A Spectral Method for Nonlinear Elliptic Equations

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1 A Spectral Method for Nonlinear Elliptic Equations Kendall Atkinson, David Chien, Olaf Hansen July 8, 6 Abstract Let Ω be an open, simply connected, and bounded region in R d, d, and assume its boundary Ω is smooth and homeomorphic to S d. Consider solving an elliptic partial differential equation Lu = f, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball B d in R d, say Lũ = f, ũ). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials ũ n of degree n that is convergent to ũ. The transformation from Ω to B d requires a special analytical calculation for its implementation. With suffi ciently smooth problem parameters, the method is shown to be rapidly convergent. For u C Ω ) and assuming Ω is a C boundary, the convergence of ũ ũ n H to zero is faster than any power of /n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to u + γu = fu) with a zero Neumann boundary condition is also presented. Introduction Consider the nonlinear problem Lu s) = f s, us)), s Ω ) u s) =, s Ω ) with L an elliptic operator over Ω and a homogeneous Dirichlet boundary condition. Let Ω be an open, simply connected, and bounded region in R d, and Departments of Mathematics & Computer Science, The University of Iowa, Iowa City, Iowa 54, USA, kendall-atkinson@uiowa.edu, fax: 39) Department of Mathematics, California State University San Marcos, San Marcos, California 996, USA, chien@csusm.edu and ohansen@csusm.edu, fax: 76)

2 assume that its boundary Ω is suffi ciently differentiable and is homeomorphic to S d. Assume L is a strongly elliptic operator of the form Lus) d i,j= s i a i,j s) us) ) + γ s) u s), s Ω, 3) s j We present a spectral method for solving )-) based on multivariate polynomial approximation over the unit ball B d. Our numerical method is similar to that presented in earlier papers for linear problems; see [], [6]. However, the nonlinearity in ) leads to the solving of nonlinear algebraic systems. Moreover, the convergence analysis requires a new approach as the standard variational analysis applies to only the linear framework. We give a new error analysis that uses a reformulation of the problem )-) and its numerical approximation using nonlinear integral equations; see 3. In 3), the functions a i,j s), i, j d, are assumed to be several times continuously differentiable over Ω, and the d d matrix [a i,j s)] is to be symmetric and to satisfy ξ T As)ξ αξ T ξ, s Ω, ξ R d 4) for some α >. Also assume the coeffi cient γ C Ω ). Note that because the right-hand function f is allowed to depend on u, an arbitrarily large multiple of u can be added to each side of ), thus justifying an assumption that min γ s) >. 5) s Ω The problem )-) can be reformulated as a variational problem. Introduce d A v, w) = a i,j s) vs) ws) ds Ω s i,j= i s j 6) + γ s) v s) w s) ds, v, w H Ω), Ω F v)) s) = f s, vs)), s Ω, v H Ω). 7) Note that the Sobolev space H m Ω) is the closure of C m Ω ) using the norm g Hm Ω) = i m D i g L Ω), g Cm Ω ), m with i a multi-integer, i = i,..., i d ), i = i + + i d, and D i g s) = i g s) s i. si d d The space H Ω) is the closure of C Ω) using H Ω), where elements of C Ω) C Ω ) are zero on some open neighborhood of the boundary of Ω.

3 Noting 4) and 5), it can be assumed that A is a strongly elliptic operator on H Ω), namely A v, v) c v H Ω), v H Ω) for some finite c >. Reformulate )-) as the following variational problem: find u H Ω) for which A u, w) = F u), w), w H Ω). 8) Throughout this paper we assume the variational reformulation of the problem )-) has a locally unique solution u H Ω). For analyses of the existence and uniqueness of a solution to )-), see Zeidler [6, 8.5]. In the following we define our spectral method for the case that Ω = B d ; and following that we show how to reformulate the problem )-) for a general smooth region Ω as an equivalent problem over B d. This follows the earlier development in []. In 3 we present a convergence analysis for our numerical method, an approach using results from the numerical analysis of nonlinear integral equations. Implementation of the method is discussed in 4, followed by numerical examples in 5. An extension to a Neumann boundary value problem is given in 6. A spectral method Begin with the special case Ω = B d, and then move to a general region Ω. Let X n denote a finite-dimensional subspace of H ) B d, and let { } ψ,..., ψ Nn be a basis of X n. Later a basis is given by using polynomials of degree n over R d, denoted by Π d n, with N n the dimension of Π d n. An approximating solution to 8) is sought by finding u n X n such that More precisely, find A u n, w) = F u n ), w), w X n. 9) N n u n x) = α l ψ l x) ) that satisfies the nonlinear algebraic system N n d α l a i,j x) ψ lx) ψ k x) + γ x) ψ l= B x d i,j= i x l x) ψ k x) dx j ) N n = x, α l ψ l x) ψ k x) dx, k =,..., N n. B d f l= l= ) As notation, we generally use the variable x when considering B d and the variable s when considering Ω. 3

4 To obtain a space for approximating the solution u of )-), proceed as follows. Denote by Π d n the space of polynomials in d variables that are of degree n: p Π d n if it has the form px) = i n a i x i xi... xi d d, i = i,..., i d ), i = i + i d. As the approximation space over B d, choose { X n = x ) } px) p Π d n H B d ) ) Let N n = dim X n = dim Π d n. For d =, N n = n + ) n + ) /. Practical implementation of the numerical method 9)-) is discussed in 4.. Transformation of the domain Ω For the more general problem )-) over a general region Ω, we reformulate it as a problem over B d. Begin by reviewing some ideas from [], to which the reader is referred for additional details. Assume the existence of a function Φ : B d onto Ω 3) with Φ a twice differentiable mapping, and let Ψ = Φ v L Ω), let and conversely for ṽ L B d), : Ω onto B d. For ṽx) = v Φ x)), x B d 4) Assuming v H Ω), it is straightforward to show vs) = ṽ Ψ s)), s Ω. 5) x ṽ x) = J x) T s v s), s = Φ x) with J x) the Jacobian matrix for Φ over the closed unit ball B d, [ Φi x) Jx) DΦ) x) = x j ] d, x B d. 6) i,j= To use our method for problems over a region Ω, it is necessary to know explicitly the functions Φ and J. The creation of such a mapping Φ is taken up in [5] for cases in which only a boundary mapping is known, from S d B d to Ω, a common way to define the region Ω. Assume det Jx), x B d. 7) 4

5 Similarly, s vs) = Ks) T x ṽx), x = Ψs) with Ks) the Jacobian matrix for Ψ over Ω. By differentiating the identity Ψ Φ x)) = x, x B d it follows that K Φ x)) = J x). Assumptions about the differentiability of ṽ x) can be related back to assumptions on the differentiability of vs) and Φx). Lemma Let Φ C m B d). If v C k Ω ), then ṽ C q B d) with q = min {k, m}. Similarly, if v H k Ω), then ṽ H q B d). A proof is straightforward using 4). A converse statement can be made as regards ṽ, v, and Ψ in 5). Moreover, the differentiability of Φ over B d is exactly the same as that of Ψ over Ω.. Reformulation from Ω to B d Applying this transformation to the equation ), it follows that where and d i,j= x i det Jx)) ã i,j x) ũx) ) + γ x) ũx) x j = f x, ũx)), x B d, 8) f x, ũx)) = det Jx)) f Φ x), ũx)), x B d 9) γ x) = det Jx)) γ Φ x)) ) Ã x) = J x) AΦ x))j x) T [ã i,j x)] d i,j=. ) A derivation of this is given in [, Thm. 3]. With 8), also impose the Dirichlet condition ũx) =, x B d. ) The problem of solving 8)-) is completely equivalent to that of solving )-). Also, the differential operator in 8) will be strongly elliptic. As noted earlier, the creation of such a mapping Φ is discussed at length in [5] for extending a boundary mapping ϕ : S d Ω to a mapping Φ satisfying 3) and 7). 5

6 3 Error analysis In [4] Osborn converted a finite element method for solving an eigenvalue problem for an elliptic partial differential equation to a corresponding numerical method for approximating the eigenvalues of a compact integral operator. He then used results for the latter to obtain convergence results for his finite element method. We use his construction to convert the numerical method for 8) to a corresponding method for finding a fixed point of a completely continuous nonlinear integral operator, and this latter numerical method will be analyzed using the results given in [, Chap. 3] and []. Important results about polynomial approximation have been given recently by Li and Xu [], and they are critical to our convergence analysis. Theorem Li and Xu) Let r. Given v H r B d), there exists a sequence of polynomials p n Π d n such that v p n H B d ) ε n,r v Hr B d ), n. 3) The sequence ε n,r = O n r+) and is independent of v. Theorem 3 Li and Xu) Let r. Given v H ) B d H r B d), there exists a sequence of polynomials p n X n such that v p n H B d ) ε n,r v H r B d ), n. 4) The sequence ε n,r = O n r+) and is independent of v. These two results are Theorems 4. and 4.3, respectively, in []. For the second theorem, also see the comments immediately following [, Thm. 4.3]. For the convergence analysis, we follow closely the development in Osborn [4, 4a)]. We omit the details, noting only those different from [4, 4a)]. Taking f to be a given function in L B d ), the element u H Ω) for which A u, w) = f, w), w H Ω), can be written as u = T f with T : L ) B d H ) B d H B d) and bounded, T f H B d ) C f L B d ), f L B d ). The operator is the Green s integral operator for the associated Dirichlet problem. More generally, for r, T : H r B d) H ) B d H r+ B d), T f H r+ B d ) C r f Hr B d ), f Hr B d). In addition, T is a compact operator on L ) B d into H ) B d, and more generally, it is compact from H r B d) into H ) B d H r+ B d). With our assumptions, T is self-adjoint on L ) B d, although Osborn allows more general 6

7 non-symmetric operators L. The same argument is applied to the numerical method 9) to obtain a solution u n = T n f with T n having properties similar to T and also having finite rank with range in X n. The major assumption of Osborn is that his finite element method satisfies an approximation inequality see [4, 4.7)]), and the above theorems of Li and Xu are the corresponding statements for our numerical method. The argument in [4, 4a)] then shows T T n L L c n. 5) Our variational problems 8) and 9) can now be reformulated as u = T F u), 6) u n = T n F u n ), 7) and we regard these as equations on some subset of L B d ), dependent on the form of the function f defining F. The operator F of 7) is sometimes called the Nemytskii operator; see [, Chap., ] for its properties. It is necessary to assume that F is defined and continuous over some open subset D L B d ) : v D = f, v) L B d ), v n v in L B d ) = f, v n ) f, v) in L B d ). 8) These are somewhat restrictive. As an example in one variable, if b, v) = v and if v L, ) then b, v) may not belong to L, ). The function v s) / 3 s is in L, ), whereas v s) = / 3 s does not belong to L, ). An analysis of when 8) is true can be based on [3]. Generally, if f, v) is bounded by a linear function of v, then 8) is true. Experimentally, the spectral method 9) works well for cases with f, v) increasing at greater than a linear rate in v. The operators T and T n are linear, and the Nemytskii operator F provides the nonlinearity. The reformulation 6)-7) can be used to give an error analysis of the spectral method 9). The mapping T F is a compact nonlinear operator on an open domain D of a Banach space X, in this case L ) B d. Let V D be an open set containing an isolated fixed point solution u of 6). We can define the index of u or more properly, the rotation of the vector field v T F v) as v varies over the boundary of V ); see [, Part II]. More generally, let K be a completely continuous operator, and let it have an isolated fixed point u of nonzero index. This fixed point is stable in the sense that small compact perturbations of K, say K, lead to one or more fixed points for K with those fixed points all close to u. For an overview of the concepts of index and rotation, see [, Properties P-P5, pp. 8-8]. Property P4 gives a way of computing the index of u, and Property P5 gives further intuition as to the stability implications of a fixed point having a nonzero index. 7

8 Theorem 4 Assume the problem 8) with Ω = B d has a solution u that is unique within some open neighborhood V of u ; further assume that u has nonzero index. Then for all suffi ciently large n, 9) has one or more solutions u n within V, and all such u n converge to u as n. Proof. This is an application of the methods of [, Chap. 3, Sec. 3] or [, Thm. 3]. A suffi cient requirement is the norm convergence of T n to T, given in 5); [, Thm. 3] uses a weaker form of 5). The most standard case of a nonzero index involves a consideration of the Frechet derivative of F; see [4, 5.3]. In particular, the linear operatorf v) is given by F f x, z) v) w) x) = z wx) z=vx) Theorem 5 Assume the problem 8) with Ω = B d has a solution u that is unique within some open neighborhood V of u ; and further assume that I T F u ) is invertible over L B d ). Then u has a nonzero index. Moreover, for all suffi ciently large n there is a unique solution u n to 7) within V, and u n converges to u with u u n LB d ) c T T n) F u ) LB d ) c n F u ) LB d ). 9) Proof. Again this is an immediate application of results in [, Chap. 3, Sec. 3] or [, Thm. 4]. Remark. To give some intuition to our assumption that I T F u ) is invertible, consider a rootfinding problem for a real-valued function f x) with x R, letting α denote the root being sought. Then our invertibility assumption is the analogue of assuming f α). To improve upon this last result 9), we need to bound T T n ) g LB d ) when g H r B d) for some r. Adapting the proof of [4, 4.9)] to our polynomial approximations and using Theorem 3, Using the conservative bound we have T T n ) g H B d ) c n r+ g H r B d ). v LB d ) v H B d ), T T n ) g LB d ) c n r+ g H r B d ). 3) Corollary 6 For some r, assume F u ) H r B d). Then u u n LB d ) O n r+)) F u ) Hr B d ). 3) 8

9 We conjecture that this bound and 3) can be improved to O n r+)). For the case r =, an improved result is given by 9). A nonhomogeneous boundary condition. Consider replacing the homogeneous boundary condition ) with the nonhomogeneous condition u s) = g s), s Ω, in which g is a continuously differentiable function over Ω. One possible approach to solving the Dirichlet problem with this nonzero boundary condition is to begin by calculating a differentiable extension of g, call it G : Ω R, with G C Ω ), G s) = g s), s Ω. With such a function G, introduce v = u G where u satisfies )-). Then v satisfies the equation Lv s) = f s, vs) + Gs)) LG s), s Ω, 3) v s) =, s Ω. 33) This problem is in the format of )-). Sometimes finding an extension G is straightforward; for example, g over Ω has the obvious extension G s). Often, however, we must compute an extension. We begin by first obtaining an extension G using a method from [5], and then we approximate it with a polynomial of some reasonably low degree. For example, see the construction of least squares approximants in [3]. 4 Implementation We consider how to set up the nonlinear system of 9)-) and how to solve it. Because we intend to apply the method to problems defined initially over a region Ω other than B d, we re-write 9)-) for this situation. The transformed equation we are considering is the equation 8). We look for a solution N n ũ n x) = α l ψ l x), l= and u n s) is to be the equivalent solution considered over Ω: ũ n x) u n Φ x)), x B d. The coeffi cients {α l l =,,..., N n } are the solutions of N n d α k det J x) ã i,j x) ψ kx) ψ l x) + γx)ψ k= B x d i,j= j x k x)ψ l x) i ) 34) N n = f x, α k ψ k x) ψ l x) dx, l =,..., N n. B d k= 9

10 For the definitions of γ, f, and à x) [ã i,jx)] d i,j=, recall 9)-). When solving the nonlinear system 34), it is necessary to have an initial guess ũ ) n x) = N n l= α) l ψ l x). In our examples, we begin with a very small value for n say n = ), use ũ ) n =, and then solve 34) by some iterative method. Then increase n, using as an initial guess the final solution obtained with a preceding n. This has worked well in our computations, allowing us to work our way to the solution of 34) for much larger values of n. For the iterative solver, we have used the M program fsolve, but will work in the future on improving it. 4. Planar problems The dimension of Π n is N n = n + ) n + ). For notation, we replace x with x, y). We create a basis for X n by first choosing an orthonormal basis for Π n, say { ϕ m,k k =,,..., m; m =,,..., n }. Then define ψ m,k x, y) = x y ) ϕ m,k x, y). 35) How do we choose the orthonormal basis {ϕ l x, y)} N l= for Π n? Unlike the situation for the single variable case, there are many possible orthonormal bases over B, the unit disk in R. We have chosen one that is convenient for our computations. These are the "ridge polynomials" introduced by Logan and Shepp [] for solving an image reconstruction problem. A choice that is more effi cient in calculational costs is given in [3]; but we continue to use the ridge polynomials because we are re-using and modifying computer code written previously for use in [], [3], [6], and [7]. We summarize here the results needed for our work. For general d, let V n = { P Π d n P, Q) = Q Π d n }, the polynomials of degree n that are orthogonal to all elements of Π d n. Then Π d n = V V V n 36) is a decomposition of Π d n into orthonormal subspaces. It is standard to construct orthonormal bases of each V n and to then combine them to form an orthonormal basis of Π d n using this decomposition. For d =, V n has dimension n +, n. As an orthonormal basis of V n we use ϕ n,k x, y) = π U n x cos kh) + y sin kh)), x, y) D, h = π n + 37)

11 for k =,,..., n. The function U n is the Chebyshev polynomial of the second kind of degree n: U n t) = sin n + ) θ, t = cos θ, t, n =,,... sin θ The family { ϕ n,k } n k= is an orthonormal basis of V n. As a basis of Π n, we order { ϕ m,k } lexicographically based on the ordering in 37) and 36): {ϕ l } Nn l= = { ϕ,, ϕ,, ϕ,, ϕ,,..., ϕ n,,..., ϕ n,n }. From 35), the family { ψ m,k } is ordered the same. To calculate the first order partial derivatives of ψ n,k x, y), we need U nt). The values of U nt) and U nt) are evaluated using the standard triple recursion relations U n+ t) = tu n t) U n t), U n+t) = U n t) + tu nt) U n t). For the numerical approximation of the integrals in 34), which are over B, the unit disk, we use the formula B gx, y) dx dy π q + q q l= m= ĝ r l, ) π m ω l r l 38) q + with ĝ r, θ) g r cos θ, r sin θ). Here the numbers r l and ω l are the nodes and weights of the q + )-point Gauss-Legendre quadrature formula on [, ]. Note that q px)dx = pr l )ω l, for all single-variable polynomials px) with deg p) q +. The formula 38) uses the trapezoidal rule with q + subdivisions for the integration over B in the azimuthal variable. This quadrature 38) is exact for all polynomials g Π q. l= 4. The three dimensional case We change our notation, replacing x B 3 with x, y, z). In R 3, the dimension of Π 3 n is ) n + 3 N n = = n + ) n + ) n + 3). 3 6

12 Here we choose orthonormal polynomials on the unit ball as described in [8], ϕ n,j,k x) = C j+k+ 3 n j k h x) x ) j n,j,k ) ) C k+ y j x y ) k/ C k z, 39) x x y j, k =,..., n, j + k n, n N. The function ϕ n,j,k x) is a polynomial of degree n, h n,j,k is a normalization constant, and the functions Ci λ are the Gegenbauer polynomials. The orthonormal base {ϕ n,j,k } n,j,k and its properties can be found in [8, Chapter ]. We can order the basis lexicographically. To calculate these polynomials we use a three term recursion whose coeffi cients are given in [3]. For the numerical approximation of the integrals in 34), we use a quadrature formula for the unit ball B 3, B 3 gx) dx = π π q Q q [g] := q q i= j= k= ĝr, θ, φ) r sinφ) dφ dθ dr Q q [g], π q ω ζk + j ν k ĝ, π i ) q, arccosξ j). Here ĝr, θ, φ) = gx) is the representation of g in spherical coordinates. For the θ integration we use the trapezoidal rule, because the function is π periodic in θ. For the r direction we use the transformation ) ) t + t + dt r vr) dr = v = ) t + t + ) v dt 8 q ) ζk + 8 ν kv, }{{} k= =:ν k where the ν k and ζ k are the weights and the nodes of the Gauss quadrature with q nodes on [, ] with respect to the inner product v, w) = + t) vt)wt) dt. The weights and nodes also depend on q but we omit this index. For the φ direction we use the transformation π sinφ)vφ) dφ = varccosφ)) dφ q ω j varccosξ j )), j=

13 where the ω j and ξ j are the nodes and weights for the Gauss Legendre quadrature on [, ]. For more information on this quadrature rule on the unit ball in R 3, see [5]. Finally we need the gradient to approximate the integral in 34). To do this one can modify the three term recursion in [3] to calculate the partial derivatives of ϕ n,j,k x). 5 Numerical examples We begin with a planar example. Consider the problem u s, t) = f s, t, u s, t)), s, t) Ω, u s, t) =, s, t) Ω. 4) Note the change in notation, from s R to s, t) R. As an illustrative region Ω, we use the mapping Φ : B Ω, s, t) = Φ x, y), s = x y + ax, t = x + y, 4) with < a <. It can be shown that Φ is a - mapping from the unit disk B. In particular, the inverse mapping Ψ : Ω B is given by x = [ + ] + a s + t) a y = [ at + )] 4) + a s + t) a In Figure a), the mapping for a =.95 is illustrated by giving the images in Ω of the circles r = j/, j =,..., and the radial lines θ = jπ/, j =,...,. An alternative polynomial mapping Φ II of degree for this region is computed using the integration/interpolation method of [5, 3]; and Φ II = Φ on the boundary. Ω as defined by 4). It is illustrated in Figure b). This boundary mapping Φ II results in better error characteristics for our spectral method as compared to the transformation Φ. As discussed earlier, we solve the nonlinear system 34) for a lower value of the degree n, usually with an initial guess associated with u ) n =. As we increase n, we use the approximate solution from a preceding n to generate an initial guess for the new value of n. We use the M program fsolve to solve the nonlinear system. In the future we plan to look at other numerical methods that take advantage of the special structure of 34). To estimate the error, we use as a true solution a numerical solution associated with a larger value of n. For a particular case, consider f s, t, z) = 3 cos π st) + z. 43)

14 s s s s a) Φ b) Φ II Figure : Illustrations of mappings on B for the region Ω given by 4) A graph of the solution is shown in Figure, along with numerical results for n = 5, 6,...,, with the solution u 5 taken as the true solution. We use both the mapping Φ of 4) and the mapping Φ II. Using either of the mappings, Φ or Φ II, the graphs indicate an exponential rate of convergence for the mappings {u n }. The mapping Φ II is better behaved, as can be seen by visually comparing the distortion in the graphs of Figure. This is the probable reason for the improved convergence of the spectral method when using Φ II in comparison to Φ. As a second planar example we consider the stationary Fisher equation where the function f in 4) is given by fs, t, u) = u u), s, t) Ω. Fisher s equation is used to model the spreading of biological populations and from f we see that u = and u = are stationary points for the time dependent equation on an unbounded domain; see [9, Chap. 7]. The original Fisher equation does not contain the term, but for small domains the Fisher equation might have no nontrivial solution and the factor corresponds to a scaling by a factor to guarantee the existence of a nontrivial solution on the domain Ω. The domain Ω is the interior of the curve ϕt) = 3 + cost) + sint)) cos t, sin t) 44) We studied this domain in earlier papers see [5]) where we called this domain a Limacon domain. In the article [5] we also describe how we use equation 44) to create a domain mapping Φ : B Ω by two dimensional interpolation. Similar to the previous example we calculate the numerical solutions u n for n =,..., 4, where we use the coeffi cients of u n as a starting value u ) n for 4

15 UsingΦ UsingΦ II t n 5 5 s The solution u The maximum error Figure : The solution u to 4) with right side 43) and its error n =,..., 4 and for u ) we use coeffi cients which are non zero all equal to ), so the iteration of fsolve does not converge to the trivial solution. As a reference solution we calculated u 45 ; see Figure 3. The shape of the solution is very much like we expect it, the function is close to inside the domain Ω and drops off very steeply to the boundary value. By looking at the reference solution in Figure 3 we also see that the function will be harder to approximate by polynomials than the function in the previous example, because of the sharp drop off. This becomes clear when we look at the convergence, also shown in Figure 3. The final error is in the range of 3 4 with a polynomial degree of 4, so the error is in the same range as in the previous example where we only used polynomials up to degree for the approximation. Still the graph suggests that the convergence is exponential as predicted by 3) for the L norm. A three dimensional example. In the following we present a three dimensional example. We use the mapping Φ : B 3 Ω, s, t, v) = Φ x, y, z), defined by s = x y + ax, t = x + y, 45) v = z + bz, where a = b =.5. We have used this mapping in a previous article, see [], where one finds plots of the surface Ω. On Ω we solve u s, t, v) = f s, t, v, u s, t, v)), u s, t, v) =, s, t, v) Ω s, t, v) Ω 46) 5

16 s s Solution for Fisher s equation n Error for Fisher s equation Figure 3: The reference solution and maximum error for Fisher s equation where f is defined by fs, t, v, u) = cos6s + t + v) + u, s, t, v) Ω. We calculated approximate solutions u,..., u and used u 5 as a reference solution. In Figure 4 we see the convergence in the maximum norm on a grid in B 3. As in our previous examples the graph suggests that we have exponential convergence. In our final Figure 5 we show the graph of the reference solution u 5 on B 3 P ν where P ν is a plane in R 3 normal to the vector ν. We have used several normal vectors ν =,, ) T, so P ν is the xy plane, ν =,, ) T, so P ν is the xz plane, ν 3 =,, ) T, so P ν3 is the yz plane, and ν 4 =,, ) T, so P ν4 is a diagonal plane. Figure 5 shows that the solution reflects the periodic character of the nonlinearity f. In the yz plane the oscillation of f is much slower which is also visible in the plot along the yz plane. 6 A Neumann boundary value problem Consider the boundary value problem u s) + γ s) u s) = f s, us)), s Ω, 47) u s) n s =, s Ω, 48) with n s the exterior unit normal to Ω at the boundary point s. Later we discuss an extension to a nonzero normal derivative over Ω. A necessary condition for 6

17 5 5 n Figure 4: For the problem 46), the convergence of the errors u u n the unknown function u to be a solution of 47)-48) is that it satisfy f s, u s)) ds = γ s) u s) ds. 49) Ω With our assumption that 47)-48) has a locally unique solution u, 49) is satisfied. Proceed in analogy with the earlier treatment of the Dirichlet problem. Use integration by parts to show that for arbitrary functions u H Ω), v H Ω), vs) [ us) + γs)u] ds Ω = [ us) vs) + γs)us)vs)] ds v s) us) ds. 5) n s Ω Introduce the bilinear functional A v, v ) = [ v s) v s) + γs)v s)v s)] ds. Ω The variational form of the Neumann problem 47)-48) is as follows: find u H Ω) such that Ω A u, v) = F u), v), v H Ω) 5) Ω 7

18 ν =,, ), P is xy-plane ν =,, ), P is xz-plane ν 3 =,, ), P 3 is yz-plane ν 4 =,, ), P 4 is diagonal Figure 5: The solution ũ x, y, z) over P B 3 with P a plane passing through the origin and orthogonal to ν 8

19 with, as before, the operator F defined by F u)) s) = fs, u s)). The theory for 5) is essentially the same as for the Dirichlet problem in its reformulation 8). Because of changes that take place in the normal derivative under the transformation s = Φ x), we modify the construction of the numerical method. In the actual implementation, however, it will mirror that for the Dirichlet problem. For the approximating space, let X n = { q q Φ = p for some p Π d n}. For the numerical method, we seek u n X n for which A u n, v) = F u n), v), v X n. 5) A similar approach was used in [6] for the linear Neumann problem. To carry out a convergence analysis for 5), it is necessary to compare convergence of approximants in X n to that of approximants from Π d n. For simplicity in notation, we assume Φ C B d). Begin by referring to Lemma and its discussion in., linking differentiability in H m Ω) and H m B d). In particular, for m, c,m v Hm Ω) ṽ H m B d ) c,m v Hm Ω), v Hm Ω), 53) with ṽ = v Φ, with constants c,m, c,m >. Also recall Theorem concerning approximation of functions ṽ H r B d) and link this to approximation of functions v H r Ω). Lemma 7 Let Φ C B d). Assume v H r Ω) for some r. Then there exist a sequence q n X n, n, for which v q n H Ω) ε n,r v Hr Ω), n. 54) The sequence ε n,r = O n r+) and is independent of v. Proof. Begin by applying Theorem to the function ṽ x) = v Φ x)). Then there is a sequence of polynomials p n Π d n for which ṽ p n H B d ) ε n,r ṽ H r B d ), n. Let q n = p n Φ. The result then follows by applying 53). The theoretical convergence analysis now follows exactly that given earlier for the Dirichlet problem. Again we use the construction from [4, 4a)], but 9

20 now use the integral operator T arising from the zero Neumann boundary condition. As with the Dirichlet problem, it is necessary to have A be strongly elliptic, and for that reason and without any loss of generality, assume min γ s) >. s Ω The solution of 5) can be written as u = T F u) with T : L B d ) H B d) and bounded. Use Theorem in place of Theorem 3 for polynomial approximation error, as in the derivation of 9). Theorems 4 and 5, along with Corollary 6 are valid for the spectral method for the Neumann problem 47)-48). 6. Implementation As in 4, we look for a solution to 5) by looking for N n u n s) = α l ψ l s) 55) l= with {ψ l j N n } a basis for X n. The system associated with 5) that is to be solved is N n d α l a i,j s) ψ ls) ψ k s) + γ s) ψ l= Ω s i,j= i s l s) ψ k s) ds j ) 56) N n = f s, α l ψ l s) ψ k s) ds, k =,..., N n. Ω l= For such a basis {ψ l }, we begin with an orthonormal basis for Π n, say {ϕ j j N n }, and then define ψ l s) = ϕ l x) with s = Φ x), l N. The function ũ n x) u n Φ x)), x B d, is to be the equivalent solution considered over B d. Using the transformation of variables s = Φ x) in the system 56), the coeffi cients {α l l =,,..., N n } are the solutions of N n d α k ã i,j x) ϕ kx) ϕ l x) + γφ x))ϕ x j x k x)ϕ l x) det J x) dx i k= B d i,j= = B d f ) N n x, α k ϕ k x) ϕ l x) det J x) dx, l =,..., N n. k= 57) For the equation 47) the matrix A s) is the identity, and therefore from ), Ã x) = J x) J x) T. The system 57) is much the same as 34) for the Dirichlet problem, differing only by the basis functions being used for the solution ũ n. We use the same numerical integration as before, and also the same orthonormal basis for Π d n.

21 6. Numerical example Consider the problem u s, t) + u s, t) = f s, t, u s, t)), s, t) Ω, u s) =, n s s, t) Ω, 58) with Ω the elliptical region s a ) + t b ). The mapping of B onto Ω is simply Φ x, y) = ax, by), x, y) B. As before, note the change in notation, from s Ω to s, t) Ω, and from x B to x, y) B. The right side f is given by f s, t, u) = e u + f s, t) 59) with the function f determined from the given true solution and the equation 58) to define f s, t, u). In our case, s ) ) ) t u s, t) = cos s + t ). 6) a b Easily this has a normal derivative of zero over the boundary of Ω. The nonlinear system 57) was solved using fsolve from M, as earlier in 5. Our region Ω uses a, b) =, ). Figure 6 contains the approximate solution for n = 8 and also shows the maximum error over Ω. Again, the convergence appears to be exponential. 6.3 Handling a nonzero Neumann condition Consider the problem u s) + γ s) u s) = f s, us)), s Ω, 6) u s) n s = gs), s Ω 6) with a nonzero Neumann boundary condition. Let u s) denote the solution we are seeking. A necessary condition for solvability of 6)-6) is that f s, u s)) ds = γ s) u s) ds g s) ds. 63) Ω Ω Ω

22 t s 8 n Solution 6) Maximum error Figure 6: The solution u to 58) with right side 59) and true solution 6) There are at least two approaches to extending our spectral method to solve this problem. First, consider the problem v s) = c, s Ω, 64) v s) n s = gs), s Ω, 65) with c a constant. From 63), solvability of 64)-65) requires c ds = g s) ds 66) to be satisfied. To achieve this, choose Ω c = Vol Ω) Ω Ω g s) ds. A solution v s) exists, although it is not unique. The solution of 64)-65) can be approximated using the method given in [6]. Then introduce w = u v. Substituting into 6)-6), the new unknown function w satisfies w s) + γ s) w s) = f s, ws) + v s)) γ s) v s) c, s Ω, 67) w s) n s =, s Ω. 68)

23 The methods of this section can be used to approximate w ; and then use u = w + v. A second approach is to use 5) to reformulate 6)-6) as the problem of finding u = u for which with Thus we seek for which A u, v) = F u), v) + l v), v H Ω) 69) l v) = Ω v s) g s) ds. N n u n s) = α l ψ l s) l= A u n, v) = F u), v) + l v), v X n. 7) The first approach, that of 6)-68), is usable, and the convergence analysis follows from combining this paper s analysis with that of [6]. Unfortunately, we do not have a convergence analysis for this second approach, that of 69)-7), as the Green s function approach of this paper does not seem to extend to it. References [] K. Atkinson. The numerical evaluation of fixed points for completely continuous operators, SIAM J. Num. Anal. 973), pp [] K. Atkinson, D. Chien, and O. Hansen. A spectral method for elliptic equations: The Dirichlet problem, Advances in Computational Mathematics, 33 ), pp [3] K. Atkinson, D. Chien, and O. Hansen. Evaluating polynomials over the unit disk and the unit ball, Numerical Algorithms 67 4), pp [4] K. Atkinson and W. Han. Theoretical Numerical Analysis: A Functional Analysis Framework, 3 rd ed., Springer-Verlag, 9. [5] K. Atkinson and O. Hansen. Creating domain mappings, Electronic Transactions on Numerical Analysis 39 ), pp. -3. [6] K. Atkinson, O. Hansen, and D. Chien. A spectral method for elliptic equations: The Neumann problem, Advances in Computational Mathematics 34 ), pp [7] K. Atkinson,O. Hansen, and D. Chien. A spectral method for parabolic differential equations, Numerical Algorithms 63 3), pp [8] C. Dunkl and Y. Xu. Orthogonal Polynomials of Several Variables, Cambridge Univ. Press,. 3

24 [9] Mark Kot. Elements of Mathematical Ecology, Cambridge University Press,. [] Huiyuan Li and Yuan Xu. Spectral approximation on the unit ball, SIAM J. Num. Anal. 5 4), pp [] B. Logan and L. Shepp. Optimal reconstruction of a function from its projections, Duke Mathematical Journal 4 975), [] M. Krasnose l, skii. Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, 964. [3] M. Marcus and V. Mizel. Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rat. Mech. & Anal ), pp [4] John Osborn. Spectral approximation for compact operators, Mathematics of Computation 9 975), pp [5] A. Stroud. Approximate Calculation of Multiple Integrals, Prentice-Hall, Inc., 97. [6] E. Zeidler. Nonlinear Functional Analysis and Its Applications: II/B, Springer-Verlag, 99. 4

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