A Numerical Algorithm for Ambrosetti-Prodi Type Operators

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1 A Numerical Algorithm for Ambrosetti-Prodi Tpe Operators José Teieira Cal Neto and Carlos Tomei March 9, Abstract We consider the numerical solution of the equation u f(u) = g, for the unknown u satisfing Dirichlet conditions in a bounded domain Ω. The nonlinearit f has bounded, continuous derivative. The algorithm uses the finite element method combined with a global Lapunov-Schmidt decomposition. Kewords: Semilinear elliptic equations, finite element method, Lapunov-Schmidt decomposition. MSC-class: 35B3, 35J9, 65N3. Introduction We consider the partial differential equation F (u) = u f(u) = g, u Ω =, on domains Ω R n, taken to be open, bounded, connected subsets of R n with piecewise smooth boundar Ω, assumed to be at least Lipschit at all points. There is a vast literature concerning the number of solutions for general and positive solutions for different kinds of nonlinearit f and right-hand side g (to cite a few, [], [], [3], [4], [5], [6], [7], [8], [9]). Here we assume that the nonlinearit f : R R has a bounded, continuous, derivative, a f () b. We show how a global Lapunov-Schmidt decomposition introduced b Berger and Podolak [] in their proof of the Ambrosetti-Prodi theorem ([3], [4]) gives rise to a satisfactor solution algorithm using the finite element method. The decomposition was rediscovered b Smile [], who realied its potential for numerics: our results advance along these lines. Write D for the Dirichlet Laplacian in Ω. The algorithm is especiall convenient when the number d of eigenvalues of D in the range of f is small: the infinite dimensional equation reduces to the inversion of a map from R d to itself. The subject of semilinear elliptic equations is sufficientl mature that algorithms should stand side b side with theor. The situation ma be compared to the stud of functions of one variable in a basic calculus course. Some functions, like parabolas, ma be handled without substantial computational effort, but understanding increases with graphs, which are obtained b following a standard procedure. We do not handle the difficulties and opportunities related to the finite dimensional inversion: a generic solver (as in [] and, for d =, [3]) should be replaced b an algorithm which makes use of features inherited b the original map F. Here jose.calneto@uniriotec.br, DME, UNIRIO, Rio de Janeiro, Brail tomei@mat.puc-rio.br, Departamento de Matemática, PUC-Rio, Rio de Janeiro, Brail

2 we onl deal with eamples for which d = and, and there is some craftsmanship in handling the -dimensional eample. It is in this step of the PDE solver that delicate issues like nonresonance and lack of properness come up. The results are part of the PhD thesis of the first author [4]. Complete proofs are presented elsewhere. The authors are grateful to CAPES, CNPq and Faperj for support. The basic estimate We consider the semilinear elliptic equation presented in the introduction for a nonlinearit f() : R R with bounded, continuous derivative. With these hpotheses, it is not hard to see that F (u) = u f(u) is a C map between the Sobolev spaces H (Ω) and L (Ω) = H (Ω) and between H (Ω) and H (Ω) H (Ω). We concentrate on the second scenario, which is natural for the weak formulation of the problem. Still, the geometric statements below hold in both cases. To fi notation, set F : X Y, where X = H (Ω) and Y = H (Ω). The basic estimate is given in Proposition. Its proof is a simple etension of the argument in []. Define f (R) = [a, b] (a allowed to be ) and a larger interval [ã, b] [a, b]. Label the eigenvalues of D in non-decreasing order. The inde set J associated to [ã, b] is the collection of indices of eigenvalues of D in that interval. The set J is associated to the nonlinearit f if [ã, b] = f (R). An inde set defined this wa is complete: it contains all indices labeling an eigenvalue in the interval. Denote the vertical subspaces b V X X and V Y Y the spans of the normalied eigenfunctions φ j, j J in X and Y respectivel, with orthogonal complements W X and W Y. Let P and Q be the orthogonal projections on V and W. Clearl, the dimension of the vertical subspaces equals J, the cardinalit of J. Let v+w X X be the horiontal affine subspace of vectors v + w, w W X and consider a projected restriction F v : v + W X W Y, the restriction of P Y F to v + W X. Proposition. Let J be the inde set associated to the nonlinearit f (or to an interval [ã, b] containing f (R)). Then the derivatives DF v : v + W X W Y are uniforml bounded from below. More precisel, there eists C > such that v V X w v + W X h W X, DF v (w)h Y C h X. () All such maps are invertible. A direct application of Hadamard globaliation theorem ([5]) implies that the projected restrictions are diffeomorphisms, for each v V X. 3 The underling picture The geometric implications are ver natural. The image under F of each horiontal affine subspace v + W X is a sheet, i.e., a surface which projects under P Y diffeomorphicall to the horiontal subspace W Y. In particular, ever vertical affine subspace w + V Y intercepts each sheet eactl at a single point. It is not hard to see that the intersection is transversal: tangent spaces of sheet and affine subspace form a direct sum decomposition of Y. A fiber is the inverse image of a vertical affine subspace. In a similar fashion, fibers are surfaces of dimension J which meet ever horiontal affine subspace v + W X at a single point again, the intersection is transversal. Thus, a vertical subspace parameteries diffeomorphicall each fiber, or, said differentl, each fiber has a single point of a given height.

3 Recall a ke idea in [] and []. It is clear that X and Y are respectivel foliated b fibers and vertical affine subspaces. B definition, all the solutions of F (u) = g must lie in the fiber α g = F (g + V Y ). So, in principle, one might solve the equation b first identifing α g R J and then facing the finite dimensional inversion of F : α g g + V Y. Horiontal affine subspaces are taken diffeomorphicall to sheets, but fibers are not taken diffeomorphicall to vertical affine subspaces. In a sense, the nonlinearit of the problem was reduced to a finite dimensional issue. Figure : Horiontal affine subspace, fiber; sheet, vertical affine subspace 4 Finding the fiber Recall that each horiontal affine subspace v + W X contains eactl one element of each fiber. So, to identif α g, choose v + W X and search in it for an element of α g. Said differentl, one ma think of F v : v + W X W Y as being a diffeomorphism between fibers (represented b points in v + W X ) and vertical affine subspaces (represented b points in W Y ). The situation is ideal for an application of Newton s method: local improvements are performed b lineariation of the diffeomorphism. There is one difficult, however, related to implementation issues. The functional spaces X and Y give rise to finite dimensional vector spaces generated b finite elements. We provide some detail: an ecellent reference is [6]. First of all, triangulate the domain Ω, i.e., split it into disjoint simplices in R n. In the eamples in Section 7, Ω is the uniforml triangulated rectangle [, ] [, ]. A nodal function is a continuous function that is affine linear on each simple and has value one at a given verte and ero at the remaining vertices. These functions form a nodal basis, which spans a finite dimensional subspace of H (Ω). Figure shows an eample of a triangulation of Ω and one nodal function. Inner products of nodal functions, both in H and L are often ero, a fact which simplifies the numerics associated to the weak formulation of the equation F (u) = g. The vertical subspaces V X and V Y are spanned b eigenfunctions φ j, j J and are well approimated b a few linear combinations on the nodal basis. On the other hand, obtaining a similar basis for the approimation of the orthogonal subspaces W X and W Y requires much more numerical effort and should be avoided. To circumvent this problem, etend the Jacobian of F v : v + W X W Y at a point u to an invertible operator L u : X Y which is eas to handle and appl Newton s method to L u instead. Setting L u = P Y f (u)p X, it is clear that L u has the required properties: it takes W X to W Y and V X to V Y and the restriction to v + W X equals DF v, which is invertible. Moreover, the restriction 3

4 Figure : Uniform triangulations of [, ] [, ] and a nodal function to V X coincides with. This map is no longer a differential operator, due to the integrals needed to compute the projections P. But those new terms are innocuous in the finite element formulation the sparsit of the underling matrices is preserved, together with the possibilit of standard preconditioning routines. F V X V Y g u u F 3 u u 3 F F W X W Y Figure 3: Finding the right fiber We search for a point of a horiontal affine subspace v + W X which belongs to α g, g Y.The algorithm is straightforward: see Figure 3. Choose a starting point u and consider its image F. All would be well if the projections of F and g on the horiontal subspace W Y were equal or at least ver near. When this does not happen, proceed b a continuation method to join both projections. Notice that the algorithm searches for the fiber (i.e., for a point in the fiber) b moving horiontall in the domain. A direct Newton iteration does not work necessaril: think of finding the (trivial) root of arctan() = starting sufficientl far from the origin. 5 Moving along the fiber The necessar ingredients for a simple predictor-corrector method to move along a fiber are now available. Sa u α g and we want to find another point in α g. Recall that fibers are parameteried b height v V X. Take u + v, which is probabl not in α g, as a starting point for the algorithm in Section 4 to obtain the point of α g in the same horiontal affine subspace of u + v (see Figure 4 for two such steps). We don t know much about the behavior of F restricted to a fiber: the hpotheses on the nonlinearit f are not sufficient to impl properness of F, for eample. 4

5 u F F V X u V Y u F W X W Y Figure 4: Mapping a -D fiber In particular, it is not clear that the restrictions of F to a fiber are also proper. 6 Stabilit issues Proposition in Section ensures geometric stabilit, in the sense that the global Lapunov-Schmidt decompositions preserve their properties under perturbations. This is convenient when replacing the vertical subspaces spanned b eigenfunctions b their finite elements counterparts. As for the algorithm itself, the identification of the fiber is robust, being a standard continuation method associated to a diffeomorphism between horiontal affine spaces. The numerical analsis along a fiber is a different matter, and the fundamental issue was addressed b Smile and Chun []: the showed that the finite element approimations to the restriction of the function F to (compact sets of) the fiber can be made arbitraril close to the original map in the appropriate Sobolev norm. Here one must proceed with caution: small metric perturbation ma induce variations in the number of solutions, as when changing from, R to ɛ, which is a perturbation of order ɛ for arbitrar C k norms. Still, solutions of F which are regular points are stable: the correspond to nearb solutions of sufficientl good approimations F h. A different approach might be to interpret the algorithm as a provider of good starting points for Newton s iteration or at least a continuation method. As stated in [7], computer assisted arguments require good approimations for the eventual validation of solutions. 7 Some eamples For the eamples that follow, F (u) = u f(u) = g, with Dirichlet conditions on Ω = [, ] [, ]. Here, D has simple eigenvalues and λ = 5 4 π.34, λ = π 9.74, λ 3 = 7 4 π Denote b φ X k and φy k the eigenfunctions of D normalied in X and Y. The first eample is a nonlinearit f satisfing the hpotheses of the Ambrosetti- Prodi theorem with f() = and derivative f () = α arctan() + β with ( Ran(f 3λ λ ) =, λ ) + λ >. The graph of f is shown on the left of Figure 5. Here, the inde set associated to f is J = {}: V X and V Y are spanned b φ X, φ Y. For right-hand side set 5

6 f () F u Figure 5: The derivative of f and the image of α g g() = ( )( ), shown on the left of Figure 6, which has a large negative component along the ground state. We search for an element of the fiber α g in the horiontal subspace W X, starting from u =, in the notation of Section 4. The result is the function on the right of Figure 6. Now move along α g, as in Section 5. The graph on the right of Figure Figure 6: A right-hand side and a function on its fiber 5 plots u, φ X X (the height of u α g ) versus F (u), φ Y Y (the height of F (u)). The horiontal line indicates the height of g: the solutions of the original PDE correspond to the intersections between the curve and this line. The two solutions found in this case are presented in Figure Figure 7: Ambrosetti-Prodi Solutions For the net eample, J = {} but f is a nonconve function whose derivative is 6

7 depicted on the left of Figure 8. We consider the fiber through u () = 5 φ X ()+ φ X (), i.e., α F (u ). According to Figure (right) 8, moving up the fiber ields three distinct solutions. f () F u Figure 8: Non-conve f As a concluding eample, we take a nonlinearit f for which J = {, }: here the vertical spaces are spanned b the first two eigenfunctions. The function f is of the same form as the first eample and its derivative is shown in Figure 9. We stud the fiber through the point u =, which is α, since F () =. Recall from f () Figure 9: The range of f contains λ and λ Section 4 that there is eactl one point of α for each height, i.e., given a point u V X, there is a unique point ζ(u) α in the same horiontal affine subspace as u. For a circle C centered at the origin in V X, ζ(c) α. The image F (ζ(c)) is shown in the right side of Figure : here, we must project F (ζ(u)) along directions φ Y and φ Y. Clearl, there is a double point Z in F (ζ(c)) and it is not hard to identif in C its two pre-images, U and D marked in the left of Figure. 3 U u F Z D 5 5 u F Figure : Two preimages on the circle We now obtain two additional preimages of Z in a rather naive fashion. The images under F ζ of the four half-aes of V X are drawn on the right of Figure 7

8 5 n 4 6 n 3 4 u 5 w 4 w 3 L w w n n e R e e 3 e 4 F e 4 e 3 n 4 n 3 n n w 4 w 3 w w e e Z s 5 s s 4 s 3 s s s 3 s u F Figure : Two preimages along the horiontal ais. It is clear, then, that the horiontal ais contains two preimages L and R of Z, which are easil computed. For the sake of completeness, Figure 7 displas the four solutions Figure : The four solutions References [] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math., vol. 54, no., pp. 7 76, 93. [] C. L. Dolph, Nonlinear integral equations of the Hammerstein tpe, Trans. Amer. Math. Soc., vol. 66, pp , 949. [3] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4), vol. 93, pp. 3 46, 97. 8

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