The purpose of this paper is to present a wavelet–Galerkin scheme for solving
nonlinear elliptic partial differential equations. We select as trial spaces a nested
sequence of spaces from an appropriate biorthogonal multiscale analysis. This gives
rise to a nonlinear discretized system. To overcome the problems of nonlinearity, we
apply the machinery of interpolating wavelets to obtain knot oriented quadrature
rules. Finally, Newton’s method is applied to approximate the solution in the given
ansatz space. The results of some numerical experiments with different biorthogonal
systems, confirming the applicability of our scheme, are presented.Instituto de Cooperação Científica e Tecnológica Internacional - Acções Integradas Luso-Alemãs (DAAD/ICCTI) - Projecto DAAD/ICCTI nº 01141

Cerejeiras, P.

Dahlke, S.

Kähler, U.

Lindemann, M.

Soares, M. J.

Teschke, G.

Zhariy, M.

Universidade do Minho: RepositoriUM

A Wavelet Based Numerical Method for NonlinearPartial Differential EquationsS. Dahlke§, M. Lindemann†, G. Teschke†, M. Zhariy†,M. J. Soares‡, P. Cerejeiras¶ and U. Ka¨hler¶AbstractThe purpose of this paper is to present a wavelet–Galerkin scheme for solvingnonlinear elliptic partial differential equations. We select as trial spaces a nestedsequence of spaces from an appropriate biorthogonal multiscale analysis. This givesrise to a nonlinear discretized system. To overcome the problems of nonlinearity, weapply the machinery of interpolating wavelets to obtain knot oriented quadraturerules. Finally, Newton’s method is applied to approximate the solution in the givenansatz space. The results of some numerical experiments with different biorthogonalsystems, confirming the applicability of our scheme, are presented.1 IntroductionIn recent years, the interest on using wavelets as a tool for the numerical treatmentof partial differential equations has been growing considerably. So far, the most far-reaching results were obtained for linear, boundedly invertible operator equations; see, e.g.[4, 3, 5, 6]. (This list is clearly not complete.) Once these problems are well–understoodand almost completely solved, the next challenging task is the numerical treatment ofnonlinear problems. This paper can be viewed as one small step in this direction. Weshall be concerned with the numerical treatment of nonlinear partial differential equationsof the typeF (u) := Lu+G(u) = 0, (1)on some bounded domain Ω ⊂ Rd, where L is an elliptic linear differential operator ofsecond order and G is a nonlinear operator.As a typical example we will focus here on the model problemLu = −∆u+ u and G(u) = u3 − f, (2)for a given function f ∈ L2(Ω).§ Universita¨t Marburg, Fachbereich Mathematik, Germany† Universita¨t Bremen, Fachbereich 3, Germany‡ Universidade do Minho, Departamento de Matema´tica, Campus de Gualtar, 4710-154 Braga Portugal¶ Universidade de Aveiro, Departamento de Matema´tica, Campus de Santiago, 3810-193, Aveiro, Portugal1To simplify the description of the method and to avoid possible boundary effects, wewill concentrate on the 1D case and on periodic problems, i.e. we will take Ω to be theunidimensional torus Ω = T := R/Z. This might sound artificial at first sight. However,one of our aims is to apply our scheme to problems in image processing where the use ofperiodic boundary conditions is quite standard. Also, the generalization to problems inhigher dimensions, via tensor product techniques, is straightforward.By modifying in an obvious manner the analysis contained in [1], one can show that theabove problem has a unique solution, which we now aim to approximate numerically. Inorder to do this, we first consider the associated weak formulationa(u, v) +∫Tg(u)v dx−∫Tfvdx = 0 , for all v ∈ H1 , (3)where a(., .) is the bilinear form induced by L and the prescribed boundary conditionsa(u, v) =∫T∇u · ∇v dx+∫Tuv dx. (4)Here, H1 := H1(T ) denotes the first order Sobolev space on T equipped with the usualnorm ‖ · ‖1 and g(u) = u3. We can then use a classical Galerkin approach, i.e., consider anested sequence of finite–dimensional approximation spaces {Vj}j≥0, Vj ⊂ H1, and lookfor an approximation uj ∈ Vj to the solution u of the problem by solvinga(uj, vj) +∫Tg(uj)vj dx−∫Tfvj dx = 0 , for all vj ∈ Vj. (5)2 Multiresolution AnalysisOur goal is to derive efficient Galerkin methods for the approximate solution of (3). Here,in contrast to conventional finite element discretizations, we will work with trial spacesthat, not only exhibit the usual approximation and good localization properties, but inaddition lead to expansions of any element in the underlying Hilbert spaces in terms ofmultiscale or wavelet bases with certain stability properties.We now briefly review some essential features of wavelet approximation that will be impor-tant in the sequel; for more details the reader is referred to, e.g. [2]. We start by recallingthe concept of multiresolution. A multiresolution analysis (MRA) (Vj, φ) of L2(R) is asequence of closed subspaces of L2(R) and an associated function φ, called the generatoror scaling function, satisfying:Vj ⊂ Vj+1,⋃j∈ZVj = L2(R),⋂j∈ZVj = {0}, (6)f ∈ Vj ⇔ f(2·) ∈ Vj+1, (7){φ(· − k) : k ∈ Z} is a Riesz basis of V0. (8)It then follows that, for each j, the set of functions {φj,k := 2j/2φ(2j. − k) : k ∈ Z} is aRiesz basis for the space Vj (the so-called nodal basis). Wavelets are associated with detailspaces, i.e. with complementary spaces Wj satisfying Vj+1 = Vj ⊕Wj, where ⊕ denotes2a direct sum. The properties of a multiresolution analysis imply that⊕j∈ZWj = L2(R).Hence, if we can find a function ψ whose integer translates form a Riesz basis of W0, thenthe collection {ψj,k := 2j/2ψ(2j · −k) : j, k ∈ Z} will be a good candidate for a Rieszbasis for the space L2(R) (a so-called wavelet basis). Of course, there is a continuumof possible choices of such complement spaces. Orthogonal decompositions would leadto orthogonal wavelets. However, orthogonality often interferes with locality and theactual computation of orthonormal bases might be too expensive. Moreover, in certainapplications orthogonal decompositions are actually not best possible. These limitationsmotivated the search for biorthogonal wavelet bases, i.e., for bases {ψj,k} and {ψ˜j,k} whichsatisfy 〈ψj,k, ψ˜j′,k′〉= δj,j′δk,k′ , for all j, j′, k, k′ ∈ Z, (9)with 〈·, ·〉 denoting the usual inner product in L2(R). These biorthogonal wavelet basescan be associated with two multiresolution analyses (Vj, φ) and (V˜j, φ˜) which are connectedin the following manner:L2(R) = V0 ⊕ (V˜0)⊥, (10)where X⊥ denotes the orthogonal complement of X in L2(R). Two such multiresolutionanalyses are said to form a dual or biorthogonal multiresolution analysis of L2(R). Inthis case, we can define the detail spaces Wj and W˜j by Wj = Vj+1 ∩ V˜ ⊥j and W˜j =V˜j+1∩V ⊥j . Each function vj ∈ Vj will have a nodal basis representation representation, asvj =∑kcj,k φj,k (11)and also a so-called multilevel representationvj =∑kcj0,kφj0,k +∑j≥j0∑kdj,kψj,k, (12)where j0 denotes a certain chosen coarsest level and cj,k =〈vj, φ˜j,k〉and dj,k =〈vj, ψ˜j,k〉.The results above where given for spaces defined on the whole of R. To adapt thisconstruction to the periodic setting, we can simply use a well-known procedure of pe-riodization, as described in detail in, e.g., [2]. For a given pair of dual multiresolutionanalyses of L2(R), with dual scaling functions φ and φ˜, we can consider the periodizedfunctions[φj,k] = 2j/2∑l∈Zφ(2j(·+ l)− k)and define the spaces[Vj] := Span({[φj,k] : k = 0, 1 . . . , 2j − 1}), j ∈ N0.(Similar constructions, naturally, associated with φ˜.) Then, it can easily be shown thatthese finite dimensional spaces are nested and dense in L2(T ). If ψ and ψ˜ are the basicwavelets associated with the biorthogonal multiresolution analyses and we consider also3the periodized functions [ψj,k] and [ψ˜j,k], j ∈ N0, k = 0, 1 . . . , 2j−1, then, it can be shownthat the orthogonality conditions (9) carry over to analogous conditions relative to theinner product defined on L2(T ).In this paper, we propose to adopt for test spaces in the Galerkin method, the (periodized)spaces [Vj] of a dual multiresolution analysis generated by scaling functions selected fromthe family of the so called Deslauriers-Dubuc fundamental functions. These functions,which are obtained as auto-correlation of the well-known compact support orthogonalDaubechies scaling functions, have very attractive properties, particularly important forour purposes; see, e.g., [10, 9]. In particular, each of these functions is compactly sup-ported and is also interpolating, i.e., satisfies φ(k) = δ0k, k ∈ Z. Also, several dualfunctions φ˜ to φ, with certain smoothness properties, can be obtained; see. e.g. [11] and[8] for details.Recalling the Galerkin approach (5) to solve our problem, we see that we obtain thefollowing system for determining the coefficient vector cj := (cj,k) of the approximatesolution uj ∈ Vj in the nodal basis∑k∈Λjcj,ka([φj,k], [φj,k′ ]) +∫Tg(∑k∈Λjcj,k[φj,k])[φj,k′ ]dx−∫Tf [φj,k′ ]dx = 0 k′ ∈ Λj,where Λj := {0, 1, . . . , 2j−1}. Due to the nonlinear structure of the function g, we are facedwith the problem of adequately estimating the integral∫T g(∑k∈Λj cj,k[φj,k])[φj,k′ ]dx. Inthis paper, we suggest to approximate the above integral by a version of the so called knotoriented quadrature rules used in the finite element setting. We replace∫T f(x)dx by∫T (Ijf)(x)dx, where Ij is an interpolation projector induced by the interpolating scalingfunction φ and defined as follows:(Ijf) :=∑kf(2−jk)φ(2j · −k). (13)Some properties of this interpolation operator can be seen in [7]. We thus obtain thefollowing approximation∫Tg∑λ∈Λjcj,k[φj,k] [φj,k′ ]dx ≈ 2−j/2g(2j/2cj,k′).This leads to a modified Galerkin system, which can be written asAj · cj + g˜j(cj)− f = 0. (14)Here, we propose to use of Newton’s method for solving the above system. With J = 2j,let Fj : RJ → RJ be the mapping defined byFj(ξ) := Aj ξ + g˜j(ξ)− f . (15)Then, we apply the following iterative scheme, starting with an appropriate initial vectord(0)j :  F′j(d(n)j ) ζ(n+1) = Fj(d(n)j )d(n+1)j = d(n)j − ζ(n+1).(16)4The Jacobian matrix of Fj is naturally given byF′j(ξ) = Aj + g′(2j/2ξ1) 0. . .0 g′(2j/2ξJ) (17)In every Newton step one has to solve a linear system of equations. Hence, we have tobe careful about the condition of F′j. A detailed discussion of this problem is given in[7]. We just refer here that it is necessary to use a wavelet preconditioner that realizes achange of basis from nodal to multilevel; the numerical results presented were obtainedby doing this change of basis.3 Numerical ResultsTo confirm the quality of our approach we present some test examples that describee.g. a chemical intermixture process. Numerical results were obtained by choosing the(periodized) Deslauriers-Dubuc interpolating scaling functions φ2N ; N = 2, 3, 4, with adual φ˜2N,1, whose construction is described in detail in [11] and [8]. We assume that thefunction u is given by u(x) = sin(2pix) which is a sufficiently smooth periodic function onΩ = [0, 1]. Then, we specify f as f(x) = sin3(2pix) + ((2pi)2 + 1) sin(2pix). The numericalapproximations are displayed in the figure below.Figure 1: Approximations on scales j = 5 through j = 9 with trial functions ofvarying smoothness and H1-approximation error (below).5Remark : A more thorough investigation of the method here proposed, including a detailedanalysis of convergence and more numerical results are given in a forthcoming paper; see [7].References[1] G. Caloz and J. Rappaz. Numerical Analysis for Nonlinear and Bifurcation Problems.In P. G. Ciarlet and J. L. Lions, editors, Handbook of Numerical Analysis, volume V,pages 487–637. Elsevier, Amsterdam, 1997.[2] A. Cohen. Wavelet Methods in Numerical Analysis. In P. G. Ciarlet and J. L.Lions, editors, Handbook of Numerical Analysis, volume VII, pages 417–711. Elsevier,Amsterdam, 2000.[3] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods ii:Beyond theelliptic case. IGPM Report # 199, RWTH Aachen, 2000. to appear in Found.Comput. Math.[4] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods for elliptic oper-ator equations - convergence rates. Math. Comput., 70(233):27–75, 2001.[5] S. Dahlke, W. Dahmen, R. Hochmuth, and R. Schneider. Stable multiscale bases andlocal error estimation for elliptic problems. Appl. Numer. Math., 23:21–47, 1997.[6] S. Dahlke, W. Dahmen, and K. Urban. Adaptive wavelet methods for saddle pointproblems - convergence rates. Technical Report IGPM # 204, RWTH Aachen, 2001.to appear in SIAM J. Numer. Anal.[7] S. Dahlke, M. Lindemann, G. Teschke, M. Zhariy, M. J. Soares, P. Cerejeiars, andU. Ka¨hler. A convergent numerical scheme for nonlinear elliptic partial differentialequations. Technical report.[8] S. Dahlke, P. Maass, and G. Teschke. Interpolating scaling functions with duals.Report 00-08, Zentrum fu¨r Technomathematik, Universita¨t Bremen, 2000.[9] G. Deslauriers and S. Dubuc. Interpolation dyadique. In G. Cherbit, editor, Fractals,Dimensions non Entie`res et Applications, pages 44–55. Masson, Paris, 1987.[10] S. Dubuc. Interpolation through an iterative scheme. J. Math. Anal. Appl., 114:185–204, 1986.[11] S. Riemenschneider and Z. Shen. Multidimensional interpolatory subdivisionschemes. SIAM J. Numer. Anal., 34:2357–2381, 1997.Acknowlegment: This work has been supported by Project Based Personnel Exchange Pro-gramme with Portugal-Acc¸o˜es Integradas Luso-Alema˜s (DAAD/ICCTI).6

A wavelet based numerical method for nonlinear partial differential equations

http://repositorium.sdum.uminho.pt/bitstream/1822/1857/1/DLTZSCK_WaveletBN_ActasConfWeimar2003.pdf

A wavelet based numerical method for nonlinear partial differential equations

Abstract

Similar works

Full text

Available Versions

Universidade do Minho: RepositoriUM