a k (η, θ) γ(k) η Hδ (D) θ Hs (D) were ( ) ( 1 (50) γ(k) := sin(π /(4k)) + 1 ) max δ s, e π /(4k). s 6. Truncated Domain Approximations To develop furter approximation to problem (5) based on te sinc quadrature approximation (39), we replace (3) wit problems on bounded domains Approximation on Bounded Domains. Let Ω be a convex bounded domain containing D and te origin. Witout loss of generality, we assume tat te diameter of Ω is 1. Tis auxiliary domain is used to generate suitable truncation domains to approximate te solution of (3). We introduce a domain parameter M > 0 and define te dilated domains { (51) Ω M {y = (1 + t(1 + M))x : x Ω}, t 1, (t) := {y = ( + M)x : x Ω}, t < 1.

14 14 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Te approximation of a k (, ) in (39) reads (5) a k,m (η, θ) := c sk N + wit t j := t(y j ) = e yj/, according to (38), and j= N e βyj (w M ( η, t j ), θ) D, (53) w M (t) := w M ( η, t) = η Ω M (t) + v M ( η, t), were v M (t) := v M ( η, t) solves (54) (v M (t), φ) Ω M (t) + t d Ω M (t)(v M (t), φ) = (η, φ) D, for all φ H 1 0 (Ω M (t)); compare wit (3). Te domains Ω M (t j ) are constructed for te truncation error to be exponentially decreasing as a function of M. Tis is te subject of next section. 6.. Consistency. Te main result of tis section provides an estimate for a k a k,m. It relies on decay properties of v( η, t) satisfying (3). In fact, Lemma.1 of [] guarantees te existence of universal constants c and C suc tat (55) t v( η, t) L (B M (t)) + v( η, t) L (B M (t)) Ce max(1,t)cm/t η L (D), provided η L (D) and v(t) := v( η, t) is given in (3). Here B M (t) := {x Ω M (t) : dist(x, Ω M (t)) < t} so tat te minimal distance between points in D Ω and B M (t) is greater tan M max(1, t). An illustration of te different domains is provided in Figure 1. Ω M (t) Ω D B M (t) Figure 1. Illustration of te different domains in R. Te domain of interest D is a L-saped domain, Ω Ω M (t) are interior of discs, and B M (t) is te filled portion of Ω M (t). Lemma 6.1 (Truncation error). Let η L (D), e(t) := v( η, t) v M ( η, t) and c be te constant appearing in (55). Tere is a positive constant C not depending on M and t satisfying (56) e(t) L (Ω M (t)) Ce max(1,t)cm/t η L (D).

15 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 15 Proof. In tis proof C denotes a generic constant only depending on Ω. Note tat e(t) satisfies te relations (57) (e(t), φ) + t d Ω M (t)(e(t), φ) = 0, φ H0 1 (Ω M (t)), e(t) = v(t), on Ω M (t). Let χ(t) 0 be a bounded cut off function satisfying χ(t) = 1 on Ω M (t) and χ(t) = 0 on Ω M (t) \ B M (t). Witout loss of generality, we may assume tat χ(t) L (R d ) C/t. Tis implies χ(t)v(t) L (B M (t)) + t (χ(t)v(t)) L (B M (t)) C( v(t) L (B M (t)) + t v(t) L (B M (t))) Ce max(1,t)cm/t η L (D). Here we use te decay estimate (55) for last inequality above. Now, setting e(t) := χ(t)v(t) + ζ(t), we find tat ζ(t) H 1 0 (Ω M (t)) satisfies (ζ(t), φ) Ω M (t) + t d Ω M (t)(ζ(t), φ) = (χ(t)v(t), φ) Ω M (t) t d Ω M (t)(χ(t)v(t), φ) for all φ H 1 0 (Ω M (t)). Taking φ = ζ(t), we deduce tat ζ(t) L (Ω M (t)) + t ζ(t) L (Ω M (t)) χ(t)v(t) L (B M (t)) + t (χ(t)v(t)) L (B M (t)) Ce max(1,t)cm/t η L (D). Tus, combining te estimates for ζ(t) and χ(t)v(t) completes te proof. Lemma 6.1 above is instrumental to derive exponentially decaying consistency error as M. Indeed, we ave te following teorem. Teorem 6. (Truncation error). Let c be te constant appearing in (55) and assume M > (s + 1)/c. Ten, tere is a positive constant C not depending on M nor k satisfying (58) a k (η, θ) a k,m (η, θ) Ce cm η L (D) θ L (D), for all η, θ L (D). Proof. In tis proof C denotes a generic constant only depending on Ω. Let η, θ be in L (D). It suffices to bound N E := c s k + e syj (w(t j ) w M (t j ), θ) D j= N 1 N + C k e syj (v(t j ) v M (t j ), θ) D + k e syj (v(t j ) v M (t j ), θ) D j= N j=0 =: E 1 + E wit v(t) = v( η, t) defined by (3) and v M (t) = v M ( η, t) defined by (54). estimate E 1 and E separately, starting wit E 1. We

16 16 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK From te definition t j = e yj/, we deduce tat wen j < 0, t j > 1 so tat (56) gives E 1 Cke cm 1 j= N e syj η L (D) θ L (D) ke sk Ce cm 1 e sk η L (D) θ L (D) Ce cm η L (D) θ L (D). Similarly, for j 0, i.e. t j < 1, using (56) again, we ave E Ck e syj e cm/tj η L (D) θ L (D) N + j=0 Ck e syj e cm(1+yj/) η L (D) θ L (D) N + j=0 + N = Cke cm e (s cm/)yj η L (D) θ L (D) j=0 Ce cm k 1 exp(k(s cm/)) η L (D) θ L (D) Ce cm cm/ s η L (D) θ L (D) Ce cm η L (D) θ L (D), were we ave also used te property cm/ s > 1 guaranteed by te assumption M > (s + 1)/c Uniform Norm Equivalence on Convex Domains. Since te domains Ω M (t) are convex, we know tat te norms in Ḣr (Ω M (t)) are equivalent to tose in H r (Ω) H 1 0 (Ω M (t)) for r [1, ], see e.g. [7]. However, as we mentioned in Remark 3.1, te equivalence constants depend a-priori on Ω M (t) and terefore on M and t. We sow in tis section tat tey can be bounded uniformly independently of bot parameters. To simplify te notation introduced in Section 3. We sall denote T ΩM (t) by T t, L Ω M (t) by L t and Ḣs (Ω M (t)) by Ḣs. We recall tat Ω M (t) is a dilatation of te convex and bounded domain Ω containing te origin, see (51). We ten ave te following lemma. Lemma 6.3 (Ellipitic Regularity on Convex Domains). Let f L (Ω M (t)). Ten θ := T t f is in H (Ω M (t)) H 1 0 (Ω M (t)) and satisfies (59) θ H (Ω M (t)) C f L (Ω M (t)), were C is a constant independent of t and M. Proof. It is well known tat te convexity of Ω and ence tat of Ω M (t) implies tat te unique solution θ of () wit ω replaced by Ω M (t) is in H (Ω M (t)) H 1 0 (Ω M (t)). Terefore, te crucial point is to sow tat te constant in (59) does not depend on M or t. To see tis, te H elliptic regularity on convex domains implies tat for ˆθ H 1 0 (Ω) wit ˆθ L (Ω) ten ˆθ H (Ω) and tere is a constant C only depending on Ω suc tat (60) ˆθ H (Ω) C ˆθ L (Ω).

17 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 17 Here H (Ω) denotes te H (Ω) seminorm. Let γ be suc tat Ω M (t) = {γx, x Ω} (see (51)) and ˆθ(ˆx) = θ(γˆx) for ˆx Ω. Once scaled back to Ω M (t), estimate (60) gives (61) θ H (Ω M (t)) C θ L (Ω M (t)) = C f θ L (Ω M (t)). Now () immediately implies tat θ H 1 (Ω M (t))) f L (Ω M (t)) and (59) follows by te triangle inequality and obvious manipulations. Remark 6.1 (Intermediate Spaces). Lemma 6.3 implies tat D(L t ) = Ḣ = H (Ω M (t)) H 1 0 (Ω M (t)) wit norm equivalence constants independent of M and t. As D(L 1/ t ) = Ḣ1 = H 1 0 (Ω M (t)), for s [1, ] Ḣ s = (H 1 0 (Ω M (t)), H (Ω M (t)) H 1 0 (Ω M (t))) s 1, = H s (Ω M (t)) H 1 0 (Ω M (t)) wit wit norm equivalence constants independent of M and t. Lemma 6.4 (Norm Equivalence). For β [1, 3/), let θ be in Ḣβ and θ denote its extension by zero outside of Ω M (t). Ten θ is in H β (R d ) and wit C not depending on t or M. θ Ḣβ C θ H β (R d ) Proof. Given θ H 1 (Ω M (t)), we denote Rθ to be te elliptic projection of θ into H 1 0 (Ω M (t)), i.e., Rθ H 1 0 (Ω M (t)) is te solution of (Rθ, φ) ΩM (t) + d ΩM (t)(rθ, φ) It immediately follows tat = (θ, φ) Ω M (t) + d Ω M (t)(θ, φ), for all φ H 1 0 (Ω M (t)). Rθ Ḣ1 = Rθ H1 (Ω M (t)) θ H1 (Ω M (t)). Also, if θ H (Ω M (t)), Lemma 6.3 (see also Remark 6.1) implies Rθ Ḣ C Rθ H (Ω M (t)) C θ H (Ω M (t)) wit C not depending on t or M. Hence, it follows by interpolation tat (6) Rθ Ḣβ C β θ (H1 (Ω M (t)),h (Ω M (t))) β 1,. Now wen θ Ḣβ H 1 (Ω M (t)), Rθ = θ so tat in view of (6), it remains to sow tat θ (H1 (Ω M (t)),h (Ω M (t))) β 1, C θ Hβ (R d ), for a constant C independent of M and t. To see tis, note tat θ is in H 1 (R d ) and te extension of θ by zero is in H β 1 (R d ) for β < 3/. We refer to Teorem of [6] for a proof wen d = 1 proof and te tecniques used in Lemma 4.33 of [19] for te extension to te iger dimensional spaces. Tis implies tat θ belongs to H β (R d ). Moreover, te restriction operator is simultaneously bounded from H j (R d ) to H j (Ω M (t)) for j = 1,. Hence, by interpolation again, we ave tat θ (H 1 (Ω M (t)),h (Ω M (t))) β 1, θ H β (R d ). Tis completes te proof of te lemma.

18 18 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK 7. Finite Element Approximation In tis section, we turn our attention to te finite element approximation of eac subproblems (54) in a k,m (, ). Trougout tis section, we omit wen no confusion is possible te subscript j in t j, i.e. we consider a generic t keeping in mind tat te subsequent statements only old for t = t j wit j = N,..., N +. We also make te additional unrestrictive assumption tat Ω used to define Ω M (t) (see (51)) is polygonal. In turn, so are all te dilated domains Ω M (t) Finite Element Approximation of a k,m (, ). For any polygonal domain ω, let {T (ω)} >0 be a sequence of conforming subdivisions of ω made of simplexes of maximal size diameter 1. We use te notation T M (t) := T (Ω M (t)) for t = t j, j = N,..., N +, given by (38). We assume tat te subdivisions on D are sape-regular and quasi-uniform. Tis means tat tere exist universal constants σ, ρ > 0 suc tat ( ) diam(t ) (63) sup max σ, T T (D) r(t ) >0 ( ) maxt T (D) diam(t ) (64) sup ρ, >0 min T T (D) diam(t ) were diam(t ) stands for te diameter of T and r(t ) for te radius of te largest ball contained in T. We also assume tat tese conditions old as well for T M (t j) wit constants σ, ρ not depending on j. We finally require tat all te subdivisions matc on D, i.e. (65) T (D) T M (t j ) for eac j. We discuss in Section 8 ow to generate subdivisions meeting tese requirements. Define V (ω) H 1 0 (ω) to be te space of continuous piecewise linear finite element functions associated wit T (ω) wit ω = D or Ω M (t). Also, we use te sort notation V M (t) := V (Ω M (t)). We are now in position to define te fully discrete/implementable problem. For η and θ in V (D), te finite element approximation of a k,m (, ) given by (5) is (66) a k,m (η, θ ) := c sk wit N + j= N e syj (w M ( η, t j ), θ ) D (67) w M ( η, t) := η ΩM (t) + v M (t) and were v M (t) VM (t) solves (68) (v M (t), φ ) Ω M (t) + t d Ω M (t)(v M (t), φ ) = ( η, φ ) Ω M (t), φ V M (t). Remark 7.1 (Assumption (65)). Two critical properties follow from (65). On te one and, our analysis below relies on te fact tat te extension by zero ṽ of v V (D) belongs to all V M (t). Tis property greatly simplifies te computation of (w M ( η, t j ), θ ) D in (66).

19 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 19 Te finite element approximation of te problem (4) is to find u V (D) so tat (69) a k,m (u, θ ) = (f, θ ) D for all θ V (D). Analogous to Lemma 4., we ave te following representation using K-functional. Te proof of te lemma is similar to tat of Lemma 4. and is omitted. Lemma 7.1 (K-functional formulation on te discrete space). For η V (D), tere olds were (w M ( η, t), η ) D = (w M ( η, t), η ) ΩM (t) = K ( η, t), K ( η, t) := ( ) min η ϕ ϕ V M (t) L (Ω M (t)) + t d Ω M (t)(ϕ, ϕ ). We empasize tat for v V M (t), its extension by zero η belongs to H 1 (R d ) and terefore (70) K (ṽ, t) K(ṽ, t). Tis property is critical in te proof of next teorem, wic ensures te V (D)- ellipticity of te discrete bilinear for a k,m. Before describing tis next result, we recall tat according to (49) a(η, θ ) a k (η, θ ) γ(k) η Hδ (D) θ Hs (D) wit δ between s and min( s, 3/) (since V (D) H 3/ ɛ (D) for any ɛ > 0) and γ(k) Ce π /(4k). Also, we note tat from te quasi-uniform (63) and saperegular (64) assumptions, tere exists a constant c I only depending on σ and ρ suc tat for r r + < 3/, tere olds (71) v Hr + (D) c I r r + v Hr (D), v V M (t). Teorem 7. (V (D)-ellipticity). Let δ in Teorem 5.1 between s and min( s, 3/), k be te quadrature spacing and c I be te inverse constant in (71). We assume tat te quadrature parameters N and N + are cosen according to (48). Let γ(k) be given by (50) and assume tat k is cosen sufficiently small so tat c I γ(k) s δ < 1. Ten, tere is a constant c independent of, k and M suc tat a k,m (η, η ) c η Hs (D), for all η V (D). Proof. Let η V M (t) so tat η H 1 (R d ). We use te equivalence relations provided by Lemmas 4. and 7.1 togeter wit te monotonicity property (70) to write a k,m (η, η ) = c sk N + e syj K ( η, t j ) c sk j= N N + j= N e syj K( η, t j ) = a k (η, η ). Te quadrature consistency bound (49) supplemented by an inverse inequality (71) yields a k,m (η, η ) a(η, η ) γ(k) η Hδ (D) η Hs (D) a(η, η ) c I γ(k) s δ η Hs (D).

20 0 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Te desired result follows from assumption c I γ(k) s δ < 1 and te coercivity of a(, ), see (1). 7.. Approximations on Ω M (t). Te fully discrete sceme (69) requires approximations by te finite element metods on domains Ω M (t). Standard finite element argumentations would lead to estimates wit constants depending on Ω M (t) and terefore M and t. In tis section, we exibit results were tis is not te case due to te particular definition (51) of Ω M (t). We can use interpolation to develop approximation results for functions in te intermediate spaces wit constants independent of M and t. Te Scott-Zang interpolation construction [40] gives rise to an approximation operator π sz : H1 0 (Ω M (t)) V M (t) satisfying η π sz η H1 (Ω M (t)) C η H1 (Ω M (t)), for all η H0 1 (Ω M (t)) = Ḣ1 and η π sz η H1 (Ω M (t)) C η H (Ω M (t)), for all η H (Ω M (t)) H 1 0 (Ω M (t)) = Ḣ. Te Scott-Zang argument is local so te constants appearing above depend on te sape regularity of te triangulations but not on t or M. Interpolating te above inequalities sows tat for all r [0, 1] (7) inf χ V M (t) η χ H1 (Ω M (t)) C r η Ḣ1+r, for all η Ḣ1+r wit C not depending on t or M. Let T t, denote te finite element approximation to T t given by (3), i.e., for F Ḣ 1, T t, F := w wit w V M (t) being te unique solution of (w, φ) Ω M (t) + d Ω M (t)(w, φ) = F, φ, for all φ V M (t). Te approximation result (7) and standard finite element analysis tecniques implies tat for any r [0, 1], (73) T t F T t, F L (Ω M (t)) C 1+r T t F Ḣ1+r C 1+r F Ḣ 1+r, were te last inequality follows from interpolation since T t F H1 (Ω M (t)) F H 1 (Ω M (t)) and (59) old. For f L (Ω M (t)), we define te operator (74) S t f := η H 1 0 (Ω M (t)) satisfying, d Ω M (t)(η, φ) = (f, φ) Ω M (t), for all φ H 1 0 (Ω M (t)) and let S t, f V M (t) denote its finite element approximation; compare wit T t and T,t. Altoug te Poincaré constant depends on te diameter of Ω M (t), we still ave te following lemma. Lemma 7.3. Tere is a constant C independent of, t, or M satisfying S t f S t, f L (Ω M (t)) C f L (Ω M (t)). Proof. For f L (Ω M (t)), set e := (S t S t, )f. Te elliptic regularity estimate (61) on convex domain and Cea s Lemma imply e H 1 (Ω M (t)) = inf χ V M (t) S t f χ H 1 (Ω M (t)) C S t f H (Ω M (t)) C S t f L (Ω M (t)) = C f L (Ω M (t)),

21 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 1 were C is a constant independent of, t and M. Galerkin ortogonality and te above estimate give e L (Ω M (t)) = d Ω M (t)(e, S t e ) = d ΩM (t)(e, (S t S t, )e ) e H 1 (Ω M (t)) (S t S t, )e H 1 (Ω M (t)) C e H 1 (Ω M (t)) e L (Ω M (t)). Combining te above two inequalities and obvious manipulations completes te proof of te lemma. We sall also need norm equivalency on discrete scales. Let (V M (t), L (Ω M (t))) and (V M (t), H 1 (Ω M (t))) denote V M (t) normed wit te norms in L (Ω M (t)) and H 1 (Ω M (t)), respectively. We define Ḣr (Ω M (t)), or simply Ḣ r, to be te norm in te interpolation space ( V M (t), L (Ω M (t))), (V M (t), H 1 (Ω M (t))) ) r,. For r [0, 1], as te natural injection is a bounded map (wit bound 1) from V M (t) into L (Ω M (t)) and H 1 0 (Ω M (t)), respectively, v Ḣr v Ḣr, for all v V M (t). For te oter direction, one needs a projector into V M (t) wic is simultaneously bounded on L (Ω M (t)) and H0 1 (Ω M (t)). In te case of a globally quasi uniform mes, it was sown by Bramble and Xu [11] tat te L (Ω M (t)) projector π satisfies tis property. Teir argument is local, utilizing te inverse inequality (71) and ence leads to constants depending on tose appearing in (63) and (64) but not t,, or M. Interpolating tese results gives, for r [0, 1], (75) c v Ḣr v Ḣr v Ḣr, for all v V M (t), were c is a constant independent of, M and t. Te spaces for negative r are defined by duality and te stability of te L (Ω M (t))-projection π yields (76) c v Ḣ r v Ḣ r v Ḣ r. for all v V M (t). We finally note tat a discrete version of Lemma 3.1 olds. Its proof is essentially te same and is omitted for brevity. Lemma 7.4. Let a be in [0, ] and b be in [0, 1] wit a + b. Ten for any µ (0, ), (µi + T t, ) 1 η Ḣ b µ (a+b)/ 1 η Ḣa, for all η V M (t) Consistency. Te next step is to estimate te consistency error between a k,m (, ) and a k,m (, ) on V (D). Its decay depends on a parameter β (s, 3/), wic will be related later to te regularity of te solution u to (4). Teorem 7.5 (Finite Element Consistency). Let β (s, 3/). We assume tat te quadrature parameters N and N + are cosen according to (48). Tere exists a constant C independent of, k and M satisfying (77) for all η, θ V (D). a k,m (η, θ ) a k,m (η, θ ) C(1 + ln( 1 )) β s η Hβ (D) θ Hs (D)

22 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Proof. In tis proof, C denotes a generic constant independent of, M, k and t. Fix η V (D) and denote by η its extension by zero outside D. We first observe tat for θ V (D) and θ its extension by zero outside D, we ave (78) (w M ( η, t j ), θ ) D = (π w M ( η, t j ), θ ) ΩM (t), were π denotes te L projection onto V (Ω M (t)). Using te above identity and recalling tat t j = e yj/, we obtain a k,m (η, θ ) a k,m (η, θ ) = c s k e syj (π w M ( η, t j ) w M ( η, t j ), θ ) Ω M (t) t j 1 } {{ } =:E 1 + c s k e syj (π w M ( η, t j ) w M ( η, t j ), θ ) Ω M (t). t j> 1 } {{ } =:E We bound te two terms separately and start wit te latter. 1 In view of te definitions (53) of w M (t) and (67) of w M (t), we ave (79) π w M ( η, t) w M ( η, t) = π v M ( η, t) v M ( η, t). We recall tat T t = T Ω M (t) and S t are defined by (3) and (74) respectively. Using tese operators and te relations satisfied by v M (t) and v M (t) (see (54) and (68)), we arrive at (80) Tus, π w M ( η, t) w M ( η, t) = [S t, (S t, + t I) 1 π S t (S t + t I) 1 ] η = t (S,t + t I) 1 π (S t, S t )(S t + t I) 1 η. π w M ( η, t) w M ( η, t) L (Ω M (t)) t (S t, + t I) 1 π (S t, S t )(S t + t I) 1 η L (Ω M (t)) t (S t, + t I) 1 π S t, S t (S t + t I) 1 η L (Ω M (t)). Here we ave used to denote te operator norm of operators from L (Ω M (t)) to L (Ω M (t)). Combining and Lemma 7.3 gives Wence, E C k (S t, + t I) 1 π t, (S t + t I) 1 t π w M ( η, t) w M ( η, t) Ct η L (Ω M (t)). t j> 1 e (s+1)jk η L (Ω M (t)) θ L (Ω M (t)) C η L (D) θ L (D) k jk< ln e (s+1)jk C η L (D) θ L (D). We now focus on E 1 wic requires a finer analysis using intermediate spaces. Also, we argue differently for β (1, 3/) and for β (s, 1]. In eiter case, we define ɛ := min{1 s, 1/ ln(1/)}

23 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 3 and note tat (81) ɛ 1 c(1 + ln(1/)) and ɛ c wit c depending on s but not. Wen β (1, 3/), we invoke (79) again to deduce (8) E 1 k e syj π v M ( η, t j ) v M ( η, t j ) Ḣ s θ Ḣs. t j 1 We set µ(t) := t 1 and compute (83) π v M ( η, t) v M ( η, t) = t [(I + µ(t)t t, ) 1 T t, π T t (I + µ(t)t t ) 1 ] η = (tµ(t)) (µ(t) 1 I + T t, ) 1 π (T t, T t )(µ(t) 1 I + T t ) 1 η, wic is now estimated in tree parts. Lemma 3.1 guarantees tat (µ(t) 1 I + T t ) 1 Ḣβ Ḣβ 1, were we recall tat Ḣ s stands for Ḣ s (Ω M (t)). For te second part, te error estimate (73) wit 1 + r = β reads T t, T t Ḣβ L (Ω M (t)) Cβ. We estimate te last term of te product in te rigt and side of (83) by (µ(t) 1 + T t, ) 1 π L (Ω M (t)) Ḣ s C (µ(t) 1 + T t, ) 1 Ḣs+ɛ Ḣ s Tus, Lemma 7.4, te inverse estimate and (81) yield (µ(t) 1 + T t, ) 1 π L (Ω M (t)) Ḣ s Note tat for t (0, 1/], 0 < t µ(t) t 1 3 (tµ(t)) 16t 9. Combining te above estimates wit (83) gives (84) π v M ( η, t) v M ( η, t) Ḣ s π L (Ω. M (t)) Ḣs+ɛ C s ɛ t (s+ɛ ) C s t (s+ɛ ). so tat Ct s+ɛ β s η Ḣβ, Since t j = e yj/, e syj t s+ɛ j = e ɛyj/. Estimates (8), (84) and (75) ten yield E 1 C β s k e ɛyj/ η Ḣβ θ Ḣs (85) ky j ln C β s ɛ 1 η Ḣβ θ Ḣs. 3 We bound te norms on Ω M (t) by norms on D using (5) wit r = s and Lemma 6.4 to arrive at E 1 C β s ɛ 1 η H β (R d ) θ Ḣs (D). Applying te norm equivalence (4) gives (86) E 1 C β s ɛ 1 η Hβ (D) θ Hs (D).

24 4 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK 4 Wen β (s, 1], we bound (83) using different norms. In fact, we ave (µ(t) 1 I + T t ) 1 Ḣβ Ḣ 1 t β 1, T t, T t Ḣ 1 L (Ω M (t)) C, and by Lemma 7.4, (µ(t) 1 + T t, ) 1 π L (Ω M (t)) Ḣ s (µ(t) 1 + T t, ) 1 π Ḣ1 β+s+ɛ π Ḣ s L (Ω M (t)) Ḣ1 β+s+ɛ C 1+β s t (s+ɛ β 1). Tese estimates lead (84) and ence (85) as well wen β (s, 1]. Te remainder of te proof is te same as in te case β (1, 3/) except tat te norm equivalence (4) is invoked in place of Lemma Te proof of te teorem is complete upon combining te estimates for E 1 and E Error Estimates. Now tat te consistency error between a(, ) and a k,m (, ) is obtained, we can apply Strang s lemma to deduce te convergence of te approximation u towards u in te energy norm. To acieve tis, we need a result regarding te stability and approximability of te Scott-Zang interpolant π sz [40] in te fractional spaces H β (D). Tis is te subject of te next lemma. Its proof is somewat tecnical and given in Appendix A. Lemma 7.6 (Scott-Zang Interpolant). Let β (1, 3/). Ten, tere is a constant C independent of suc tat (87) π sz v Hβ (D) C v Hβ (D) and for s [0, 1], (88) π sz v v Hs (D) Cβ s v Hβ (D), for all v H β (D). We note tat te above lemma olds for β (0, 1) and s (0, β) provided tat π sz is replaced by π, te L projection onto V (D); see e.g. Lemma 5.1 of [8]. In order to consider bot case simultaneously in te following proof, we set Π = π wen β [0, 1] and Π = π sz wen β (1, 3/). Teorem 7.7. Assume tat te solution u of (5) belongs to H β (D) for β (s, 3/). Let δ := min( s, β) be as in Teorem 5.1, k be te quadrature spacing and c I be te inverse constant in (71). We assume tat te quadrature parameters N and N + are cosen according to (48). Let γ(k) be given by (50) and assume tat k is cosen sufficiently small so tat c I γ(k) s δ < 1. Moreover, let u V (D) be te solution of (69). independent of, M and k satisfying Ten tere is a constant C (89) u u Hs (D) C(γ(k) + e cm + (1 + ln ( 1 )) β s ) u Hβ (D).

25 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 5 Proof. In our context, te first Strang s lemma (see e.g. Teorem in [14]) reads ( ) u u Hs (D) C inf u v v V (D) Hs (D) + sup (a a k,m )(v, w ), w V (D) w Hs (D) were C is a constant independent of, k and M. From te consistency estimates (49), (58) and (77), we deduce tat u u Hs (D) C u Π u Hs (D) + C(γ(k) + e cm + (1 + ln ( 1 )) β s ) Π u Hβ (D) Te desired estimate follows from te approximability and stability of Π. 8. Numerical implementation and results In tis section, we present detailed numerical implementation to solve te following model problems Model Problems. One of te difficulties in developing numerical approximation to (5) is tat tere are relatively few examples were analytical solutions are available. One exception is te case wen D is te unit ball in R d. In tat case, te solution to te variational problem (90) a(u, φ) = (1, φ) D, for all φ H s (D) is radial and given by, (see [3]) (91) u(x) = s Γ(d/) Γ(d/ + s)γ(1 + s) (1 x ) s It is also possible to compute te rigt and side corresponding to te solution u(x) = 1 x in te unit ball. Te corresponding rigt and side can be derived by first computing te Fourier transform of ũ, i.e., F(ũ) = J ( ζ )/ ζ, were J n is te Bessel function of te first kind. Wen 0 < s < 1, we obtain (9) f(x) = F 1 ( ζ s J ( ζ )) = s Γ(d/ + s) Γ(d/)Γ( s) F 1 ( d/ + s, s 1, d/, x ), were F 1 is te Gaussian or ordinary ypergeometric function. Remark 8.1 (Smootness). Even toug te solution u(x) = 1 x is infinitely differentiable on te unit ball, te rigt and side f as limited smootness. Note tat f is te restriction of ( ) s ũ to te unit ball. Now ũ H 3/ ɛ (R d ) for ɛ > 0 but is not in H 3/ (R d ). Tis means tat ( ) s ũ is only in H 3/ s ɛ (R d ) and ence f is only in H 3/ s ɛ (Ω). Tis is in agreement wit te singular beavior of F 1 (d/ + s, s 1, d/, t) at t = 1 (see [36], Section 15.4). In fact, Γ(d/ + s)γ(1 s) F 1 (d/ + s, s 1, d/, 1) = Γ(d/ + 1 s)γ( s) wen 0 < s < 1/, F 1 (d/ + s, s 1, d/, t) Γ(d/) lim = t 1 log(1 t) Γ( 1/)Γ(1/) wen s = 1/, F 1 (d/ + s, s 1, d/, t) Γ(d/) lim t 1 (1 t) s+1 = Γ( 1/)Γ(1/) wen 1/ < s < 1.

26 6 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK Tis implies tat for s 1/, te trace on x = 1 of f(x) given by (9) fails to exist (as for generic functions in H 3/ s (R d )). Tis singular beavior affects te convergence rate of te finite element metod wen te finite element data vector is approximated using standard numerical quadrature (e.g. Gaussian quadrature). 8.. Numerical Implementation. Based on te notations in Section 6, we set Ω = D to be eiter te unit disk in R or D = ( 1, 1) in R. Let Ω M (t) be corresponding dilated domains. In one dimensional case, we consider T (D) to be a uniform mes and V (D) to be te continuous piecewise linear finite element space. For te two dimensional case, T (D) a regular (in te sense of page 47 in [14]) subdivision made of quadrilaterals. In tis case, V (D) is te set of continuous piecewise bilinear functions. Non-uniform Meses for Ω M (t). We extend T (D) to non-uniform meses T M (t), tereby violating te quasi-uniform assumption. For t 1, we use a quasi-uniform mes on Ω M (t) = Ω M (1) wit te same mes size. Wen t > 1 and D = ( 1, 1), we use an exponentially graded mes outside of D, i.e. te mes points are ±e i0 for i = 1,..., M/ wit 0 = (ln γ)/m, were γ is te radius of Ω M (t) (see (51)). Terefore, we maintain te same number of mes points for all Ω M (t). Wen D is a unit disk in R, we start wit a coarse subdivision of Ω M (t) as in te left of Figure (te coarse mes of D in grey). Note tat all vertices of a square ave te same radial coordinates. We also point out tat te position of te vertices along te radial direction and outside of D follow te same exponential distribution as in te one dimensional case. Ten we refine eac cell in D by connecting te midpoints between opposite edges. For te cells outside of D, we consider te same refinement in te polar coordinate system (ln r, θ) wit r > 1 and θ [0, π]. Tis guarantees tat mes points on te same radial direction still follows te exponential distribution after global refinements and te number of mes points in T M (t) is uncanged for all t > 0. Te figure on te rigt of Figure sows te exponentially graded mes after tree times global refinement. Figure. Coarse gird (left) and tree-times-refined non-uniform grid (rigt) of Ω M (t) wit M = 4 and t = 1. Grids of D are in grey.

27 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 7 Matrix Aspects. To express te linear system to be solved, we denote by U to be te coefficient vector of u and F to be te coefficient vector of te L projection of f onto V (D). Let M (t) and A (t) be te mass and stiffness matrix in V M (t). Denote M D, to be te mass matrix in V (D). Te linear system is given by (93) sin (πβ)k π N + i= N e syi M D, (e yi M (t i ) + A (t i )) 1 A (t i )U = F wit y i = ik and t i = e yi/. Here M D,, U and F are all extended by zeros so tat te dimension of te system is equal to te dimension of V M (t). Preconditioner. Since te linear system is symmetric, we apply te Conjugate Gradient metod to solve te above linear system. Due to te norm equivalence between (L (D), H 1 0 (D)) s, and H s (D), te condition number of te system matrix is bounded by C s. In order to reduce te number of iterations in one dimensional space, we use fractional powers of te discrete Laplacian L D, as a preconditioner, were L D, : H 1 0 (Ω) L (D) is defined by d D (L D, w, φ ) = d D (w, φ ), for all φ V (D). Tis can be computed by te discrete sine transform similar to te implementation discussed in [6]. More precisely, te matrix representation of L D, is given by (M D, ) 1 A D,, were A D, is stiffness matrix in V (D). Te eigenvalues of A D, and M D, (for te same eigenvectors) are a j := ( + cos(jπ))/ and m j := (4 + cos(jπ))/6 for j = 1,..., dim(v (D)), respectively. Terefore, te eigenvalues of L are given by λ j, := a j /m j. We use B := SΛS as a preconditioner, were S ij := sin(ijπ) and Λ is te diagonal matrix wose diagonal entries are λ s j, /m j. We also note tat S 1 = S. In two dimensional space, we use te multilevel preconditioner advocated in [10] Numerical Illustration for te Non-smoot Solution. We first consider te numerical experiments for te model problem (90) and study te beavior of te L (D) error. Influence from te Sinc Quadrature and Domain Truncation. Wen D = ( 1, 1), we approximate te solution on te fixed uniform mes wit te mes size = 1/819. Te domain truncation parameter M is also fixed to be 0. Tus, is small enoug and M is large enoug so tat te L (D)-error is dominant by te sinc quadrature spacing k. Te left part of Figure 3 sows tat te L (D)-error quickly converges to te error dominant by te Galerkin approximation wen k approaces zero. Similar results are observed from te rigt part of Figure 3 wen te domain truncation parameter M increases. In tis case, te mes size = 1/819 and te quadrature step size k = 0.. Error Convergence from te Finite Element Approximation. We note tat we implement te numerical algoritm for te two dimensional case using te deal.ii Library [4] and we invert matrices in (93) using te direct solver from UMFPACK [18]. Figure 4 sows te approximated solutions for s = 0.3 and s = 0.7, respectively. Table 1 reports errors u u L (D) and rates of convergence wit s = 0.3, 0.5 and

28 8 ANDREA BONITO, WENYU LEI, AND JOSEPH E. PASCIAK 0.7. Here te quadrature spacing (k = 0.5) and te domain truncation parameter (M = 4) are fixed so tat te finite element discretization dominates te error. We note tat Teorem 7.1 togeter wit Teorem 5.4 in [7] (see also Proposition.7 in [9]) guarantees tat wen D is of class C and f is in L (D), te solution of (5) is in H s+α (D) were (94) α := min{s, 1/} and α denotes any number strictly smaller tat α. Tis indicates tat te expected rate of convergence in L (D) norm sould be β+α s if te solution u is in H β (D). Since te solution u is in H s+1/ ɛ (D) (see [1] for a proof), Table 1 matces te expected rate of convergence min(1, s + 1/). 1e- s=0.3 s=0.5 s=0.7 1e-1 s=0.3 s=0.5 s=0.7 1e-3 1e- L error 1e-4 L error 1e-3 1e-5 1e-4 1e k 1e M Figure 3. Te above figures report te L (D)-error beavior wen D = ( 1, 1). Te left one sows te error as a function of te quadrature spacing k for a fixed mes size ( = 1/819) and domain truncation parameter (M = 0). Te rigt plot reports te error as a function of te domain truncation parameter M wit fixed mes size ( = 1/819) and quadrature spacing (k = 0.). Te spatial error dominates wen k is small (left) and M is large (rigt). #DOFS s = 0.3 s = 0.5 s = Table 1. L (D)-errors for different values of s versus te number of degree of freedom used for te -D nonsmoot computations. #DOFS denotes te dimension of te finite element space V M (t).

29 APPROXIMATION OF THE FRACTIONAL LAPLACIAN 9 Figure 4. Approximated solutions of (91) for s = 0.3 (left) and s = 0.7 (rigt) on te unit disk Numerical Illustration for te Smoot Solution. Wen te solution is smoot, te finite element error (assuming te exact computation of te stiffness entries, i.e. no consistency error) satisfies u u L (D) c s+α, were α is given by (94). In contrast, because of te inerent consistency error, our metod only guarantees (c.f., Teorem 7.7) (95) u u L (D) c 3/ s+α. Table reports L (D)-errors and rates for te problem (5) wit te smoot solution u(x) = 1 x and te corresponding rigt and side data (9) in te unit disk. To see te error decay, ere we coose te quadrature step size k = 0. and te domain truncation parameter M = 5. Te observed decay in te error does not matc te expected rate (95). We tink tis loss of accuracy may be due eiter to te deterioration of te sape regularity constant in generating te subdivisions of Ω M (t) (see Section 8.) or to te imprecise numerical integration of te singular rigt and side in (9). To illustrate tis, we consider te one dimensional problem. Instead of using (9) to compute te rigt and side vector, similar to (7), we compute (96) (f, φ j ) = a(u, φ j ) = ( s 1 L φ j, u ) D + ( s 1 L u, φ j ) D cos(sπ) wit D = ( 1, 1). We note tat wen s < 1/, te fractional derivative wit te negative power s 1 still makes sense for te local basis function φ j. Te rigt and side of (96) can now be computed exactly.