Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

Transcription

1 Controllability of a one-dimensional fractional eat equation: teoretical and numerical aspects Umberto Biccari, Víctor Hernández-Santamaría To cite tis version: Umberto Biccari, Víctor Hernández-Santamaría. Controllability of a one-dimensional fractional eat equation: teoretical and numerical aspects. 17. <al v> HAL Id: al ttps://al.arcives-ouvertes.fr/al v Submitted on 9 Nov 17 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 Controllability of a one-dimensional fractional eat equation: teoretical and numerical aspects Umberto Biccari Víctor Hernández-Santamaría Abstract We analyze te controllability problem for a one-dimensional eat equation involving te fractional Laplacian ( d x ) s on te interval (, 1). Using classical results and tecniques, we sow tat, acting from an open subset ω (, 1), te problem is null-controllable for s > 1/ and tat for s 1/ we only ave approximate controllability. Moreover, we deal wit te numerical computation of te control employing te penalized Hilbert Uniqueness Metod (HUM) and a finite element (FE) sceme for te approximation of te solution to te corresponding elliptic equation. We present several experiments confirming te expected controllability properties. 1 Introduction and main results Let ω (, 1) be an open and nonempty subset. In tis work, we consider te following nonlocal one-dimensional eat equation defined on te domain (, 1) (, T) z t + ( dx ) s z = g1 ω, (x, t) (, 1) (, T) z =, (x, t) [ R \ (, 1) ] (, T) (1.1) z(x, ) = z (x), x (, 1), were z L (, 1) is a given initial datum. In (1.1), for all s (, 1), ( dx ) s denotes te one-dimensional fractional Laplace operator, wic is defined as te following singular integral ( dx ) s u(x) u(y) u(x) = c 1,s P.V. dy. (1.) x y 1+s Here, c 1,s is a normalization constant given by R c 1,s = ss Γ ( ) 1+s, πγ(1 s) were Γ is te usual Gamma function. Moreover, we ave to mention tat, for aving a completely rigorous definition of te fractional Laplace operator, it is necessary to introduce also te class of functions u for wic computing ( d x ) s u makes sense. We postpone tis discussion to te next section. Te analysis of non-local operators and non-local PDEs is a topic in continuous development. A motivation for tis growing interest relies in te large number of possible applications in te modeling of several complex penomena for wic a local approac turns up to be inappropriate or limiting. Indeed, tere is an ample spectrum of situations in wic a non-local equation gives a significantly better description tan a PDE of te problem one wants to analyze. Among oters, we mention applications in turbulence ([5]), anomalous transport and diffusion ([8, 37]), elasticity ([19]), image processing ([7]), porous media flow ([48]), wave propagation in eterogeneous ig contrast media DeustoTec, University of Deusto, 487 Bilbao, Basque Country, Spain. Facultad Ingeniería, Universidad de Deusto, Avda Universidades 4, 487 Bilbao, Basque Country, Spain. s: umberto.biccari@deusto.es, u.biccari@gmail.com, victor.santamaria@deusto.es 1

3 ([49]). Also, it is well known tat te fractional Laplacian is te generator of s-stable processes, and it is often used in stocastic models wit applications, for instance, in matematical finance ([35, 4]). One of te main differences between tese non-local models and classical Partial Differential Equations is tat te fulfillment of a non-local equation at a point involves te values of te function far away from tat point. In te present paper, we are interested in studying control properties for equation (1.1). In more detail, we aim to give an answer to te following question: given any T > and any initial datum z L (, 1), can we find a control function g L (ω (, T)) suc tat te corresponding solution to (1.1) satisfies z(x, T) =? It is well-known tat te classical local eat equation (as well as many oter general variants) is null-controllable for any time T > (see, e.g., [1, 4, 33]). Neverteless, to te best of our knowledge, tere are few results in te literature on te null-controllability of te fractional eat equation, and none of tem is for a problem involving te fractional Laplacian in its integral form (1.). Te existing results ([39, 4]), instead, deal wit te so-called spectral fractional Laplace operator, wose definition will be given later. In tis paper, we deal wit te controllability of (1.1), bot from te teoretical and te numerical point of view. Employing spectral analysis tecniques based on te works [31, 3], te first main result tat we obtain is te following Teorem 1.1. Given any z L (, 1), te parabolic problem (1.1) is null-controllable at time T > wit a control function g L (ω (, T)) if and only if s > 1/. Furtermore, even if for s 1/ null controllability for (1.1) fails, we still ave te following result of approximate controllability, as a consequence of unique continuation properties for te fractional Laplace operator ([]). Teorem 1.. Let s (, 1). Given any z L (, 1), tere exists a control function g L (ω (, T)) suc tat te unique solution z to te parabolic problem (1.1) is approximately controllable at time T >. Teorems 1.1 and 1. will ten find a confirmation in te study of te corresponding numerical control problem. For tis purpose, we will employ te penalized Hilbert Uniqueness Metod, wic relies in te classical works of Glowinski and Lions ([8, 9]). Tis metod is very general and it can be applied to a broad class of PDE control problems, see, for instance, [1, 13, 15, 3]. However, all of te previous works ave one ting in common: tey are devoted to te study of equations of local nature. Here, we will see tat, wen dealing wit a nonlocal equation as (1.1), new issues arise during te numerical implementation. In tis context, for te resolution of te numerical control problem, we will need a finite element (FE) approximation of te solution to te following non-local Poisson equation ( dx ) s u = f, x (, 1) (1.3) u, x R \ (, 1). In te recent past, te fractional Laplacian as been widely analyzed also from te point of view of numerical analysis. We refer, for instance, to te works [, 3, 11]. Tere, te autors present a FE sceme for implementing te solution of (1.3) in a bounded domain Ω R. In particular, tey provide appropriate quadrature rules in order to solve numerically te variational formulation associated to te problem. Moreover, in [3, 11] it is also developed an accurate analysis of te efficiency of te FE metod, employing several existing results. Te tecniques of te aforementioned works ave ten been applied in [1], combined wit a convolution quadrature approac, for solving evolution equations involving te fractional Laplacian. For te sake of completeness, we also mention [1], were it is presented a discretization of te spectral fractional Laplacian and its application to te evolutionary case [9], and [41], were te same problem is treated applying te well known extension of Caffarelli and Silvestre ([16]). In te present paper, we propose a FE approximation for te fractional Poisson equation (1.3) wic does not require any quadrature rule. Indeed, exploiting te one-dimensional nature of te problem, eac entry of te stiffness matrix can be computed explicitly in terms of its position, te parameter s and te mes size. Tis, in particular, allows a quick and simple implementation of te control problem. Tis paper is organized as follows. In Section, we briefly present te functional setting and some existing teory related to te problems tat we are going to analyze. In particular, we give a more accurate definition of te fractional

4 Laplace operator, we discuss te known results on te controllability of fractional parabolic equations and we introduce te variational formulation associated to (1.3) (needed for te development of te FE sceme). Section 3 is devoted to te proof of Teorems 1.1 and 1.. In Section 4, we describe te penalized HUM wit its application to our control problem. Moreover, we present our FE metod for te elliptic equation (1.3), tat ten will be employed for te numerical control of (1.1). In Section 5 we present and comment te results of our numerical simulations. Finally, in Appendix A we include te complete details for computing te stiffness matrix associated to our FE sceme. Preliminary results on te fractional Laplace operator In tis Section, we introduce some preliminary result tat will be useful in te remainder of te paper. We start by giving a more rigorous definition of te fractional Laplace operator, as we ave anticipated in Section 1. Let L 1 s(r) := { u : R R : u measurable, R } u(x) dx <. (1 + x ) 1+s For any u L 1 s and ε > we set ( d x ) s ε u(x) = c 1,s x y >ε u(x) u(y) x y 1+s dy, x R. Te fractional Laplacian is ten defined by te following singular integral ( dx ) s u(x) u(y) u(x) = c 1,s P.V. dy = lim x y 1+s ε + x) εu(x), s x R, (.1) R provided tat te limit exists. We notice tat if < s < 1/ and u is a smoot function, for example bounded and Lipscitz continuous on R, ten te integral in (.1) is in fact not really singular near x (see e.g. [18, Remark 3.1]). Moreover, L 1 s(r) is te rigt space for wic v := ( dx ) ε s u exists for every ε >, v being also continuous at te continuity points of u. It is by now well-known (see, e.g., [18]) tat te natural functional setting for problems involving te Fractional Laplacian is te one of te fractional Sobolev spaces. Since tese spaces are not as familiar as te classical integral order ones, for te sake of completeness, we recall ere teir definition. Given s (, 1), te fractional Sobolev space H s (, 1) is defined as H s (, 1) := u L (, 1) : u(x) u(y) L ( (, 1) (, 1) ) x y 1 +s. It is classical tat tis is a Hilbert space, endowed wit te norm (derived from te scalar product) were te term u H s (,1) := [ u L (,1) + u H s (,1) ( 1 u H s (,1) := is te so-called Gagliardo seminorm of u. We set 1 ] 1, u(x) u(y) ) 1 dxdy x y 1+s H s (, 1) := C (, 1) Hs (,1) te closure of te continuous infinitely differentiable functions wit compact support in (, 1) wit respect to te H s (, 1)-norm. Te following facts are well-known. 3

5 For < s 1, te identity Hs (, 1) = Hs (, 1) olds. Tis is because, in tis case, te C (, 1) functions are dense in H s (, 1) (see, e.g., [36, Teorem 11.1]). For 1 < s < 1, we ave Hs (, 1) = {u Hs (R) : u = in R \ (, 1)} ([3]). Finally, in wat follows we will indicate wit H s (, 1) = (H s (, 1)) te dual space of H s (, 1) wit respect to te pivot space L (, 1). A more exaustive description of fractional Sobolev spaces and of teir properties can be found in several classical references (see, e.g., [4, 18, 36]). Let us now discuss te parabolic equation (1.1). First of all, we mention tat te issues of existence, uniqueness and regularity of te solutions ave been studied by several autors. Among oters, we mention te works [7,, 34]. In particular, in [34, Teorem 6] it is sowed tat, assuming z L (Ω) and g L (, T; H s (Ω)), ten equation (1.1) admits a unique weak solution z L (, T; H s (Ω)) C([, T]; L (Ω)) wit z t L (, T; H s (Ω)). Notice tat taking as in our case g L (ω (, T)), te same result olds due to te continuous injection of L into H s. In tis paper we are mainly interested in te study of control properties for te parabolic system (1.1). For te sake of completeness, we include below te definitions of null and approximate controllability. Definition.1. System (1.1) is said to be null-controllable at time T > if, for any z L (, 1), tere exists g L (ω (, T)) suc tat te corresponding solution z satisfies z(x, T) =. Definition.. System (1.1) is said to be approximately controllable at time T > if, for any z, z T L (, 1) and any δ >, tere exists g L (ω (, T)) suc tat te corresponding solution z satisfies z(x, T) z T L (,1) δ. We already mentioned tat, to te best of our knowledge, tere are no results in te literature concerning te controllability of te fractional eat equation involving te integral operator (.1). Te existing ones deal wit te spectral definition of te fractional Laplace operator, wic is given as follows. Let {ψ k, λ k } k N H 1(, 1) R+ be te set of normalized eigenfunctions and eigenvalues of te Laplace operator in (, 1) wit omogeneous Diriclet boundary conditions, so tat {ψ k } k N is an ortonormal basis of L (, 1) and d xψ k = λ k ψ k, x (, 1), ψ k () = ψ k (1) =. Ten, te spectral fractional Laplacian ( dx ) S s is defined by ( dx ) S s u(x) = u, ψ k λ s k ψ k(x), (.) k 1 firstly for u C (, 1) and ten for u Hs (, 1) employing a density argument. It is important to notice tat te spectral fractional Laplacian and te fractional Laplacian defined as in (.1) are two different operators. For instance, definition (.) depends on te coice of te domain, wile te integral definition does not. For a complete discussion on te differences of tese two operators, we refer to [46]. Te control problem for te fractional eat equation involving te operator ( dx ) S s as been analyzed in [39], were te autors proved null controllability provided tat s > 1/. For s 1/, instead, null controllability does not old, not even for T large. Tis negative result is based on te equivalence (consequence of Müntz Teorem, see, e.g., [45, Page 4]) between te controllability property (more specifically, te possibility of proving an observability inequality), and te following condition for te eigenvalues of te considered operator k 1 1 λ k <, (.3) 4

6 wic is clearly not satisfied for te spectral fractional Laplacian wen s 1/, since in tat case te eigenvalues are λ k = (kπ) s. Finally, in [4], te same result as in [39] is obtained in a multi-dimensional setting, by means of a spectral observability condition for a negative self-adjoint operator, wic allows to prove te null-controllability of te semi-group tat it generates. As we anticipated in Section 1, te same null controllability result olds for te parabolic equation (1.1). Tis will be obtained by means of classical tools ([1]) and by an explicit approximations of te eigenvalues and te eigenfunctions te fractional Laplacian wit omogeneous Diriclet boundary conditions. We stress tat (1.1) is a different model wit respect to te ones analyzed in [38, 4], since te operators (.1) and (.) are not equivalent. We conclude tis section by introducing te variational formulation associated to equation (1.3), wic is at te eart of our numerical metod. Tat is, find u H s (, 1) suc tat a(u, v) = 1 for all v H s (, 1), were te bilinear form a(, ) : Hs (, 1) Hs (, 1) R is given by a(u, v) = c 1,s (u(x) u(y))(v(x) v(y)) dxdy. (.4) x y 1+s R R Since te bilinear form a is continuous and coercive, Lax-Milgram Teorem immediately implies existence and uniqueness of solutions to te Diriclet problem (1.3). In more detail, if f H s (, 1), ten (1.3) admits a unique weak solution u H s (, 1) (see, e.g., [6, Proposition.1]). Furtermore, in te literature it is possible to find improved regularity results for te solution to (1.3), bot in Hölder and Sobolev spaces. Te interested reader may refer, for instance, to [3, 6, 34, 43, 44]. 3 Proof of te controllability properties Tis section is devoted to study te control properties for te parabolic system (1.1). We begin by proving te null controllability result. Proof of Teorem 1.1. First of all, for all ϕ T L (, 1), let ϕ(x, t) be te unique weak solution to te adjoint system ϕ t + ( dx ) s ϕ =, (x, t) (, 1) (, T) ϕ =, (x, t) [ R \ (, 1) ] (, T) (3.1) ϕ(x, T) = ϕ T (x), x (, 1). Multiplying (1.1) by ϕ and integrating over (, 1) (, T), it is straigtforward to ceck tat z(x, T) = if and only if T 1 f v dx, ϕ(x, t)g(x, t)1 ω (x) dxdt = 1 z (x)ϕ(x, ) dx, (3.) In turn, it is classical tat (3.) is equivalent to te existence of a constant C > suc tat te following observability inequality olds T ϕ(x, ) L (,1) C 1 ϕ(x, t)g(x, t)1 ω (x) dx dt, (3.3) Notice tat ϕ can be expressed in te basis of te eigenfunctions of te fractional Laplacian on (, 1) wit zero Diriclet boundary conditions. Namely, ϕ(x, t) = ϕ k e λk(t t) ϱ k (x), (3.4) k 1 5

7 were ϕ k = ϕ T, ϱ k and, for k 1, ϱ k (x) are te solutions to te following eigenvalue problem ( dx ) s ϱ k = λ k ϱ k, x (, 1), k N ϱ k =, x R \ (, 1). Now, plugging (3.4) into (3.3), using te ortonormality of te eigenfunctions ϱ k as a basis of L (, 1) and employing te cange of variables T t t, te observability inequality becomes T ϕ k e λkt C ϕ k g k (t)e λ kt dt, (3.5) k 1 were g k = g1 ω, ϱ k. By means of te classical moment metod ([1]), inequalities of te form (3.5) are well known to be true if and only if (.3) olds and te eigenfunctions ϱ k satisfy te following lower bound: k 1 ϱ k L (ω) C >, k 1, (3.6) were te constant C is independent of k. Te proof of (3.6) is an easy adaptation of te one of [3, Lemma ]. Moreover, according to [31, 3] we ave λ k = ( kπ ) s ( ) (1 s)π 1 + O. 4 k Terefore, we easily see tat te condition (.3) is satisfied if and only if s > 1/. If s 1/, instead, te series diverges, since it beaves like an armonic series. In conclusion, te observability inequality (3.3) olds true wen s > 1/, but it is false wen s 1/. Tis concludes te proof. Even if for s 1/ null controllability for (1.1) fails, Teorem 1. ensures tat, for all s (, 1), we still ave approximate controllability. Tis is consequence of a unique continuation property for te fractional Laplacian, wic as been obtained in []. Proof of Teorem 1.. It is classical (see, e.g., [38, Teorem 5.]) tat te result is true as soon as one as te following unique continuation property for te solution to te adjoint equation (3.1). Given s (, 1) and ϕ T L (, 1), let ϕ be te unique solution to te system (3.1). Let ω (, 1) be an arbitrary open set. If ϕ = on ω (, T), ten ϕ = on (, 1) (, T). (P) Terefore, we are reduced to te proof of te property (P). To tis end, let us recall tat ϕ can be expressed in te form (3.4) and let us assume tat ϕ = in ω (, T). (3.7) Let {ψ k j } 1 k mk be an ortonormal basis of ker(λ k ( dx ) s ). Ten, (3.4) can be rewritten as m k ϕ(x, t) = ϕ k j ψ k j (x) e λ k(t t), (x, t) (, 1) (, T). k 1 j=1 Let z C wit η := R(z) > and let N N. Since te functions ψ k j, 1 j m k, 1 k N are ortonormal, we ave tat N m k ϕ k j ψ k j (x) ez(t T) e λ N m k k(t t) ϕ k j eη(t T) e λ k(t t) k=1 j=1 L (,1) k=1 6 j=1

8 m k ϕ k j eη(t T) e λk(t t) Ce η(t T) ϕ T. L (,1) k 1 j=1 Hence, letting N m k w N (x, t) := ϕ k j ψ k j (x) ez(t T) e λk(t t), k=1 j=1 we ave sown tat w N (x, t) L (,1) Ce η(t T) ϕ T L. Moreover, we ave (,1) T e η(t T) ϕ T L dt = 1 ϕ T L (,1) η (,1) + e τ dτ = 1 ϕ T L η. (,1) Terefore, we can apply te Dominated Convergence Teorem, obtaining T T T m k lim w N (x, t) dt = lim w N(x, t) dt = e z(t T) N + N + ϕ k j ψ k j (x) e λ k(t t) dt k 1 j=1 + m k T + m k + = ϕ k j ψ k j (x) e z(t T) e λk(t t) dt = ϕ k j ψ k j (x) e (z+λk)τ dτ It follows from (3.7) and (3.8) tat = k 1 j=1 + m k k 1 j=1 + m k k 1 k 1 j=1 ϕ k j z + λ k ψ k j (x), x (, 1), R(z) >. (3.8) j=1 ϕ k j z + λ k ψ k j (x) =, x ω, R(z) >. Tis olds for every z C \ { λ k } k N, using te analytic continuation in z. Hence, taking a suitable small circle around λ l not including { λ k } k l and integrating on tat circle we get tat m l w l := ϕ l j ψ l j (x) =, x ω. j=1 According to [, Teorem 1.4], ( dx ) s as te unique continuation property in te sense tat if λ k is an eigenvalue of ( dx ) s on (-1,1) wit Diriclet boundary conditions, and (( dx ) s λ k )ϱ k = in (, 1) wit ϱ k = in ω, ten ϱ k = in (, 1). Tis can applied to w l, in order to conclude w l = in (, 1) for every l. Since {ψ l j } 1 j ml are linearly independent in L (, 1), we get ϕ l j =, 1 j m k, l N. It follows tat ϕ T = and ence, ϕ = in (, 1) (, T), meaning tat ϕ enjoys te property (P). As an immediate consequence, we ave tat our original equation (1.1) is approximately controllable. Our proof is ten concluded. Remark 3.1. According to [], te elliptic unique continuation property for te fractional Laplacian olds in any space dimension. In view of tat, Teorem 1. may be extended also to te case N > 1. On te oter and, te same does not applies to Teorem 1.1. Indeed, te proof of tis result use arguments tat are designed specifically for one-dimensional problems ([1]). If one would like to analyze te null-controllability in a general multi-dimensional setting, oter tools (for instance Carleman estimates) are needed. As far as we know, tese tecniques ave not been fully developed yet for problems involving te fractional Laplacian on a domain. Remark 3.. For te sake of simplicity, in te results presented above we focused on te interval (, 1). Neverteless, everyting tat we did in tis section actually olds in te more general case x ( L, L) and te extension is immediate. 7

9 4 Te penalized HUM and its implementation We devote tis Section to te description of te numerical sceme tat we are going to employ for solving te control problem. Let us start wit a brief description of te so called penalized Hilbert Uniqueness Metod (HUM in wat follows) tat we sall employ for computing te controls for equation (1.1). Here, we will mostly refer to te work of Boyer [1]. Let (E,, ) be a Hilbert space wose norm is denoted by. Let (A, D(A)) be an unbounded operator in E suc tat A generates an analytic semi-group in E tat we indicate by t e ta. Also, we denote (A, D(A )) te adjoint of tis operator and by t e ta te corresponding semi-group. Let (U, [, ]) be anoter Hilbert space wose norm is denoted by. Let B be an unbounded operator from U to D(A ) and let B : D(A ) U be its adjoint. Let T > be given and, for any y E and v L (, T; U), let us consider te non-omogeneous evolution problem y t + Ay = Bv, t [, T] (4.1) y() = y. Te well posedness of (4.1) is guaranteed by [17, Teorem.37]. From now on, we will denote te solution at time T corresponding to te initial datum y and te control v by y v,y (T) = L T (v, y ). (4.) Te linear operator L T (, ) is ten continuous from L (, T; U) E into E. In te framework of bot controllability notions tat we introduced in Section, if one control exists it is certainly not unique. In te penalized version of te HUM, we look for a control tat is solution to a suitable optimization problem. In particular, for any ε >, we sall find were v ε = min F ε (v) (4.3) v L (,T;U) F ε (v) := 1 T v(t) dt + 1 ε L T (v, y ), v L (, T; U). Observe tat, for any ε >, te functional F ε as a unique minimizer in L (, T; U) tat we denote by v ε. Tis is due to te fact tat F ε is strictly convex, continuous and coercive. However, te space L (, T; U) in wic one as to minimize F ε is a quite big one and it depends on te time variable. Tis makes te optimization problem computationally expensive. Tis issue can be circumvented by considering te following problem, defined on te smaller space E. Namely, we consider te minimization problem q T ε = min q E J ε(q T ) (4.4) were J ε (q T ) := 1 T B e (T t)a q T dt + ε q T + LT (y, ), q T, q T E. (4.5) In fact, it is classical to prove tat (4.3) and (4.4) are equivalent since, for any ε >, te minimizers v ε and q T ε of te functionals F ε and J ε, respectively, are related troug te formula v ε = B e (T t)a q T ε, for a.e. t (, T). Notice also tat we can express te approximate and null controllability properties of te system, for a given initial datum y, in terms of te beavior of te penalized HUM approac described above. In particular, we ave 8

10 Teorem 4.1 (Teorem 1.7 of [1]). Problem (4.1) is approximately controllable from te initial datum y if and only if we ave L T (v ε, y ) = y vε,y (T), as ε. Problem (4.1) is null-controllable from te initial datum y if and only if we ave ( In tis case, we ave M y := sup ε> ) inf F ε < +. L (,T;U) v ε L (,T;U) M y, L T (v ε, y ) M y ε. Observe tat te fractional Laplacian ( d x ) s satisfies te properties required for te operator A (see, e.g., [5, Teorem.14]). Terefore, te penalized HUM approac tat we just described can be applied to te control problem (1.1). Following te discussion in [1], we expect tat, under discretization, te solution z to (1.1) retains te properties of controllability stated in Teorem 4.1. Tis will be in accordance wit te teoretical results obtained in Teorems 1.1 and 1.. To tis end, let us study te fully-discrete version of (1.1). For any given mes M and any integer M >, we set δt = T/M and we consider an implicit Euler metod, wit respect to te time variable. More precisely, we consider z n+1 z n M δt z = z, + A z n+1 = v n+1 1 ω, n {1,..., M 1} were z R M, M is te classical mass matrix and A is a suitable stiffness matrix approximating te fractional Laplacian. We are going to present more details on te construction of (4.6) in te next section. In system (4.6), v,δt = (v n ) 1 n M is a fully-discrete control function wose cost, tat is te discrete L δt (, T; RM )- norm, is defined by M v δt L δt (,T;R M ) := δt v n L (R M ) and were L (R M ) stands for te norm associated to te L -inner product on R M i=1 (u, v) L (R M ) = N u i v i. Wit te above notation and according to te penalized HUM strategy, we introduce, for some penalization parameter ε >, te following primal fully-discrete functional i=1 1/, (4.6) F ε,,δt (v δt ) = 1 v δt Lδt (,T;RM ) + 1 ε L T (v δt, y ) L (R M ), v δt Lδt (, T; RM ), tat we wis to minimize onto te wole fully-discrete control space Lδt (, T; RM ) and were z M is te final value of te controlled problem (4.6). Here L T (, ) stands for te discrete version of te operator (4.). We can apply Fencel-Rockafellar teory results to obtain te corresponding dual functional, wic reads as follows J ε,,δt (ϕ T ) = 1 L T (, ) ϕ T + ε Lδt (,T;RM ) ϕt L (R M ) + ( ϕ T, L T (, y ) ), L (R M ) ϕt L,δt (, 1) (4.7) 9

11 were L T (, ) is te adjoint of L T (, ) and L T (, ) ϕ T = (1 ω ϕ n ) 1 n M+1 wit ϕ = (ϕ n ) 1 n M+1 solution to te adjoint system ϕ n ϕ n+1 M + A ϕ n =, n {1,..., M} δt ϕ M+1 = ϕ T. (4.8) Notice tat (4.7) is te fully-discrete approximation of (4.5). Moreover, it can be readily verified tat tis functional as a unique minimizer witout any additional assumption on te problem. Terefore, by minimizing (4.7), and from duality teory, we obtain a control function v ε,,δt = ( ) 1 ω ϕ n ε,,δt 1 n M, were ϕ ɛ is te solution to (4.8) evaluated in te optimal datum ϕ T ε. Tus, te optimal penalized control always exists and is unique. Deducing controllability properties amounts to study te beavior of tis control wit respect to te penalization parameter ε, in connection wit te discretization parameters. It is well known tat, in general, we cannot expect for a given bounded family of initial data tat te fully-discrete controls are uniformly bounded wen te discretization parameters, δt and te penalization term ε tend to zero independently. Instead, we expect to obtain uniform bounds by taking te penalization parameter ε = φ() tat tends to zero in connection wit te mes size not too fast (see [1]) and a time step δt verifying some weak condition of te kind δt ζ() were ζ tends to zero logaritmically wen (see [14]). Tis fact will be confirmed by te numerical simulations tat we are going to present in Section 5.1 below, by observing te beavior of te norm of te control, te optimal energy inf F ε, and te norm of te solution at time T. In tis way, wit te elp of Teorem 4.1, we obtain numerical evidences of te properties of null and approximate controllability for equation (1.1), wic are in accordance wit te teoretical results in Section. 4.1 Finite element approximation According to te formulation of systems (4.6) and (4.8), in order to solve numerically our control problem, a proper approximation A of te operator ( d x ) s is needed. Furtermore, observe tat, for every every time step n, (4.6) and (4.8) are actually discrete equations of te form ( I + δtm A ) u = f. Te matrices appearing in te above equation will be computed employing a finite element sceme on a uniform mes. Computation of te stiffness matrix A. First of all, we recall te variational formulation associated to te elliptic equation (1.3): find u H s (, 1) suc tat a(u, v) = for all v H s (, 1), were a(u, v) is te bilinear form introduced in (.4). Let us take a partition of te interval (, 1) as follows: 1 = x < x 1 <... < < +1 <... < x N+1 = 1, f v dx, (4.9) 1

12 wit +1 = +, i =,... N. We call M te mes composed by te points { : i = 1,..., N}, wile te set of te boundary points is denoted M := {x, x N+1 }. Now, define K i := [, +1 ] and consider te discrete space V := { v H s (, 1) v Ki P 1}, (4.1) were P 1 is te space of te continuous and piece-wise linear functions. Hence, we approximate (4.9) wit te following discrete problem: find u V suc tat c 1,s (u (x) u (y))(v (x) v (y)) 1 dxdy = f v x y 1+s dx, R R for all v V. If now we indicate wit { } φ N i i=1 a basis of V, it will be sufficient tat te above equality is satisfied for all te functions of te basis, since any element of V is a linear combination of tem. Terefore te problem takes te following form c 1,s R Clearly, since u V, we ave R (u (x) u (y))(φ i (x) φ i (y)) 1 dxdy = f v x y 1+s dx, i = 1,..., N. (4.11) u (x) = N u j φ j (x), j=1 were te coefficients u j are, a priori, unknown. In tis way, (4.11) is reduced to solve te linear system A u = F, were te stiffness matrix A R N N as components a i, j = c 1,s (φ i (x) φ i (y))(φ j (x) φ j (y)) dxdy, (4.1) x y 1+s wile te vector F R N is given by F = (F 1,..., F N ) wit R R F i = f, φ i = 1 f φ i dx, i = 1,..., N. Moreover, te basis { φ i } N i=1 tat we will employ is te classical one in wic eac φ i is te tent function wit supp(φ i ) = (, +1 ) and verifying φ i (x j ) = δ i, j. In particular, for x {,, +1 } te i t function of te basis is explicitly defined as (see Figure 1) φ i (x) = 1 x. (4.13) y (, 1) φ i (x) (, ) (, ) (+1, ) x Figure 1: Basis function φ i (x) on its support (, +1 ). Let us now describe our algoritm for computing te coefficients a i, j. Before tat, we sall make te following preliminary comments. 11

13 Remark 4.1. Te following fact are wort noticing. 1. Te constant c 1,s in te definition (.1) of te fractional Laplacian is common for all te entries of te matrix. For te sake of simplicity, we will drop tis constant in te following computations.. It is evident from te definition (4.1) tat A is symmetric. Terefore, in our algoritm we will only need to compute te values a i, j wit j i. 3. Due to te non-local nature of te problem, te matrix A is full. However, wile computing its components, we will encounter many simplifications, due to te fact tat supp(φ i ) supp(φ j ) = for j i Wile computing te values a i, j, we will only work on te mes M, not considering te points of te set M. In tis way, we will ensure tat te basis functions φ i satisfy te zero Diriclet boundary conditions. In oter words, in our FE approximation we are considering only te functions from φ 1 to φ N. Instead, if we considered te points x and x N+1, ten we would need to introduce in our discretization also te basis functions φ and φ N+1, wic take value one at te boundary, and tis would not be consistent wit te continuous problem. Figure provides a grapical explanation of tis last discussion. y 1 φ 1 φ φ 3 φ N x = x 1 x x 3 x 4 x N x N x N+1 = 1 x Figure : Basis functions φ i (x) on te wole interval ( L, L). We now start building te stiffness matrix A. Tis will be done it in tree steps, since te values of te matrix can be computed differentiating among tree well defined regions: te upper triangle, corresponding to j i +, te upper diagonal corresponding to j = i + 1 and te diagonal, corresponding to j = i (see Figure 3). In fact, as it will be clear during our computations, in eac of tese regions te intersections among te support of te basis functions are different, tus generating different values of te bilinear form. In wat follows, we will briefly present wic will be te contributions to te matrin eac of tese tree steps, including te complete computations as an appendix at te end of te paper. Step 1: j i + As we mentioned in Remark 4.1(3), in tis case we ave supp(φ i ) supp(φ j ) = (see also Figure 4). Hence, (4.1) is reduced to computing only te integral x j+1 φ i (x)φ j (y) a i, j = dxdy. (4.14) x j x y 1+s Taking into account te definition of te basis function (4.13), from (4.14) we obtain x j+1 ( ) ( xi+1 1 x y x 1 j ) a i, j = dxdy. x j x y 1+s Finally, tis last integral can be computed explicitly employing te following cange of variables: x = ˆx, y = ŷ. (4.15) 1

14 a 1,1 a 1, a 1, a 1,N a, a,3 a, a,n an,n a N,N a N,N a N,N Figure 3: Structure of te stiffness matrix A. y (, 1) (x j, 1) φ i (x) φ j (x) (, ) (, ) (+1, ) (x j, ) (x j, ) (x j+1, ) x Figure 4: Basis functions φ i (x) and φ j (x) for j i + 1. In tis case, te supports are disjoint. In tis way, for te elements a i, j, j i +, we get te following values: 1 s 4(k + 1)3 s + 4(k 1) 3 s 6k 3 s (k + ) 3 s (k ) 3 s, k = j i, s 1 s(1 s)(1 s)(3 s) a i, j = 4( j i + 1) log( j i + 1) 4( j i 1) log( j i 1) s = 1, j > i + +6( j i) log( j i) + ( j i + ) log( j i + ) + ( j i ) log( j i ), 56 ln() 36 ln(3), s = 1, j = i +. Step : j = i + 1 Tis is te most cumbersome case, since it is te one wit te most interactions between te basis functions (see Figure 5). According to (4.1), and using te symmetry of te integral wit respect to te bisector y = x, we ave (φ i (x) φ i (y))(φ i+1 (x) φ i+1 (y)) a i,i+1 = dxdy x y 1+s = R R x i+1 xi dxdy dxdy + +1 xi... dxdy +... dxdy + + xi +1 xi xi... dxdy... dxdy 13

15 :=Q 1 + Q + Q 3 + Q 4 + Q 5 + Q 6. Tese contributions will be calculated separately, employing canges of variables analogous to (4.15). After several y (, 1) (x j, 1) φ i (x) φ j (x) (, ) (, ) (+1, ) (x j, ) (x j, ) (x j+1, ) x Figure 5: Basis functions φ i (x) and φ i+1 (x). In tis case, te intersection of te supports is te interval [, +1 ]. computations, we obtain 1 s 3 3 s 5 s + 7 a i,i+1 = s(1 s)(1 s)(3 s), s 1 9 ln 3 16 ln, s = 1. Step 3: j = i As a last step, we fill te diagonal of te matrix A, wic collects te values corresponding to te case φ i (x) = φ j (x) (see Figure 6). We ave a i,i = = R R (φ i (x) φ i (y)) dxdy x y 1+s x i+1 xi dxdy + x i xi xi... dxdy xi... dxdy +... dxdy xi... dxdy +1 xi +... dxdy := R 1 + R + R 3 + R 4 + R 5 + R 6 + R 7. xi+1... dxdy y (, 1) = (x j, 1) φ i (x) = φ j (x) (, ) (, ) (+1, ) (x j, ) (x j, ) (x j+1, ) x Figure 6: Basis functions φ i (x) and φ j (x). In tis case, te two functions coincide. 14

16 Once again, te terms R i, i = 1,..., 7 will be computed separately, and summing tem we will obtain 1 s 3 s 4 s(1 s)(1 s)(3 s), s 1 a i,i = 8 ln, s = 1. Conclusion Summarizing, we ave te following values for te elements of te stiffness matrix A : for s 1/ 4(k + 1) 3 s + 4(k 1) 3 s 6k 3 s (k + ) 3 s (k ) 3 s, k = j i, k s(1 s)(1 s)(3 s) a i, j = 1 s 3 3 s 5 s + 7 s(1 s)(1 s)(3 s), j = i + 1 For s = 1/, instead, we ave 3 s 4 s(1 s)(1 s)(3 s), j = i. 4( j i + 1) log( j i + 1) 4( j i 1) log( j i 1) +6( j i) log( j i) + ( j i + ) log( j i + ) + ( j i ) log( j i ), j > i + 56 ln() 36 ln(3), j = i +. a i, j = 9 ln 3 16 ln, j = i ln, j = i. Remark 4.. We point out te following facts: 1. Te matrix A as te structure of a N-diagonal matrix, meaning tat value of its elements remain constant along its diagonals. Tis is in analogy wit te tridiagonal matrix approximating te classical Laplace operator. Notice, owever, tat in our case we obtain a full matrix. Tis is consistent wit te nonlocal nature of te operator tat we are discretizing.. Te value of eac element a i, j is given explicitly, and it only depends on i, j, s and. In oter words, wen approximating te left and side of (4.11), no numerical integration is needed. 3. For s = 1/, te elements a i, j do not depend on te value of wic, in turn, is a function of N. Tis implies tat, in tis particular case, no matter ow many points we consider in our mes, te matrix A will always ave te same entries. 4. A quick computation sows tat taking te limit s 1 in A we recover te tridiagonal matrix of te classical FE approximation of te local Poisson equation. Tis is in accordance wit te same beavior of te continuous operator, as sown, for instance, in [18, Proposition 4.4]. 15

17 Computation of te mass matrix M. Te matrix M appears wen deriving te FE sceme for te parabolic problem (1.1). Adapting te same construction tat we presented before for te elliptic equation (1.3), te approximate solution z sall be written in te form z (x, t) := N z j (t)φ j (x), j=1 were te basis functions {φ j } N j=1 are te same as before wile te coefficients {z j(t)} N j=1 ave to be computed. Ten, te corresponding variational formulation reads as N 1 z j (t) φ j φ i dx +z j (t) a(φ j, φ i ) = gφ i dx, i = 1,..., N. j=1 } {{ } ω } {{ } a i, j } {{ } m i, j g i (t) Terefore, denoting g(t) := (g 1 (t),..., g N (t)) T, te unknown vector z(t) := (z 1 (t),..., z N (t)) T is obtained by solving te following linear system M z (t) + A z(t) = g(t), (4.16) were M is te mass matrix wit entries m i, j and A is te stiffness matrix composed by te entries a i, j previously computed. Solving (4.16) by means of an implicit Euler metod in time, we obtain exactly te formulation (4.6). Moreover, (4.8) is obtained wit te same procedure, taking into account tat tis time, te original problem being omogeneous, te vector g is actually zero. 5 Numerical results In tis Section, we present te numerical simulations corresponding to te algoritm previously described, and we provide a complete discussion of te results obtained. First of all, we test numerically te accuracy of our metod for te resolution of te elliptic equation (1.3), by applying it to te following problem { ( d x ) s u = 1, x (, 1) (5.1) u, x R \ (, 1). In tis particular case, te unique solution to (5.1) can be computed exactly and it is given in [6]. It reads as follows, s π u(x) = Γ ( ( ) ) 1 x s 1(,1). (5.) 1+s Γ(1 + s) In Figure 7, we sow a comparison for different values of s between te exact solution (5.) and te computed numerical approximation. Here we consider N = 5. One can notice tat wen s =.1 (and also for oter small values of s), te computed solution is to a certain extent different from te exact solution. However, one sould be careful wit suc result and a more precise analysis of te error sould be carried. In te same spirit as in [], te computation of te error in te space H s (, 1) can be readily done by using te definition of te bilinear form, namely u u H s (,1) = a(u u, u u ) = a(u, u u ) = 1 f (x) (u(x) u (x)) dx, 16

18 Numerical solution Real solution (a) s = (b) s = (c) s = (d) s =.8 Figure 7: Plot for different values of s. were ave used te ortogonality condition a(v, u u ) = v V. For tis particular test, since f 1 in (, 1), te problem is terefore reduced to ( 1 u u H s (,1) = ) 1/ (u(x) u (x)) dx were te rigt-and side can be easily computed, since we ave te closed formula 1 π u dx = s Γ(s + 1 )Γ(s + 3 ) and te term corresponding to 1 u can be carried out numerically. In Figure 8, we present te computational errors evaluated for different values of s and. Te rates of convergence sown are of order (in ) of 1/. Tis is in accordance wit te following result: Teorem 5.1 ([, Teorem 4.6]). For te solution u of (4.9) and its FE approximation u given by (4.11), if is sufficiently small, te following estimates old u u H s (,1) C 1/ ln f C 1/ s (,1), if s < 1/, u u H s (,1) C 1/ ln f L (,1), if s = 1/ u u H s (,1) C s 1/ ln f C β (,1), if s > 1/, β > 17

19 1 slope.5 s =.1 s =.3 s =.5 s =.7 s = Figure 8: Convergence of te error. were C is a positive constant not depending on. Moreover, Figure 8 sows tat te convergence rate is maintained also for small values of s. Tis confirms tat te beavior sown in Figure 7a is not in contrast wit te known teoretical results. Indeed, since it is well-known tat te notion of trace is not defined for te spaces H s (, 1) wit s 1/ (see [36, 47]), it is someow natural tat we cannot expect a point-wise convergence in tis case. 5.1 Control experiments To address te actual computation of fully-discrete controls for a given problem, we use te metodology described, for instance, in [9]. We apply an optimization algoritm to te dual functional (4.7). Since tese functionals are quadratic and coercive, te conjugate gradient is a natural and quite simple coice. First of all, starting from te expression (4.7), te gradient of te functional J ε,,δt (ϕ T ) can be easily computed and it reads as ( J ε,,δt (ϕ T ) = L T L T (, ) ϕ T, ) + εϕ T + L T (, y ). Hence, te computation of tis gradient at eac iteration amounts to solve first te omogeneous equation (4.8). Ten, set v n = ϕ n 1 ω and finally solve (4.6) wit zero initial datum. In tis way, te procedure to compute te gradient of (4.7) basically requires to solve two parabolic equations: a omogeneous backward one associated wit te final data ϕ T, and a non-omogeneous forward problem wit zero initial data. We present now some results obtained wit te described metodology. In accordance wit te discussion in Section 4, we use te finite-element approximation of ( d x ) s for te space discretization and te implicit Euler sceme in te time variable. We denote by N te number of points in te mes and by M te number of time intervals. As discussed in [14], te results in tis kind of problems does not depend too muc in te time step, as soon as it is cosen to ensure at least te same accuracy as te space discretization. Te same remains true ere, and terefore we always take M = in order to concentrate te discussion on te dependency of te results wit respect to te mes size and te parameter s. As we mentioned, we coose te penalization term ε as a function of. A reasonable practical rule ([1]) is to systematically coose ε = φ() p were p is te order of accuracy in space of te numerical metod employed for te discretization of te spatial operator involved (in tis case te fractional Laplacian (.1)). We recall tat te solution z to (1.1) belongs to te space L (, T; H s (, 1)) C([, T]; L (, 1)). In view of tat, we immediately ave tat z(, T) L (, 1). Terefore, we sall coose te value of p as te convergence rate in te L -norm for te discretization of te elliptic problem (1.3). Tis convergence rate is given by te following result. 18

20 Teorem 5. ([11, Proposition 3.3.]). Let s (, 1), f L (, 1) and u be te solution to (1.3). Given a uniform mes M wit mes size, and te space V defined as in (4.1), let u be te finite element solution to te corresponding discrete problem. Ten, it olds tat were α := min{s, 1/ ε}, for all ε >. u u L (,1) C(s, α) α f L (,1), By virtue of Teorem 5., te appropriate value of p tat we sall employ is s, for s < 1 p = α = 1 ε, for s 1. We present below te numerical experiments obtained applying our metod. We begin by plotting on Figure 9 te time evolution of te uncontrolled solution as well as te controlled solution. Here, we set s =.8, ω = (.3,.8) and T =.3, and as an initial condition we take z (x) = sin(πx). Te control domain is represented as igligted zone on te plane (t, x). As expected, we observe tat te uncontrolled solution is damped wit time, but does not reac zero at time T, wile te controlled solution does. 1 1 T =.3 T =.3 (a) Uncontrolled solution (b) Controlled solution ( =control domain) Figure 9: Time evolution of system (4.6). In Figure 1, we present te computed values of various quantities of interest wen te mes size goes to zero. More precisely, we observe tat te control cost v δt L δt (,T;R M ) and te optimal energy remain bounded as. On te oter and, we see tat y M L (R M ) C φ() = C 1/. (5.3) We know tat, for s =.8, system (1.1) is null controllable. Tis is now confirmed by (5.3), according to Teorem 4.1. In fact, te same experiment can be repeated for different values of s > 1/, obtaining te same conclusions. According to te discussion in Section, one can prove tat null controllability does not old for system (1.1) in te case s 1/. However approximate controllability can be proved by means of te unique continuation property of te operator ( d x ) s. We would like to illustrate tis property in Figure 11. We observe tat te results are different from wat we obtained in Figure 1. In fact, te cost of te control and te optimal energy increase in bot cases, wile te target y M tends to zero wit a slower rate tan 1/. Tis seems to confirm tat a uniform observability estimate for (1.1) does not old and tat we can only expect to ave approximate controllability (see Teorem 4.1). A Explicit computations of te elements of te matrix A We present ere te explicit computations for eac element a i, j of te stiffness matrix, completing te discussion tat we started in Section 4. 19

21 1 1 1 Cost of te control Size of y M Optimal energy 1 1 sl Figure 1: Convergence properties of te metod for te controllability of te fractional eat equation for s =.8. Step 1: j i + We recall tat, in tis case, te value of a i, j is given by te integral x j+1 a i, j = x j φ i (x)φ j (y) dxdy. (A.1) x y 1+s In Figure 1, we give a sceme of te region of interaction (marked in gray) between te basis functions in tis case. Tese are te only regions in wic (A.1) will be different tan zero. Now, taking into account te definition of te basis function (4.13), te integral (A.1) becomes x j+1 a i, j = x j Let us introduce te following cange of variables: x = ˆx, ( 1 x ) ( y x 1 j ) dxdy. x y 1+s Ten, rewriting (wit some abuse of notations since tere is no possibility of confusion) ˆx = x and ŷ = y, we get a i, j = 1 s 1 1 y = ŷ. (1 x )(1 y ) dxdy. (A.) x y + i j 1+s Te integral (A.) can be computed explicitly in te following way. First of all, for simplifying te notation, let us define k = j i. We ave a i, j = 1 s 1 = 1 s s 1 (1 x )(1 y ) dxdy = 1 s x y + i j 1+s (1 x)(1 y) dxdy 1 s (y x + k) 1+s (1 x)(1 + y) dxdy 1 s (y x + k) 1+s (1 x )(1 y ) x y k 1+s dxdy (1 + x)(1 y) dxdy (y x + k) 1+s (1 + x)(1 + y) dxdy (y x + k) 1+s

22 sl..6/.45 1 sl..4/.7 sl sl (a) s = (b) s =.5 Figure 11: Convergence properties of te metod for s < 1/. Same legend as in Figure 1 = 1 s (B 1 + B + B 3 + B 4 ). Tese terms B i, i = 1,, 3, 4, can be computed integrating by parts several times. In more detail, we ave 1 B 1 = [k 1 s (k + 1) s (k 1) s k3 s (k + 1) 3 s (k 1) 3 s ] 4s(1 s) 1 s (1 s)(3 s) 1 B = [ k 1 s + (k + 1) s k s + (k + 1)3 s k 3 s (k + ) 3 s ] 4s(1 s) 1 s (1 s)(3 s) 1 B 3 = [ k 1 s + k s (k 1) s + (k 1)3 s k 3 s (k ) 3 s ] 4s(1 s) 1 s (1 s)(3 s) 1 B 4 = [k 1 s (k + 1) s (k 1) s k3 s (k + 1) 3 s (k 1) 3 s ]. 4s(1 s) 1 s (1 s)(3 s) Terefore, we obtain a i, j = 1 s 4(k + 1)3 s + 4(k 1) 3 s 6k 3 s (k + ) 3 s (k ) 3 s. (A.3) s(1 s)(1 s)(3 s) We notice tat, wen s = 1/, bot te numerator and te denominator of te expression above are zero. Hence, in tis particular case, it would not be possible to introduce te value tat we just encountered in our code. Neverteless, tis difficulty can be overcome at least in two ways: 1. by setting s = 1/ in B i, i = 1,, 3, 4, before computing tese integrals;. by computing te limit s 1/ in (A.3). However, since we already ave te expressions of B i, i = 1,, 3, 4, for a general s, te second approac is actually straigtforward and quicker. Indeed, we can easily compute lim 1 s 4(k + 1)3 s + 4(k 1) 3 s 6k 3 s (k + ) 3 s (k ) 3 s s 1 s(1 s)(1 s)(3 s) = 4(k + 1) log(k + 1) 4(k 1) log(k 1) + 6k log(k) + (k + ) log(k + ) + (k ) log(k ), 1

23 y x j+1 x j x j +1 O +1 x j x j x j+1 x Figure 1: Interactions between te basis function φ i and φ j wen j i +. if k. Wen k =, instead, since lim (k k ) log(k ) =, te corresponding value a i, j = a i,i+ is given by a i,i+ = 56 ln() 36 ln(3). Step : j = i + 1 Tis is te most cumbersome case, since it is te one wit te most interactions between te basis functions (see Figure 5). According to (4.1), and using te symmetry of te integral wit respect to te bisector y = x, we ave (φ i (x) φ i (y))(φ i+1 (x) φ i+1 (y)) a i,i+1 = dxdy x y 1+s = R R + + x i+1 xi dxdy dxdy + :=Q 1 + Q + Q 3 + Q 4 + Q 5 + Q xi... dxdy +... dxdy + + xi +1 xi xi... dxdy... dxdy In Figure 13, we give a sceme of te regions of interactions between te basis functions φ i and φ i+1 enligtening te domain of integration of te Q i. Te regions in grey are te ones tat produce a contribution to a i,i+1, wile on te regions in wite te integrals will be zero. Le us now compute te terms Q i, i = 1,..., 6, separately. Computation of Q 1 Since φ i = on te domain of integration we ave Q 1 = φ i+1(x) φ i+1(y) x y 1+s dxdy

24 y + Q Q 1 +1 Q 3 Q 5 Q 4 Q Q 3 O +1 + x Q 6 Q 5 Figure 13: Interactions between te basis function φ i and φ i+1. = φ i+1(x) dxdy x y 1+s φ i+1(y) dxdy =. x y 1+s Te fact tat tis integral is zero is, actually, not surprising since, according to Figure 13, te region of integration for Q 1 is outside of te region of interaction of te basis functions. Computation of Q We ave Q = + +1 φ i (x)(φ i+1(x) φ i+1(y)) x y 1+s dxdy. Now, using Fubini s teorem we can excange te order of te integrals, obtaining ( + ) dy xi+ φ i (x)φ i+1(y) Q = φ i (x)φ i+1 (x) dx dxdy x y 1+s x y 1+s = 1 s = 1 s x i xi+1 +1 φ i (x)φ i+1(x) dx (+1 x) s ( ) ( 1 x 1 x +1 xi+ +1 (+1 x) s dx ) +1 φ i (x)φ i+1(y) dxdy x y 1+s xi+ +1 ( 1 x ) ( ) 1 y +1 dxdy := Q 1 x y 1+s + Q. Te two integrals above can be computed explicitly. Indeed, employing te cange of variables and ten renaming ˆx = x, Q 1 becomes Q 1 = 1 s s 1 +1 x = ˆx, x 1 s (1 x) dx = 1 s s( s)(3 s). 3

25 For computing Q, instead, we introduce te cange of variables and we obtain Q = 1 s 1 1 x = ˆx, y +1 = ŷ, (A.4) (1 x)(1 y) (y x + 1) dxdy = 1 s + s 1+s 1 s s(1 s)(3 s). Adding te two contributions, we get te following expression for te term Q Q = 1 s s + s 3 s( s)(3 s). Computation of Q 3 In tis case, we simply take into account te intervals in wic te basis functions are supported, so tat we obtain xi+ xi φ i (x)φ xi+ ( ) ( ) xi i+1(y) 1 x 1 y +1 Q 3 = dxdy = dxdy. +1 x y 1+s +1 x y 1+s Tis integral can be computed applying again (A.4), and we get Q 3 = 1 s 1 if s 1/. If s = 1/, instead, we ave Q 3 = (1 + x)(1 y) (y x + 1) dxdy = s 1 s 3 s s + s( 4 s 14) + 4s, (A.5) s(1 s)(1 s)(3 s) 1 (1 + x)(1 y) (y x + 1) dxdy = ln 3 16 ln(). Notice tat tis last value could ave been computed directly from (A.5), by taking te limit as s 1/ in tat expression, being tis limit exactly ln 3 16 ln(). Computation of Q 4 In tis case, we are in te intersection of te supports of φ i and φ i+1. Terefore, we ave Q 4 = (φ i (x) φ i (y))(φ i+1(x) φ i+1(y)) x y 1+s dxdy. Moreover, we notice tat, tis time, it is possible tat x = y, meaning tat Q 4 could be a singular integral. To deal wit tis difficulty, we will exploit te explicit definition of te basis function. We ave (see also Figure 14) Terefore, φ i (x) = 1 x, φ i+1 (x) = +1 x, x (, +1 ). and te integral becomes ( y x ) ( x y ) x y (φ i (x) φ i (y))(φ i+1 (x) φ i+1 (y)) = =, Q 4 = 1 s x y 1 s dxdy = (1 s)(3 s). 4