External approximation of nonlinear operator equations

Transcription

1 External approximation of nonlinear operator equations Etienne Emmric a a TU Berlin, Institut für Matematik, Straße des 17. Juni 136, Berlin January 4, 2009) Based upon an external approximation sceme for te underlying Banac space, a nonlinear operator equation is approximated by a sequence of coercive problems. Te equation is supposed to be governed by te sum of two nonlinear operators acting between a reflexive Banac space and its dual. Under suitable stability assumptions and if te underlying operators can be approximated consistently, weak convergence of a subsequence of approximate solutions is sown. Tis also proves existence of solutions to te original equation. Keywords: Nonlinear operator equation; monotone operator; perturbation; external approximation sceme; convergence AMS Subject Classification: 65J15; 65N12; 47J25; 47H05 1. Introduction Let V be a real reflexive Banac space wit its dual V. We are concerned wit te approximate solution of te following problem: For f V find u V suc tat Au + Bu = f in V. 1) Here, A : V V and B : V V are given operators. A standard situation is A being monotone and emicontinuous, B being a strongly continuous perturbation of A and A + B being coercive. Many boundary value problems for quasilinear partial differential equations arising in pysics, fluid mecanics and oter areas of application can be formulated as 1) see, e.g., te monograps [6, 8, 19] and te references terein). For teir approximate solution, often Galerkin-type or finite difference metods are employed. Te study of all tese metods can be unified by considering so-called external approximation scemes. Tis concept covers in particular finite differences, nonconforming finite elements as well as fully discrete finite element metods wit quadrature see [15, 16] for several examples). External approximation scemes ave been studied in [13] in te context of te convergence of finite difference approximations for quasilinear partial differential equations and A-proper operators see [16], [19, C. 35] for introductions into te concept of external approximation scemes). Later on, extensions of te results obtained in [13] ave been studied in [17, 18], te focus being on te equivalence of unique solvability of te original and approximate problem. External approximations ave also been studied for eigenvalue problems and te penomenon of superconvergence see [10 12]) and, more recently, for te approx- emmric@mat.tu-berlin.de

2 2 E. Emmric imation of variational problems in spaces of piecewise constant functions see [5]). Te concept of external approximation as furter been employed in [15] for studying different numerical metods for solving te Navier-Stokes problem tat is, in te stationary case, indeed of te type 1) wit A being linear and strongly positive, B being strongly continuous and A + B being coercive. Te assumptions on te approximating operators studied so far in [13, 17, 18] in particular, an inverse stability tat would follow from stronger uniform monotonicity assumptions) are, owever, different from te assumptions on wic our studies are based. Indeed, we try to apply te concept of external approximation in order to generalise te often found standard situation for 1) wit A being monotone and emicontinuous, B being strongly continuous and A + B being coercive. It is well-known tat in tis situation Brézis teorem on pseudomonotone operators provides existence of solutions to 1). So, our approac is not based upon te concept of A-properness see [9] and te references cited terein for a discussion of tis concept) but on weaker assumptions yielding ten only convergence of a subsequence in a weak sense. In particular, we do not assume well-posedness of te approximate problems but only solvability and, in general, we do not ave a continuous inverse of te approximate operator for tis case, see also te exaustive work [3]). Anoter approac for studying approximations of monotone operator equations employs projection metods as in [1, 2, 6], wic can be interpreted as internal approximation scemes. Te essential advantage of external approximation scemes, owever, is tat te function spaces approximating V need not to be subspaces of V. Tis allows muc more flexibility in te coice of te numerical metod and often simplifies te numerical analysis. We sould mention tat, besides te concept of external approximation, also te concept of discrete convergence and discrete approximation as introduced in [14], wic goes witout prolongation and restriction operators, provides a frame for considering rater different numerical metods in a unifying way. Tis concept as been applied in [7] to te study of nonlinear operator equations based upon te notion of approximation-regular operators, wic is a generalisation of A-properness. Te results in [7] also cover quasilinear elliptic problems leading to a coercive and strictly monotone operator equation of te type 1)) under perturbation of te domain or coefficients. Te main result in tis paper will be a general convergence result for te case tat tere is a stable and admissible external approximation sceme and tat A, B, f in 1) can be approximated in a consistent way. Te required assumption on te sequence of approximations of A can be seen as a discrete analogue of te property M). Moreover, our convergence result is based upon an a priori estimate for te sequence of approximate solutions wic follows from uniform coercivity. Te Galerkin metod in te standard situation as described above is sown to be a special case of our approximation sceme. Te paper is organised as follows: In Section 2, we describe te external approximation sceme, present te necessary notation, and prove some results on te consistency and solvability of te approximation. Te main result is stated and proven in Section 3. Section 4 provides an example. 2. Te external approximation sceme For a normed space X, we always denote its norm by X, its dual space by X wit standard norm X and te dual pairing by,. Let V be a given real reflexive Banac space. Let H be a countable infinite

3 External approximation of nonlinear operator equations 3 sequence of indices and let {V, p, r )} H be a sequence of real normed spaces V, prolongation operators p : V F and restriction operators r : V V. Te prolongation operators are assumed to be linear and bounded. Here, F is a suitably cosen real reflexive Banac space suc tat tere is a so-called syncronisation operator ω : V F tat is linear, bounded and injective. Te family {V, p, r )} H ten is said to be an external approximation sceme for V. In some situations, te restriction operators are only defined on a dense subset of V but can be extended on V see [15, Prop. 3.1 on p. 30], [16, Prop. 4 on p. 28]). Definition 2.1: An external approximation sceme {V, p, r )} H for V is said to be stable iff tere is a constant c > 0 suc tat for all H It is said to be admissible iff it fulfills i) te compatibility condition: p v F c v V v V. p r v ωv in F H ) v V, ii) te syncronisation condition: for any subsequence H H of indices and {v } H {V }, g F wit p v g in F H ) tere is an element v V suc tat ωv = g. Note tat te use of te foregoing notions is not consistent in te literature. We now consider te sequence of approximate problems: For f V find u V suc tat A u + B u = f in V. 2) Here, {A } H, {B } H and {f } H are sequences of operators and functionals approximating A, B and f, respectively. Definition 2.2: Let {V, p, r )} H be a stable and admissible external approximation sceme for V. A sequence {A, B, f )} H of operators A : V V, B : V V and functionals f V is said to be a consistent approximation of A, B, f) iff for any subsequence H H and any {v } H {V }, v V wit p v ωv in F H ) tere olds i) if tere is an element g F suc tat ten Av = g in V ; A v, r w g, w w V, lim sup A v, v g, v ii) B v, r w Bv, w w V, lim inf B v, v Bv, v ; iii) f, r w f, w w V, lim sup f, v f, v. We remark tat condition i) in te foregoing definition is a discrete counterpart of te property M) see [8, p. 173]). A standard example is given by Proposition 2.3: Let {V } H be a Galerkin sceme for V, p : V V be te identity and r : V V suc tat r v is a best approximation of v V in V. Ten {V, p, r )} H is a stable and admissible external approximation sceme wit F = V. Let A = p Ap wit p : V V denoting te dual operator of p ), B = p Bp and f = p f. If A : V V is monotone and

4 4 E. Emmric emicontinuous and B : V V is strongly continuous ten A, B, f ) H consistent approximation of A, B, f). Proof : Stability and admissibility of te external approximation sceme built by a Galerkin sceme is evident. For proving consistency, let H H and {v } H {V }, v V wit p v ωv = v in F = V H ) be given. In wat follows, all convergence is meant for H. i) Wit A = p Ap, we find from te definition of te dual operator A v, w = Ap v, p w w V. Let w V be arbitrary. Te monotonicity of A : V V ten provides is a A v, v = Ap v, p v Ap v, p v Ap v Aw, p v w = Aw, p v w + Ap v, p r w + Ap v, w p r w. 3) Te first term on te rigt-and side of 3) converges towards Aw, v w since p v v in V. Te second term Ap v, p r w = A v, r w converges towards g, w by te assumption in Definition 2.2 i). For te tird term, we observe tat {Ap v } H is bounded in V, wic follows see, e.g., [6, Folg. 1.2 on p. 65]) from te boundedness of { Ap v, p v } H by te assumption in Definition 2.2 i)), te boundedness of {p v } H in V te sequence is weakly convergent) and te monotonicity of A : V V. Te tird term tus vanises in te limit since p r w w in V compatibility). In te limit, we finally obtain from 3) togeter wit te assumption in Definition 2.2 i) g, v lim sup A v, v lim Aw, p v w + Ap v, p r w + Ap v, w p r w = Aw, v w + g, w. ) 4) Taking w = v ± sz for arbitrary z V and s 0, 1] yields Av + sz), z g, z and Av sz), z g, z, and wit s 0+, te emicontinuity of A : V V sows Av, z = g, z and ence Av = g. ii) Wit B = p Bp, we find for all w V B v, r w = Bp v, p r w. Since B : V V is strongly continuous and p v v in V, it follows Bp v Bv in V. Because of p r w w in V, we come up wit Moreover, we ave B v, r w Bv, w. B v, v = Bp v, p v Bv, v.

5 External approximation of nonlinear operator equations 5 iii) Wit respect to te approximation of f V, we observe for all w V f, r w = f, p r w f, w as well as remember p v v in V ) f, v = f, p v f, v. Note tat a best approximation r v of v V in V always exists if dim V < but r migt be nonlinear. Instead of te best approximation, one may also take a suitable projection. In [16, p. 28], te restriction operator on V is constructed from its definition on te dense subset H V. It arises te question, from wic assumptions one can derive condition i) in Definition 2.2. An answer is given by Proposition 2.4: Let {V, p, r )} H be a stable and admissible external approximation sceme for V and let A : V V be emicontinuous. Assume tat all A : V V H ) are monotone and tat for any subsequence H H and any {v } H {V }, v V wit p v ωv in F H ) lim sup A r w, v Aw, v w V. 5) Ten condition i) in Definition 2.2 is fulfilled. Proof : Te monotonicity of A H ) yields for arbitrary w V A v, v A v, v A v A r w, v r w = A r w, v r w + A v, r w. In te limit, we tus obtain by te assumption in Definition 2.2 i) and wit 5) g, v lim sup A v, v lim sup A r w, v r w + lim A v, r w Aw, v w + g, w. Here, we ave employed tat p v ωv, p r w ωw in F H ). Te emicontinuity of A implies Av = g in V as in 4). We will also make use of te following notion. Definition 2.5: Let {V } H be a sequence of normed spaces. A sequence {T } H of operators T : V V is said to be coercive uniformly in iff tere is a function γ : R + 0 R wit γz) as z suc tat for all H T v, v γ v V ) v V v V. We end tis section by presenting a criterion for te existence of solutions to te approximate problem.

6 6 E. Emmric Lemma 2.6: Let Φ : R N R N be continuous. If tere is R > 0 suc tat Φv) v 0 for all v R N wit v R N = R ten tere exists u R N wit u R N R and Φu) = 0. Proof : Te proof follows by contradiction from Brouwer s fixed point teorem see, e.g., [6, Lemma 2.1 on p. 74]). Teorem 2.7 : Let V be a normed space wit dim V = N < and let A, B : V V be continuous operators suc tat A + B is coercive. For any f V, equation 2) ten possesses a solution. Proof : Let {e i } N be a basis in V. Ten tere is a bijective mapping between V and R N given by te representation v = N v i e i V, v = [v 1,..., v N ] R N. On R N, we define te norm v R N := v V and te mapping Φv) = [Φ 1,..., Φ N ] wit Φ i v) := A v + B v f, e i i = 1,..., N). Obviously, Φ : R N R N is continuous if A, B : V V are continuous. Because of te coercivity of A + B, we find wit a function γ : R + 0 R wit γz) as z ) Φv) v = A v + B v f, v γ v V ) v V f V v V 0 if v V = v R N is sufficiently large. Lemma 2.6 now yields te existence of u R N and tus of u V suc tat Φu) = 0. But ten u solves 2). In applications, te continuity of te approximate operators in a finite dimensional space often follows already from te emicontinuity of te operators A, B. 3. Convergence Te main result can be formulated as follows. Teorem 3.1 : Suppose tere is a consistent approximation of A, B, f). Assume furter tat 2) possesses a solution u V for any H, tat te operators A + B : V V H ) are coercive uniformly in and tat te sequence { f V } H is bounded. Ten tere is a subsequence H H and an element u V suc tat p u ωu in F H ) ; te limit u satisfies 1). Proof : Wit te coercivity assumption, we immediately find γ u V ) u V A u + B u, u = f, u f V u V.

7 External approximation of nonlinear operator equations 7 Since γz) z as z and since { f V } H is bounded, tis sows also te boundedness of { u V } H. Because of te stability of te external approximation sceme, ten also te sequence {p u } H F is bounded. In view of te reflexivity of F, tere is a subsequence H H suc tat {p u } H is weakly convergent in F see, e.g., [4, Tm. III.27]). Togeter wit te syncronisation condition in Definition 2.1 ii), tere is an element u V suc tat p u ωu in F H ). Wit Definition 2.2 ii), iii), we now find for all w V f B u, r w f Bu, w, lim sup f B u, u f Bu, u. Wit 2), it follows for all w V as well as A u, r w = f B u, r w f Bu, w lim sup A u, u = lim sup f B u, u f Bu, u. Definition 2.2 i) now provides Au = f Bu in V. 4. Example In order to keep te presentation sort, we only consider a somewat simple example: a linear finite element metod wit quadrature for a one-dimensional quasilinear Diriclet problem witout perturbation suc tat B = B 0. Let ψ = ψx, t) : [0, 1] R + 0 R be a given continuous function. We suppose tat t ψx, t)t is monotonically increasing for all x [0, 1] and tat tere is a number p 1, ) and constants µ, c > 0 suc tat ψx, t)t 2 µt p, ψx, t) c1 + t p 2 ) x, t) [0, 1] R + 0. Tis setting covers, e.g., te one-dimensional p-laplacian. For a given rigt-and side f L p 0, 1) 1/p + 1/p = 1), ten consider te problem ψx, u x) )u x)) = fx) x 0, 1)), u0) = u1) = 0. 6) Te weak formulation of 6) leads to te operator equation 1) wit te standard Sobolev space V = W 1,p 1 1/p 0 0, 1) wit norm v V := 0 v x) dx) p and an operator A : V V, defined for v, w V via Av, w := 1 0 ψx, v x) )v x)w x)dx.

8 8 E. Emmric Te growt condition for ψ ensures tat A maps V into V. Te emicontinuity of A is a direct consequence of te continuity of ψ. Te rigt-and side in 1) is te functional v 1 0 fx)vx)dx. We partition [0, 1] equidistantly into M N subintervals [x i, x i+1 ] x i = i/m, i = 0,..., M) of lengt = 1/M and employ linear finite elements. Wen also applying a simple rectangular rule taking te rigt value) for te numerical evaluation of te appearing integrals, we end up wit a fully discrete approximation of 6) tat can be written in te form 2). Tis finite element metod wit quadrature is equivalent to a finite difference sceme. We take V as te space of grid functions v = [v,0,..., v,m ] T R M+1, v,0 = v,m = 0, and endow it wit te norm ) 1/p v V := Di v p, were Di v := v i v i 1 )/. Te dual V can be identified wit te M 1)- dimensional space of grid functions g = [g,1,..., g,m 1 ] T R M 1 suc tat g i = D i + w := w,i+1 w,i )/ i = 1,..., M 1) for some w V, te dual pairing is given by M 1 M 1 g, v = g,i v,i = w,i Di v. Te prolongation p v of v V is te piecewise linear interpolation of te points x 0, v,0 ),..., x M, v,m ). Te restriction is defined via r v) i := vx i ) i = 0,..., M) wic is well-defined since W 1,p 0 0, 1) is continuously embedded in C [0, 1]). It is easy to sow tat te sequence {V, p, r )} M=1/ N builds a stable and admissible external approximation sceme for V wit F = V. In particular, we find for all v V p v V = v V, and te compatibility condition, i.e. p r v v in V, is easily sown by density. If we would take a simpler prolongation suc as te piecewise constant interpolation of te values of v as well as of teir divided differences ten we would come up wit ωv = v, v ) F = L p 0, 1) L p 0, 1). A simple calculation sows for v, w V A v, w = ψx i, Di v )Di v Di w, and, loosely written, we ave A v ) i = D i + ψxi, Di v )Di v ) for i = 1,..., M 1. Te assumptions on ψ allow to prove in particular tat A maps V into V and is monotone. In view of Teorem 2.7, te discrete problem is solvable: Te continuity of A : V V is a direct consequence of te continuity of ψ. Te coercivity uniform

9 External approximation of nonlinear operator equations 9 in ) follows from A v, v = ψx i, Di v )Di v ) 2 µ Di v p = µ v p V, v V. We now prove condition 5). Let p v v in V for an arbitrary null sequence of mes sizes wit 1/ = M N) and let w V be arbitrary. We set Di w := wx i ) wx i 1 ))/ i = 1,..., M) remember w V = W 1,p 0 0, 1) C [0, 1])). A straigtforward calculation sows A r w, v Aw, v = xi x i 1 ) ψx i, Di w )D i w ψx, w x) )w x) dx Di v ψx, w x) )w x) p v ) x) v x) ) dx =: a 1, + a 2,. 7) Denoting by ψ te w.r.t. te first argument piecewise constant approximation of ψ suc tat ψ x, t) = ψx i, t) for x x i 1, x i ] i = 1,..., M) and t R + 0 and upon noting tat p r w) x) = Di w for x x i 1, x i ) i = 1,..., M), we find for te first term a 1, by applying Hölder s inequality wit 1/p + 1/p = 1 a 1, M 1 xi x i 1 xi x i 1 ) ψx i, Di w )D i w ψx, w x) )w p) 1/p x) dx v V ψx i, Di w )D i w ψx, w x) )w x) p dx ) 1/p 1 = ψ x, p r w) x) )p r w) x) ψx, w x) )w x) p dx 0 v V ) 1/p v V. Since te sequence {p v } is weakly convergent in V it is also bounded in V suc tat v V = p v V c for some c > 0. It remains to analyse te integral on te rigt-and side of te foregoing estimate. Tis integral, owever, converges towards zero, wic follows from te continuity of ψ, te continuity of te Nemyzkii operator corresponding to ψ as a mapping from L p 0, 1) into L p 0, 1) see [19, Prop on p. 561] and remember te growt condition for ψ) as well as p r w w in V = W 1,p 0 0, 1), i.e. p r w) w in L p 0, 1). For te second term a 2, in 7), we immediately ave in view of p v v in V ) a 2, = Aw, p v v 0. Wit respect to te rigt-and side, we ave to be somewat careful as f L p 0, 1) does not allow to take point values. Instead, we may take f,i = 1 xi+1 x i fx)dx, i = 1,..., M 1.

10 10 REFERENCES One can easily prove tat f is in V. Moreover, for arbitrary v V, we ave f, v = M 1 xi+1 and tus wit 1/p + 1/p = 1) x i fx)dx v,i = 1 x i fx)dx D i v f, v f V = sup v V \{0} v V 1 x i fx)dx p ) 1/p f L p 0,1), wic sows te boundedness of te sequence { f V } as required in Teorem 3.1. It remains to prove condition iii) in Definition 2.2. Let p v v in V, wic implies tat te sequence {p v } is bounded in V. Since and tus f, v f, p v = f, v f, p v we obtain 1 0 = M 1 M 1 xi x i 1 fx)v,i p v )x))dx xi x i 1 fx)p v ) x)x i x)dx fx)p v ) x) dx f L p 0,1) p v V 0, f, v f, v = f, v f, p v + f, p v v 0. After all, Proposition 2.4 and Teorem 3.1 can be applied. Tis sows te existence of a weak solution u V = W 1,p 0 0, 1) to 6) and a subsequence, denoted by, suc tat te piecewise linear interpolations of te discrete solutions u converge weakly in V towards u. References [1] P.M. Anselone and J.-G. Lei, Te approximate solution of monotone nonlinear operator equations, Rocky Mt. J. Mat ) 4, [2] P.M. Anselone and J.-G. Lei, Nonlinear operator approximation teory based on demiregular convergence, Acta Mat. Sci ), [3] J.-P. Aubin, Approximation des espaces de distributions et des opérateurs différentiels. Mémoires de la S.M.F ), [4] H. Brézis, Analyse fonctionnelle: Téorie et applications, Dunod, Paris, [5] C. Davini and R. Paroni, External approximation of first order variational problems via W 1,p estimates, ESAIM Control Optim. Calc. Var ) 4, [6] H. Gajewski, K. Gröger and K. Zacarias, Nictlineare Operatorgleicungen und Operatordifferentialgleicungen, Akademie-Verlag, Berlin, [7] R.D. Grigorieff, Über diskrete Approximationen nictlinearer Gleicungen 1. Art, Mat. Nacr ) [8] J.-L. Lions, Quelques métodes de résolution des problèmes aux limites non linéaires, Dunod, Gautier-Villars, Paris, 1969.

11 REFERENCES 11 [9] W.V. Petrysin, Approximation-solvability of nonlinear functional and differential equations, M. Dekker, New York, [10] T. Regińska, External approximation of eigenvalue problems in Banac spaces, RAIRO Anal. Numér ), [11] T. Regińska, Superconvergence of external approximation of eigenvalues of ordinary differential operators, IMA J. Numer. Anal ) 3, [12] T. Regińska, Superconvergence of external approximation for two-point boundary problems, Apl. Mat ) 1, [13] R. Scumann and E. Zeidler, Te finite difference metod for quasilinear elliptic equations of order 2m, Numer. Funct. Anal. Optimization ), [14] F. Stummel, Diskrete Konvergenz linearer Operatoren I, Mat. Ann ) [15] R. Temam, Navier-Stokes equations: Teory and numerical analysis, AMS Celsea Publ., American Matematical Society, Providence, Rode Island, [16] R. Temam, Numerical analysis, D. Reidel Publ. Company, Dordrect, [17] R.U. Verma, On te external approximation-solvability of nonlinear equations, Panam. Mat. J ) 3, [18] R.U. Verma, Stable discretization metods wit external approximation scemes, J. Appl. Mat. Stocastic Anal ) 4, [19] E. Zeidler, Nonlinear functional analysis and its applications, II/A: Linear monotone operators, II/B: Nonlinear monotone operators, Springer, New York, 1990.