1. Introduction. In this paper, we discuss the numerical solution of singular boundary value problems of the form

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1 ANALYSIS OF A NEW ERROR ESTIMATE FOR COLLOCATION METHODS APPLIED TO SINGULAR BOUNDARY VALUE PROBLEMS WINFRIED AUZINGER, OTHMAR KOCH, AND EWA WEINMÜLLER Absrac. We discuss an a-poseriori error esimae for he numerical soluion of boundary value problems for nonlinear sysems of ordinary differenial equaions wih a singulariy of he firs kind. The esimae for he global error of an approximaion obained by collocaion wih piecewise polynomial funcions is based on he defec correcion principle. We prove ha for collocaion mehods based on an even number of equidisan collocaion nodes, he error esimae is asympoically correc. As an essenial prerequisie we derive convergence resuls for collocaion mehods applied o nonlinear singular problems. Key words. Boundary value problems, singulariy of he firs kind, collocaion mehods, error esimae, defec correcion, asympoical correcness. AMS subjec classificaion. 65L5 1. Inroducion. In his paper, we discuss he numerical soluion of singular boundary value problems of he form 1.1a 1.1b 1.1c z = M z + f, z,, 1], B a z + B b z1 = β, z C[, 1], where z is an n-dimensional real funcion, M is a smooh n n marix and f is an n-dimensional smooh funcion on a suiable domain. B a and B b are consan r n marices, wih r < n. Condiion 1.1c is equivalen o a se of n r linearly independen condiions z mus saisfy. These boundary condiions are augmened by 1.1b o yield an isolaed soluion z. In his paper, we resric our aenion o he class of singular boundary value problems which can equivalenly be expressed as a wellposed singular iniial value problem, where all boundary condiions are posed a =. The search for an efficien numerical mehod o solve problems 1.1 is srongly moivaed by numerous applicaions from physics, see [9], [17], [29], chemisry, cf. [25], and mechanics buckling of spherical shells, [1], [2], or ecology, see [21] and [23]. Also, research aciviies in relaed fields, like he compuaion of connecing orbis in dynamical sysems [24], differenial algebraic equaions [22] or singular Surm- Liouville problems [7], may benefi from echniques developed for problems of he form 1.1. To compue he numerical soluion of 1.1, we use collocaion a an even number of collocaion poins spaced equidisanly in he inerior of every collocaion inerval. Collocaion has been used in one of he bes esablished sandard codes for regular boundary value problems, COLSYS COLNEW, see [1] and [2]. In COLSYS, superconvergen collocaion a Gaussian poins is used, cf. [8]. Our decision o use collocaion was moivaed by is advanageous convergence properies for 1.1, while This projec was suppored by he Ausrian Science Fund FWF under gran no. P-1572-MAT. All: Insiue for Applied Mahemaics and Numerical Analysis, Vienna Universiy of Technology, Ausria. 1

2 2 W. Auzinger, O. Koch and E. Weinmüller in he presence of a singulariy oher high order mehods show order reducions and become inefficien see for example [16]. For linear problems 1.1 which can equivalenly be posed as iniial value problems, i was shown in [15] ha he convergence order of collocaion mehods is a leas equal o he sage order see below of he mehod. We will discuss he resricions implied by he laer requiremen in 3. For he general class of problems 1.1, numerical evidence suggess ha he full convergence order holds for boh he linear and he nonlinear case 1, cf. [5]. One of he reasons why we concenrae on collocaion a an even number of equidisan poins is ha in general, we canno expec o observe superconvergence cf. [8] when collocaion is applied o 1.1. A mos, a convergence order of O lnh n 1 h, for some posiive ineger n, holds for a mehod of sage order m, see [15]. Our main aim was o consruc an efficien asympoically correc error esimae for he global error of he numerical soluion obained by collocaion. This esimae, inroduced in [5], is based on he defec correcion principle, which was firs considered in [3] for he esimaion of he global error of Runge-Kua mehods. In [3], he esimae for he error in he mesh poins is obained by applying he high order basic numerical scheme wice, o he original and o a suiably defined neighboring problem. An exension of his idea proposed in [11], [26] avoids he second applicaion of he high order scheme, using a cheap low order mehod wice insead. Again, his esimae is asympoically correc in he mesh poins only. A furher modificaion proposed by he auhors provides an error esimae which is asympoically correc a boh, he mesh and he collocaion poins. The analysis of his esimae in he conex of nonlinear regular problems was given in [5]. I could be shown ha for a collocaion mehod of order Oh m, he error of he esimae he difference beween he esimae and he exac global error is of order Oh. Numerical evidence suggess ha his is also rue for singular problems. In his paper, we will prove his asserion for he class of singular problems 1.1. The reason why we propose o conrol he global error insead of monioring he local error is he unsmoohness of he local error near he singular poin and order reducions from which i ofen suffers. For an exensive discussion of his phenomenon and numerical experimens, see [6] and [12]. I urns ou ha grids generaed via he equidisribuion of he local error are usually very fine close o he singulariy even when he soluion is smooh here. The collocaion mehod and error esimae described in his paper were also implemened in he MATLAB code sbvp designed especially o solve singular boundary value problems. The error esimae yields a reliable basis for a mesh selecion procedure which enables an efficien compuaion of he numerical soluion. A descripion of he code and experimenal evidence of is advanageous properies are given in [4]. The paper is organized as follows: The analyical properies of 1.1 which were discussed in deail in [14] are briefly recapiulaed in 3. In 4.1, he resuls for collocaion mehods according o [15] are given. Using hese resuls, we derive new, refined bounds for he errors of he numerical soluion and is derivaive, and exend hese resuls o he nonlinear case. This analysis is carried ou in 4.2. In 5 we use hese esimaes for he collocaion soluion in order o prove ha our version of he error esimae is asympoically correc for problem 1.1. Finally, in 6 we give a numerical example which illusraes he heory. 1 The analysis given in [27] for second order problems migh provide ools o prove his asserion.

3 Error esimae for singular BVPs 3 2. Preliminaries. Throughou he paper, he following noaion is used. We denoe by R n he space of real vecors of dimension n and use, x = x 1, x 2,..., x n T := max 1 i n x i, o denoe he maximum norm in R n. C p n[, 1] is he space of real vecor-valued funcions which are p imes coninuously differeniable on [, 1]. For funcions y C n[, 1], we define he maximum norm, y [,1] := max 1 y, or more generally for an inerval J [, 1], y J := max J y. C p n n [, 1] is he space of real n n marices wih columns in Cn[, p 1]. For a marix A = a ij n i,j=1, A C n n[, 1], A [,1] is he induced norm, n A [,1] = max A = max max a ij i n Where here is no confusion, we will omi he subscrips n and n n and denoe C[, 1] = C [, 1]. For he numerical analysis, we define meshes := τ, τ 1,..., τ N, and h i := τ i+1 τ i, i =,..., N 1, τ =, τ N = 1. For reasons of simpliciy, we resric he discussion o equidisan meshes, h i = h, i =,..., N 1. However, he resuls also hold for nonuniform meshes which have a limied variaion in he sepsizes. On, we define corresponding grid vecors j=1 u := u,..., u N R N+1n. The norm on he space of grid vecors is given by u := max k N u k. For a coninuous funcion y C[, 1], we denoe by R he poinwise projecion ono he space of grid vecors, R y := yτ,..., yτ N. For collocaion, m poins spaced a disances, j = 1..., m, are insered in each subinerval J i := [τ i, τ i+1 ]. This yields he fine grid 2 { } j 2.1 m := : = τ i + δ i,k, i =,..., N 1, j =,..., m + 1. k= 2 For convenience, we denoe τ i by i, i 1,, i = 1,..., N 1. Moreover, we define δ i, :=, δ i, := i, i,m. Noe ha we choose he same disribuion of collocaion poins in every subinerval J i, and ha P j= = h i holds for i =,..., N 1.

4 4 W. Auzinger, O. Koch and E. Weinmüller We resric ourselves o grids where δ i,1 > o avoid a special reamen of he singular poin =. For he analysis of collocaion mehods, we allow δ i, =. In he discussion of he error esimae, we consider only equidisan collocaion poins, where 2.2 := h i, j = 1,..., m + 1. m + 1 For a grid m, u m, m and R m are defined accordingly. {}}{ τ... τ i } {{ } h i Fig The compuaional grid τ i+1... τ N 3. Analyical resuls. In his secion we discuss he analyical properies of 1.1, cf. [14]. Here, we assume all eigenvalues of M o have nonposiive real pars. Moreover, he only eigenvalue of M on he imaginary axis is zero. These resricions are necessary o ensure he exisence of a wellposed iniial value problem equivalen o 1.1. Firs, we rea he linear case, 3.1a 3.1b 3.1c z = M z + f,, 1], B a z + B b z1 = β, z C[, 1], where B a, B b R r n, r < n, are consan marices, and β R r is a consan vecor. Throughou, we assume M C 1 [, 1]. Consequenly, we can rewrie M and obain 3.2 M = M + C wih a coninuous marix C. Le X be he eigenspace of M corresponding o he eigenvalue λ = and le R be a projecion ono X. We define S := I n R, where we denoe by I n he n n ideniy marix. The necessary and sufficien condiion for z o be coninuous on [, 1] is This yields and due o Sz =. z = S + Rz = Rz, Mz = MRz =

5 Error esimae for singular BVPs 5 i follows ha 3.1c is equivalen o z kerm. These condiions are augmened by 3.1b o yield a unique soluion. We denoe by Ẽ he n r marix consising of a maximal se of linearly independen columns of R. Moreover, le Z be he fundamenal soluion marix of he iniial value problem 3.3a 3.3b Z = M Z,, 1], Z = Ẽ. Consequenly, he necessary and sufficien condiion for problem 3.1 o have a unique soluion is ha he r r marix Q, 3.4 Q := B a Ẽ + B b Z1 is nonsingular. In his case, we can represen he soluion z of 3.1 by 3.5 r z = a k Z k + z, where z is he soluion of 3.6a 3.6b z = M z + f,, 1], z =. If f C k [, 1] and M C k+1 [, 1], hen z C k+1 [, 1] holds. Now we discuss he nonlinear problem 3 3.7a 3.7b 3.7c z = M z + f, z,, 1], B a z + B b z1 = β, Mz =. In order o formulae analogous smoohness properies for z, we make he following assumpions: 1. f : D 1 R n is a nonlinear mapping, where D 1 [, 1] R n is a suiable se. 2. Equaion 3.7 has a soluion z C[, 1] C 1, 1]. Wih his soluion and a ρ > we associae he closed balls and he ube S ρ z := {x R n : z x ρ} T ρ z := {, x : [, 1], x S ρ z}. 3. f, z is coninuously differeniable wih respec o z, and f,z z is coninuous on T ρ z. 3 Again, we assume ha M has only eigenvalues wih negaive real pars or he eigenvalue.

6 6 W. Auzinger, O. Koch and E. Weinmüller 4. The soluion z is isolaed. This means ha where u = M u + Au,, 1], B a u + B b u1 =, Mu =, A := f, z, z, z has only he rivial soluion. Under hese assumpions and for f C k T ρ z, M C k+1 [, 1], he soluion z of 3.7 saisfies z C k+1 [, 1]. For furher deails and proofs see [14]. 4. Collocaion mehods. In his secion, we derive new, refined error bounds for collocaion mehods applied o 1.1, relying on earlier resuls formulaed in [15]. Moreover, we exend he convergence analysis o he nonlinear case. Le us denoe by B he Banach space of coninuous, piecewise polynomial funcions q P m of degree m, m N m is called he sage order of he mehod, equipped wih he norm [,1]. As an approximaion for he exac soluion z of 1.1, we define an elemen of B which saisfies he differenial equaion 1.1a a a finie number of poins and which is subjec o he same boundary condiions. Since we require he numerical soluion o saisfy 1.1c, we inroduce he space B 1 B, such ha Mq =, q B 1. Thus, we are seeking a funcion p = p i, J i, i =,..., N 1, in B 1 which saisfies 4.1a 4.1b p i = M p i +f, p i, i =,..., N 1, j = 1,..., m, B a p + B b p1 = β. We consider collocaion on general grids m as defined in 1, subjec o he resricion i,1 > i,, i =,..., N Earlier resuls. In [15], collocaion mehods for linear problems were sudied. For he analysis of he nonlinear case in 4.2, bounds for he collocaion soluion p B 1 need o be specified. Here, he relevan preliminaries from [15] are recapiulaed. Thus, we consider he soluion p B 1 of 4.2a 4.2b p = M p + f, i =,..., N 1, j = 1,..., m, B a p + B b p1 = β. Lemma 4.1. For µ, β {, 1} and arbirary consans c i,j, here exiss a unique p B 1 which saisfies 4.3a 4.3b p = M p =. p + Mµ β c i,j, i =,..., N 1, j = 1,..., m, i,j

7 Furhermore, 4.4 p Ji Error esimae for singular BVPs 7 cons. 1 β i+1 lnh βn µ+ C i, i =,..., N 1, where n is he dimension of he larges Jordan block of M corresponding o he eigenvalue, { x, x, x + :=, x <, and C i := max l =,..., i j = 1,..., m c l,j. 4.5a Proof. See [15, Lemma 4.4]. The following resul is a slighly modified version of [15, Theorem 4.1]. Theorem 4.2. For µ, β {, 1}, consider he problem p = M 4.5b p = δ kerm. p + Mµ β c i,j, i =,..., N 1, j = 1,..., m, i,j There exiss a unique soluion of 4.5 when h is sufficienly small, and his soluion saisfies 4.6 p Ji cons. δ + 1 β i+1 lnh βn µ + C i, i =,..., i, for a suiable i N 1. Proof. In [15, Theorem 4.1], he esimae following [15, 4.15] can be replaced by 4.7 p Ji κ i+1 p Ji + δ + 1 β i+1 lnh βn µ + C i, i =,..., i, if he resuls of [15, Lemma 4.4] are suiably applied. Subsiuion of he bound for p derived in [15, Theorem 4.1] ino he righ-hand side of 4.7 yields he resul. Noe ha he exisence of he soluion of 4.5 and he esimae 4.6 are shown only on an inerval [, b], where b is sufficienly small. Thus, we need o use classical heory for regular problems o ensure he exisence of he soluion on he whole inerval. In he sequel, we rea he underlying singular problems only on he resriced inerval, and apply classical resuls for collocaion from [3], and he error esimae analysis for regular problems from [5], o complee he proofs New error bounds. Firs, we use Theorem 4.2 o derive bounds for he soluion p B 1 of he general linear problem 4.2 and for is derivaive p. By he superposiion principle, p can be wrien in he form r 4.8 p = b k P k + p, analogous o 3.5 for he exac soluion. Here, P is he n r marix soluion of 4.9a 4.9b P = M P, i =,..., N 1, j = 1,..., m, P = Ẽ,

8 8 W. Auzinger, O. Koch and E. Weinmüller whose columns are in B 1, and p saisfies 4.1a 4.1b p = M p + f, i =,..., N 1, j = 1,..., m, p =. I was shown in [15, Theorem 4.4] ha he represenaion 4.8 is well defined. Nex, we derive convergence resuls for he quaniies appearing in he represenaion 4.8 using argumens similar o hose given in [15, Theorem 4.2]. Consider he soluions z and q of 3.1a and 4.2a, respecively, subjec o he iniial condiions z = q = δ kerm. We define an error funcion e B 1 by e = z q, i =,..., N 1, j = 1,..., m, e =. From sandard resuls for inerpolaion, see for example [13], we conclude ha if z is sufficienly smooh, whence e = z q + Oh m 4.11a 4.11b e = M e + Oh m, i =,..., N 1, j = 1,..., m, e =. Now, Theorem 4.2 yields 4.12 e Ji i+1 Oh m, i =,..., i, and consequenly 4.13 z q Ji i+1 Oh m, i =,..., i. I follows from 4.11a and 4.12 ha e = Oh m, which implies 4.14 z q [,1] = Oh m. Finally, we show ha he residual of q w.r.. 3.1a has he same asympoic qualiy. Since q C[, 1] and q has only a finie number of jump disconinuiies in [, 1], we can use he represenaions 4.15a 4.15b o conclude ha 4.16 q M q = δ + z = δ q s ds, z s ds q f = q z + M = Oh m, [, 1]. 1 q s z s ds

9 Error esimae for singular BVPs 9 This means ha he refined bounds 4.13, 4.14 and 4.16 hold for he fundamenal modes P k and he paricular soluion p in 4.8. To show ha hese bounds also hold for he soluion p of 4.2, we have o esimae he differences a k b k for k = 1,..., r. We subsiue 3.5 and 4.8 ino 3.1b and obain a sysem of linear equaions for a k b k. This sysem is nonsingular since Q from 3.4 is nonsingular and P 1 = Z1 + Oh m. This implies 4.17 b k = a k + Oh m, k = 1,..., r, see also [15, Theorem 4.5]. Consequenly, he following resul holds. Theorem 4.3. Consider he soluion p B 1 of 4.2 as an approximaion of he sufficienly smooh 4 soluion z of 3.1. Then, for a sufficienly small sepsize h and a suiable i N 1 he following bounds hold: 4.18a 4.18b 4.18c z p = ẼOhm + i+1 Oh m, J i, i =,..., i, z p [,1] = Oh m, p M p f = Ohm, [, 1]. Proof. The resul follows immediaely on noing ha P can also be wrien in a form given by 4.15a and herefore, z p = r a k Z k P k + r a k b k P k + z p = i+1 Oh m + Ẽ + O1Ohm + i+1 Oh m, J i. The bounds 4.18b and 4.18c are direc consequences of his represenaion. To prove he analogous convergence resuls for nonlinear problems, we use echniques developed in [19]. In order o show he exisence of he soluion and derive he error bounds, we rewrie he problem in an absrac Banach space seing and apply he Banach fixed poin heorem. The argumens are similar o hose given in he proof of [19, Theorem 3.6], bu we canno use his heorem direcly, because some of he assumpions made here are violaed and also, a refined error esimae is required. Therefore, we need o repea he main seps of he proof. We wrie he collocaion problem as an operaor equaion 4.19 F p =, where F : B 1 B 2 is defined by p F p= Mi,j p f, p, i =,..., N 1, j = 1,..., m, B a p + B b p1 β and B 1 and B 2 are Banach spaces, B 1 = {q P m : Mq = }, [,1], B 2 = R Nmn+r,. 4 We require ha z C [, 1], which holds if f C m [, 1] and M C [, 1].

10 1 W. Auzinger, O. Koch and E. Weinmüller For p B 1, he Fréche derivaive DF p : B 1 B 2 of F is given by DF pq = q Mi,j q D 2 f, p q, i =,..., N 1, j = 1,..., m, B a q + B b q1 where D 2 f, z is he Fréche derivaive of f wih respec o z. If D 2 f is Lipschiz, hen DF also saisfies a Lipschiz condiion wih he same consan, DF p 1 DF p 2 q = D2 f, p 1 D 2 f, p 2 q, i, j L p 1 p 2 [,1] q [,1]. For he convergence proof, we require all assumpions from 3 o hold. In paricular, his means ha an isolaed, smooh soluion z of 1.1 exiss. Using his funcion, we now consruc an auxiliary elemen p ref B 1 for he proof of he exisence of a soluion p of 4.1. We require ha p ref saisfies 4.2a 4.2b p ref = z, i =,..., N 1, j = 1,..., m, B a p ref + B b p ref 1 = β. Since p ref is a piecewise polynomial of degree m 1, i is uniquely defined by he sysem 4.2a. Moreover, 4.21 z p ref [,1] = Oh m. Represening p ref by means of 4.15a we conclude z p ref = Ẽr 1 r 2 + Oh m, r 1, r 2 R r. Subsiuion ino 4.2b yields B a + B b Ẽr 1 r 2 = Oh m. For he furher analysis, we assume ha 4.22 Q := B a + B b Ẽ is nonsingular. This implies r 1 r 2 = Oh m, and consequenly, 4.23 z p ref = ẼOhm + Oh m. Remark. Assumpion 4.22 is quie naural. If we require ha boundary value problems consising of 1.1a posed on inervals, b], < b 1, and boundary condiions Mz = and B a z+b b zb = β, have unique, coninuous soluions, hen 4.22 follows. Moreover, we can inerpre 4.2 as he regular collocaion problem associaed wih he boundary value problem y = z,, 1], B a y + B b y1 = β, My =.

11 Error esimae for singular BVPs 11 Obviously, y = z is a soluion of his reconsrucion problem, and if we require he soluion o be unique, hen 4.22 mus hold. Noe ha 4.22 always holds for problems wih separaed boundary condiions. We now use 4.23 o derive he following relaion: p F p ref = ref Mi,j p ref f, p ref, i, j B a p ref + B b p ref 1 β = p ref z Mi,j p ref z f, p ref + f, z, i, j M = ẼOhm i,j + Oh m + Oh m, i, j 4.24 = Oh m. Finally, we give an esimae for DF 1 p ref. Noe ha q := DF 1 γi,j, i, j p ref β is he soluion of he linear collocaion problem 4.25a 4.25b q = M q + D 2 f, p ref q + γ i,j, i, j, B a q + B b q1 = β. Since for sufficienly small h, p ref is in T ρ z, his problem is well defined. Finally, from Theorem 4.2, we have 4.26 q Ji cons. β + i+1 γ i, where γ i = max l =,..., i j = 1,..., m γ l,j. Wih hese preliminary resuls we can prove he nex heorem. Theorem 4.4. Le z be an isolaed, sufficienly smooh soluion of 1.1. For sufficienly small h and ρ >, he nonlinear collocaion scheme 4.1 has a unique soluion p in he ube T ρ z around z. Moreover, he esimaes 4.18 hold. Proof. We proceed similarly as in in he proof of [19, Theorem 3.6]. Define a mapping G : B 1 B 1, 4.27 Gq := q DF 1 p ref F q. Obviously, F p = is equivalen o he fixed poin equaion Gp = p. We use he Banach fixed poin heorem o show ha his equaion has a unique soluion in a suiably chosen closed ball K := Kp ref, ρ := {q B 1 : q p ref [,1] ρ }.

12 12 W. Auzinger, O. Koch and E. Weinmüller To show ha G is a conracion, we wrie q := Gp 1 Gp 2 = DF 1 p ref DF p ref DF p 1, p 2 p 1 p 2, for p 1, p 2 K, where DF p 1, p 2 := 1 DF τp τp 2 dτ. Consequenly, q is he soluion of he scheme 4.25, where γi,j, i, j 1 = β DF p ref DF τp τp 2 dτp 1 p 2 Lρ p 1 p 2 [,1], due o he Lipschiz condiion DF saisfies. Thus, i follows from 4.26 ha G is a conracion wih consan L < 1 if ρ is sufficienly small. To show ha G maps K ino iself, we esimae for q K, p ref Gq [,1] p ref Gp ref [,1] + Gp ref Gq [,1], where p ref Gp ref = DF 1 p ref F p ref is he soluion of 4.25 wih γ i,j = Oh m and β =, cf Thus, 4.28 p ref Gq [,1] Oh m + Lρ ρ provided ha h is sufficienly small. The Banach fixed poin heorem now implies ha a soluion p B 1 of 4.1 exiss. We now prove he convergence resuls From p ref p Ji = p ref Gp Ji p ref Gp ref Ji + Gp ref Gp Ji i+1 Oh m + L p ref p Ji we have p ref p Ji i+1 Oh m, which ogeher wih 4.23 yields 4.29 z p = z p ref + p ref p = ẼOhm + Oh m + i+1 Oh m, J i. Consequenly, 4.18a follows. Nex, we choose a piecewise polynomial funcion e B 1 saisfying e = z p. Therefore, e = z p +Oh m. Moreover, 4.29 implies e = z p = M z p + f, z f, p = Oh m, i =,..., N 1, j = 1,..., m. Thus e = Oh m = z p + Oh m and 4.18b follows. Finally, 4.18c is shown by using 4.18b, 4.29, and he Lipschiz condiion for f in p M p f, p = p z + M p z f, p + f, z = Oh m, [, 1].

13 Error esimae for singular BVPs 13 Under he previous assumpions we can also show ha Newon s mehod converges quadraically when i is applied o compue he collocaion soluion p, provided ha he saring approximaion p [] is chosen sufficienly close o p ref. Theorem 4.5. Le all assumpions of Theorem 4.4 hold. Newon s mehod converges quadraically o he soluion p Kp ref, ρ of 4.1 if he saring ierae p [] is chosen in a ball Kp ref, ρ 1, ρ 1 ρ, provided ha ρ, ρ 1 and he sepsize h are sufficienly small. Proof. The proof is analogous o ha of [19, Theorem 3.7], aking ino accoun he modificaions made earlier in he proof of Theorem 4.4. We wrie 5 DF q = DF p ref I + DF 1 p ref DF q DF p ref for q Kp ref, ρ and use he bound for DF 1 p ref, he Lipschiz condiion for DF and he Banach lemma o show ha DF 1 q is bounded if ρ is sufficienly small, 4.3 where K ρ DF 1 q [,1] K ρ, is a consan depending on ρ. Furhermore, le p [] in Kp ref, ρ 1, hen p [1] p [] = DF 1 p [] F p [] = DF 1 p [] F p ref + DF 1 p [] DF p ref, p [] p ref p [] holds, wih DF p 1, p 2 specified in Theorem 4.4. Using he Lipschiz condiion for DF we obain DF 1 p [] DF p ref, p [] p ref p [] [,1] Finally, we conclude = p ref p [] + DF 1 p [] DF p ref, p [] DF p [] p ref p [] [,1] 1 + Lρ 1 2 K ρ ρ 1 =: Cρ 1. p [1] p [] [,1] K ρ Oh m + Cρ 1. Consider a ball Kp [], r. For a sufficienly small ρ 1 i is possible o choose he radius r ρ in such a way ha Kp [], r Kp ref, ρ. Moreover, le DF 1 p [] DF q 1 DF q 2 [,1] ω q 1 q 2 [,1], q 1, q 2 Kp [], r, and choose r such ha he condiion ωr = 2K ρ Lr 1/2 holds. Consequenly, p [1] p [] [,1] K ρ Oh m + Cρ 1 1 2ωrr, provided ha ρ 1 and h are sufficienly small, cf. [18, 6c]. This implies ha he assumpions of [18, Theorem 1] are saisfied and he quadraic convergence of Newon s mehod in Kp [], r follows. 5 I is he idenical mapping on he space of operaors mapping B 1 B 2, ha is, I : DF p ref DF p ref.

14 14 W. Auzinger, O. Koch and E. Weinmüller 5. The error esimae. In his secion, we analyze an error esimae based on he defec correcion principle for he numerical soluion p on he collocaion grid m. For reasons explained below, we consider equidisan collocaion, cf. 2.2, where we choose m even. However, he argumen is valid on any collocaion grid wih i,m < i,, i =,..., N 1. Our esimae was inroduced in [5], where i was shown o be asympoically correc for regular problems. The numerical soluion p obained by collocaion is used o define a neighboring problem o 1.1. The original and neighboring problems are solved by he backward Euler mehod a he poins, i =,..., N 1, j = 1,...,. This yields he grid vecors 6 ξ i,j and π i,j as he soluions of he following schemes, subjec o boundary condiions 1.1b and 1.1c, 5.1a 5.1b ξ i,j ξ i,j 1 π i,j π i,j 1 = M ξ i,j + f, ξ i,j, and = M π i,j + f, π i,j + d i,j, where d i,j is a defec erm defined by 5.2 d i,j := p p 1 Mi,k α j,k p i,k + f i,k, p i,k. i,k Here, he coefficiens α j,k are chosen in such a way ha he quadraure rules given by 1 i,j 1 ϕτ dτ α j,k ϕ i,k have precision m + 1. In he nex heorem, we show ha he difference ξ m π m is an asympoically correc esimae for he global error of he collocaion soluion, R mz R mp. Theorem 5.1. Assume ha he singular boundary value problem 1.1 has an isolaed sufficienly smooh 7 soluion z. Then, provided ha h is sufficienly small, he following esimae holds: 5.3 R mz R mp ξ m π m m = O lnh n 1 h, wih n specified in Lemma 4.1. Proof. The general idea of he proof is similar o ha for regular problems. In paricular, he smooh nonlinear par in he righ-hand of 1.1a can be reaed analogously. Therefore, we give a general ouline of he proof here, and discuss hose aspecs which are crucial for he singular case. For furher echnical deails we refer he reader o [5]. Le 5.4 ε m := ξ m R mz, ε m := π m R mp, 6 Here and in Theorem 5.1, we assume hroughou i =,..., N 1, j = 1,..., m In fac, we require z C m+2 [, 1].

15 hen he quaniy o be esimaed is Error esimae for singular BVPs ε m := R mp R mz π m ξ m = ε m ε m. Here, ε m, he error of he backward Euler scheme applied o he original problem, saisfies 5.6 ε i,j ε i,j 1 = M ξ i,j + f, ξ i,j z z 1 = M ξ i,j + f, ξ i,j Mi,k α j,k z i,k + f i,k, z i,k + Oh, i,k since he α j,k define quadraure rules of precision Oh. Moreover, ε m saisfies 5.7 ε i,j ε i,j 1 = M π i,j + f, π i,j + d i,j p p 1 = M π i,j + f, π i,j Mi,k α j,k p i,k + f i,k, p i,k. i,k Boh 5.6 and 5.7 hold for i =,..., N 1, j = 1,..., m + 1, and ε m ε m saisfy homogeneous boundary condiions. In order o proceed, we use Taylor s Theorem o conclude ha as well as f, ξ i,j f, z = 5.8 and analogously, 1 =: A ε i,j, D 2 f, z + τξ i,j z dτ ε i,j 5.9 f, π i,j f, p =: Ā ε i,j. Nex, we noe ha due o 4.18c, 5.1 d i,j = p p 1 = 1 i,j p τ dτ α j,k p i,k + 1 Mi,k α j,k p i,k + f i,k, p i,k i,k + α j, p i, M i, p i, f i,, p i, i, = Oh m. From his we conclude ha ξ i,j = π i,j + Oh m using he following argumen: The backward Euler schemes 5.1a and 5.1b can be wrien as collocaion mehods wih m = 1 and he collocaing condiion posed a he righ endpoin of

16 16 W. Auzinger, O. Koch and E. Weinmüller each inerval [ 1, ]. Thus, we discuss he collocaion soluions ξ, π of wo singular boundary value problems whose righ-hand sides differ by a erm Oh m. This erm can be assumed o be smooh, if a suiable inerpolan g of d i,j is used. More precisely, ξ is an approximaion o he soluion z of 1.1, and π is an approximaion o he soluion of z def = M z def + f, z def + g,, 1], B a z def + B b z def 1 = β, Mz def =. For 1.1, we make he assumpion ha he analyical problem is sable in he sense ha z z def [,1] cons. g [,1] = Oh m holds. For resuls on his ype of sabiliy analysis, see [28]. According o 4.2, we consruc he reference funcions p ref and p def relaed o z and z def, respecively, and have ξ π [,1] ξ p ref [,1] + p ref z [,1] + z z def [,1] + + z def p def [,1] + p def π [,1] = Oh m. Thus, ξ m π m m wrie see [5] = Oh m. Since ε i,j = Oh and ε i,j = Oh, we may finally Ā ε i,j = A ε i,j + Ā A ε i,j = A ε i,j + Oh. Now we use 5.8, 5.9 o rewrie 5.6, 5.7 and obain 5.11 and 5.12 ε i,j ε i,j 1 ε i,j ε i,j 1 = M = M ε i,j + A ε i,j + M z + f, z α j,k Mi,k i,k z i,k + f i,k, z i,k + Oh, ε i,j + A ε i,j + M p + f, p α j,k Mi,k i,k p i,k + f i,k, p i,k + Oh. Sysems 5.11 and 5.12 are a pair of parallel backward Euler schemes, wih relaed inhomogeneous erms. Le us use he shorhand noaion φ := f, p f, z. I can be shown ha for he difference in he smooh pars of he inhomogeneous erms he esimae 5.13 φ α j,k φ i,k cons. φ Ji cons. z p Ji + z p Ji Oh

17 Error esimae for singular BVPs 17 holds. To see his we use Taylor expansion of φ i,k abou and he fac ha α j,k = 1 for all j, see [5]. The esimae finally follows from Theorem 4.4. In he nex sep, we derive a represenaion for he difference in he singular erms occurring in he inhomogeneous pars of he schemes 5.11 and Wih ɛ := z p and wih σ := + τ i,k, we rewrie 5.14 M M i,k ɛ α j,k ɛ i,k i,k = M i,j ɛ = 1 d Mσ + dσ σ ɛσ α j,k i,k α j,k M 1 ɛ + dτ i,k M σ 2 ɛσ + C σɛσ + Cσɛ σ = M Oh + Oh, M σ ɛ σ dτ on noing ha 1 σ m, k = 1,..., m + 1, j = 1,..., m, τ [, 1], and using he resuls of Theorem 4.4. Alogeher, we have shown ha he error of he error esimae ε m, cf. 5.5, saisfies a linear Euler difference scheme 5.15a 5.15b 5.15c ε i,j ε i,j 1 = M ε i,j + A ε i,j + M Oh + Oh, i, j, B a ε, + B b ε N 1, =, M ε, =. This scheme can also be inerpreed as a collocaion scheme wih m = 1 where he only collocaion poin is he righ endpoin of every collocaion inerval. To esimae he soluion of 5.15 we use a represenaion according o 4.8 for ε m. Then we apply Theorem 4.2 o derive bounds for he quaniies occurring in 4.8, and conclude ha alogeher he esimae 5.3 holds for he soluion of Remark. Obviously, he argumens used o prove he las heorem remain valid when we consider general, non-equidisan collocaion grids m. The only necessary resricion is i, > i,m. However, if we consider superconvergen schemes, he error esimae is no longer asympoically correc, because he basic collocaion soluion has a higher convergence order in ha case. Therefore we resric ourselves o an even number of equidisan collocaion poins. This resricion is no severe, since in he case of singular problems, he highes convergence order ha can generally be expeced in he mesh poins τ i is O lnh n 1 h, see [15].

18 18 W. Auzinger, O. Koch and E. Weinmüller 6. Numerical resuls. To illusrae he heory, we consider he following nonlinear problem: z = a z z z1 3, b z + z1 =. 1 1/ ln3 Is exac soluion reads 1 z = ln 2 + 2, 2 2 T ln The compuaions were carried ou wih he subrouines from our MATLAB code sbvp cf. [4] on fixed, equidisan grids. For he purpose of deermining he empirical convergence orders he mesh adapaion sraegy was disabled. The ess were performed in IEEE double precision wih EPS In Table 6.1, we give he exac global errors err coll of he collocaion soluions for he respecive mesh widh h, and he convergence orders p coll compued from he errors for wo consecuive sepsizes. Moreover, he errors of he error esimae wih respec o he exac global errors, err es, are recorded, ogeher wih associaed empirical convergence orders p es. In accordance wih he heoreical resuls from 4 5, convergence orders Oh 4 for collocaion and Oh 5 for he error esimae are observed. This illusraes he asympoical correcness of he error esimae analyzed in his paper. Tes runs given in [4] demonsrae ha his error esimae can be used as a dependable basis for a mesh adapaion algorihm, providing an efficien, high precision numerical solver. Table 6.1 Convergence orders of collocaion and error esimae for 6.1 h err coll p coll err es p es e e e e e e e e e e e e e e REFERENCES [1] U. Ascher, J. Chrisiansen, and R. Russell, A collocaion solver for mixed order sysems of boundary values problems, Mah. Comp., , pp [2], Collocaion sofware for boundary value ODEs, ACM Transacions on Mahemaical Sofware, , pp [3] U. Ascher, R. Maheij, and R. Russell, Numerical Soluion of Boundary Value Problems for Ordinary Differenial Equaions, Prenice-Hall, Englewood Cliffs, NJ, [4] W. Auzinger, G. Kneisl, O. Koch, and E. Weinmüller, A collocaion code for boundary value problems in ordinary differenial equaions. To appear in Numer. Algorihms. Also available as ANUM Preprin Nr. 18/1 a hp:// [5] W. Auzinger, O. Koch, and E. Weinmüller, Efficien collocaion schemes for singular boundary value problems. To appear in Numer. Algorihms. Also available as ANUM Preprin Nr. 5/1 a hp://

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