A Spectral Method for Nonlinear Elliptic Equations
|
|
- Jordan Haynes
- 5 years ago
- Views:
Transcription
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
A Spectral Method for Elliptic Equations: The Neumann Problem
A Spectral Method for Elliptic Equations: The Neumann Problem Kendall Atkinson Departments of Mathematics & Computer Science The University of Iowa Olaf Hansen, David Chien Department of Mathematics California
More informationA Spectral Method for Elliptic Equations: The Neumann Problem
A Spectral Method for Elliptic Equations: The Neumann Problem Kendall Atkinson Departments of Mathematics & Computer Science The University of Iowa David Chien, Olaf Hansen Department of Mathematics California
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationOblique derivative problems for elliptic and parabolic equations, Lecture II
of the for elliptic and parabolic equations, Lecture II Iowa State University July 22, 2011 of the 1 2 of the of the As a preliminary step in our further we now look at a special situation for elliptic.
More informationA Spectral Method for the Biharmonic Equation
A Spectral Metho for the Biharmonic Equation Kenall Atkinson, Davi Chien, an Olaf Hansen Abstract Let Ω be an open, simply connecte, an boune region in Ê,, with a smooth bounary Ω that is homeomorphic
More informationWave equation on manifolds and finite speed of propagation
Wave equation on manifolds and finite speed of propagation Ethan Y. Jaffe Let M be a Riemannian manifold (without boundary), and let be the (negative of) the Laplace-Beltrami operator. In this note, we
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationPART IV Spectral Methods
PART IV Spectral Methods Additional References: R. Peyret, Spectral methods for incompressible viscous flow, Springer (2002), B. Mercier, An introduction to the numerical analysis of spectral methods,
More informationHIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS
HIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS JASON ALBRIGHT, YEKATERINA EPSHTEYN, AND QING XIA Abstract. Highly-accurate numerical methods that can efficiently
More informationProjected Surface Finite Elements for Elliptic Equations
Available at http://pvamu.edu/aam Appl. Appl. Math. IN: 1932-9466 Vol. 8, Issue 1 (June 2013), pp. 16 33 Applications and Applied Mathematics: An International Journal (AAM) Projected urface Finite Elements
More informationSpectra of Multiplication Operators as a Numerical Tool. B. Vioreanu and V. Rokhlin Technical Report YALEU/DCS/TR-1443 March 3, 2011
We introduce a numerical procedure for the construction of interpolation and quadrature formulae on bounded convex regions in the plane. The construction is based on the behavior of spectra of certain
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationMultivariable Calculus
2 Multivariable Calculus 2.1 Limits and Continuity Problem 2.1.1 (Fa94) Let the function f : R n R n satisfy the following two conditions: (i) f (K ) is compact whenever K is a compact subset of R n. (ii)
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More information12.0 Properties of orthogonal polynomials
12.0 Properties of orthogonal polynomials In this section we study orthogonal polynomials to use them for the construction of quadrature formulas investigate projections on polynomial spaces and their
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationCourse Summary Math 211
Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.
More informationAn Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations
An Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations Moshe Israeli Computer Science Department, Technion-Israel Institute of Technology, Technion city, Haifa 32000, ISRAEL Alexander
More informationRecurrence Relations and Fast Algorithms
Recurrence Relations and Fast Algorithms Mark Tygert Research Report YALEU/DCS/RR-343 December 29, 2005 Abstract We construct fast algorithms for decomposing into and reconstructing from linear combinations
More informationThe Dirichlet-to-Neumann operator
Lecture 8 The Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator plays an important role in the theory of inverse problems. In fact, from measurements of electrical currents at the surface
More informationHIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS
HIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS JASON ALBRIGHT, YEKATERINA EPSHTEYN, AND QING XIA Abstract. Highly-accurate numerical methods that can efficiently
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More informationimplies that if we fix a basis v of V and let M and M be the associated invertible symmetric matrices computing, and, then M = (L L)M and the
Math 395. Geometric approach to signature For the amusement of the reader who knows a tiny bit about groups (enough to know the meaning of a transitive group action on a set), we now provide an alternative
More informationAnalysis in weighted spaces : preliminary version
Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.
More informationScientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1
Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,
More informationPointwise estimates for Green s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone
Pointwise estimates for Green s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone by V. Maz ya 1 and J. Rossmann 1 University of Linköping, epartment of Mathematics, 58183
More informationREMARKS ON THE MEMBRANE AND BUCKLING EIGENVALUES FOR PLANAR DOMAINS. Leonid Friedlander
REMARKS ON THE MEMBRANE AND BUCKLING EIGENVALUES FOR PLANAR DOMAINS Leonid Friedlander Abstract. I present a counter-example to the conjecture that the first eigenvalue of the clamped buckling problem
More informationA FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS
Proceedings of ALGORITMY 2005 pp. 222 229 A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS ELENA BRAVERMAN, MOSHE ISRAELI, AND ALEXANDER SHERMAN Abstract. Based on a fast subtractional
More informationChapter 8 Integral Operators
Chapter 8 Integral Operators In our development of metrics, norms, inner products, and operator theory in Chapters 1 7 we only tangentially considered topics that involved the use of Lebesgue measure,
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationA COLLOCATION METHOD FOR SOLVING THE EXTERIOR NEUMANN PROBLEM
TUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, eptember 2003 A COLLOCATION METHOD FOR OLVING THE EXTERIOR NEUMANN PROBLEM ANDA MICULA Dedicated to Professor Gheorghe Micula at his 60 th
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More informationi=1 α i. Given an m-times continuously
1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable
More informationStability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games
Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,
More informationMcGill University Department of Mathematics and Statistics. Ph.D. preliminary examination, PART A. PURE AND APPLIED MATHEMATICS Paper BETA
McGill University Department of Mathematics and Statistics Ph.D. preliminary examination, PART A PURE AND APPLIED MATHEMATICS Paper BETA 17 August, 2018 1:00 p.m. - 5:00 p.m. INSTRUCTIONS: (i) This paper
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationDETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION
Journal of Fractional Calculus and Applications, Vol. 6(1) Jan. 2015, pp. 83-90. ISSN: 2090-5858. http://fcag-egypt.com/journals/jfca/ DETERMINATION OF AN UNKNOWN SOURCE TERM IN A SPACE-TIME FRACTIONAL
More informationA Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators
A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of
More informationSOLUTIONS TO SELECTED PROBLEMS FROM ASSIGNMENTS 3, 4
SOLUTIONS TO SELECTED POBLEMS FOM ASSIGNMENTS 3, 4 Problem 5 from Assignment 3 Statement. Let be an n-dimensional bounded domain with smooth boundary. Show that the eigenvalues of the Laplacian on with
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationChapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs
Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u
More informationShort note on compact operators - Monday 24 th March, Sylvester Eriksson-Bique
Short note on compact operators - Monday 24 th March, 2014 Sylvester Eriksson-Bique 1 Introduction In this note I will give a short outline about the structure theory of compact operators. I restrict attention
More informationTHE STOKES SYSTEM R.E. SHOWALTER
THE STOKES SYSTEM R.E. SHOWALTER Contents 1. Stokes System 1 Stokes System 2 2. The Weak Solution of the Stokes System 3 3. The Strong Solution 4 4. The Normal Trace 6 5. The Mixed Problem 7 6. The Navier-Stokes
More informationLecture Notes: African Institute of Mathematics Senegal, January Topic Title: A short introduction to numerical methods for elliptic PDEs
Lecture Notes: African Institute of Mathematics Senegal, January 26 opic itle: A short introduction to numerical methods for elliptic PDEs Authors and Lecturers: Gerard Awanou (University of Illinois-Chicago)
More informationMATH 205C: STATIONARY PHASE LEMMA
MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationDeterminant lines and determinant line bundles
CHAPTER Determinant lines and determinant line bundles This appendix is an exposition of G. Segal s work sketched in [?] on determinant line bundles over the moduli spaces of Riemann surfaces with parametrized
More informationFINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS
FINITE ENERGY SOLUTIONS OF MIXED 3D DIV-CURL SYSTEMS GILES AUCHMUTY AND JAMES C. ALEXANDER Abstract. This paper describes the existence and representation of certain finite energy (L 2 -) solutions of
More information256 Summary. D n f(x j ) = f j+n f j n 2n x. j n=1. α m n = 2( 1) n (m!) 2 (m n)!(m + n)!. PPW = 2π k x 2 N + 1. i=0?d i,j. N/2} N + 1-dim.
56 Summary High order FD Finite-order finite differences: Points per Wavelength: Number of passes: D n f(x j ) = f j+n f j n n x df xj = m α m dx n D n f j j n= α m n = ( ) n (m!) (m n)!(m + n)!. PPW =
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationMath 5520 Homework 2 Solutions
Math 552 Homework 2 Solutions March, 26. Consider the function fx) = 2x ) 3 if x, 3x ) 2 if < x 2. Determine for which k there holds f H k, 2). Find D α f for α k. Solution. We show that k = 2. The formulas
More information2 Two-Point Boundary Value Problems
2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationTOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :
TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted
More information1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0
Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More informationPreconditioned space-time boundary element methods for the heat equation
W I S S E N T E C H N I K L E I D E N S C H A F T Preconditioned space-time boundary element methods for the heat equation S. Dohr and O. Steinbach Institut für Numerische Mathematik Space-Time Methods
More informationb i (x) u + c(x)u = f in Ω,
SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 1938 1953 c 2002 Society for Industrial and Applied Mathematics SUBOPTIMAL AND OPTIMAL CONVERGENCE IN MIXED FINITE ELEMENT METHODS ALAN DEMLOW Abstract. An elliptic
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationNumerical Methods for Two Point Boundary Value Problems
Numerical Methods for Two Point Boundary Value Problems Graeme Fairweather and Ian Gladwell 1 Finite Difference Methods 1.1 Introduction Consider the second order linear two point boundary value problem
More informationNormed & Inner Product Vector Spaces
Normed & Inner Product Vector Spaces ECE 174 Introduction to Linear & Nonlinear Optimization Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 174 Fall 2016 1 / 27 Normed
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationBoundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis
Boundary Integral Equations on the Sphere with Radial Basis Functions: Error Analysis T. Tran Q. T. Le Gia I. H. Sloan E. P. Stephan Abstract Radial basis functions are used to define approximate solutions
More informationTechnische Universität Graz
Technische Universität Graz Stability of the Laplace single layer boundary integral operator in Sobolev spaces O. Steinbach Berichte aus dem Institut für Numerische Mathematik Bericht 2016/2 Technische
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationL. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS
Rend. Sem. Mat. Univ. Pol. Torino Vol. 57, 1999) L. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS Abstract. We use an abstract framework to obtain a multilevel decomposition of a variety
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationNumerical Methods for Partial Differential Equations
Numerical Methods for Partial Differential Equations Steffen Börm Compiled July 12, 2018, 12:01 All rights reserved. Contents 1. Introduction 5 2. Finite difference methods 7 2.1. Potential equation.............................
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationOctober 25, 2013 INNER PRODUCT SPACES
October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal
More informationNonlinear Analysis. Global solution curves for boundary value problems, with linear part at resonance
Nonlinear Analysis 71 (29) 2456 2467 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Global solution curves for boundary value problems, with linear
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1
More informationRNDr. Petr Tomiczek CSc.
HABILITAČNÍ PRÁCE RNDr. Petr Tomiczek CSc. Plzeň 26 Nonlinear differential equation of second order RNDr. Petr Tomiczek CSc. Department of Mathematics, University of West Bohemia 5 Contents 1 Introduction
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationat time t, in dimension d. The index i varies in a countable set I. We call configuration the family, denoted generically by Φ: U (x i (t) x j (t))
Notations In this chapter we investigate infinite systems of interacting particles subject to Newtonian dynamics Each particle is characterized by its position an velocity x i t, v i t R d R d at time
More informationThe Implicit and Inverse Function Theorems Notes to supplement Chapter 13.
The Implicit and Inverse Function Theorems Notes to supplement Chapter 13. Remark: These notes are still in draft form. Examples will be added to Section 5. If you see any errors, please let me know. 1.
More informationStabilization and Acceleration of Algebraic Multigrid Method
Stabilization and Acceleration of Algebraic Multigrid Method Recursive Projection Algorithm A. Jemcov J.P. Maruszewski Fluent Inc. October 24, 2006 Outline 1 Need for Algorithm Stabilization and Acceleration
More informationGaussian interval quadrature rule for exponential weights
Gaussian interval quadrature rule for exponential weights Aleksandar S. Cvetković, a, Gradimir V. Milovanović b a Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice
More informationAnalysis-3 lecture schemes
Analysis-3 lecture schemes (with Homeworks) 1 Csörgő István November, 2015 1 A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült Contents 1. Lesson 1 4 1.1. The Space
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationwhich arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i
MODULE 6 Topics: Gram-Schmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.
More informationEuler Equations: local existence
Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u
More informationFractional order operators on bounded domains
on bounded domains Gerd Grubb Copenhagen University Geometry seminar, Stanford University September 23, 2015 1. Fractional-order pseudodifferential operators A prominent example of a fractional-order pseudodifferential
More informationThroughout these notes we assume V, W are finite dimensional inner product spaces over C.
Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal
More informationChapter 12. Partial di erential equations Di erential operators in R n. The gradient and Jacobian. Divergence and rotation
Chapter 12 Partial di erential equations 12.1 Di erential operators in R n The gradient and Jacobian We recall the definition of the gradient of a scalar function f : R n! R, as @f grad f = rf =,..., @f
More informationVISCOSITY SOLUTIONS. We follow Han and Lin, Elliptic Partial Differential Equations, 5.
VISCOSITY SOLUTIONS PETER HINTZ We follow Han and Lin, Elliptic Partial Differential Equations, 5. 1. Motivation Throughout, we will assume that Ω R n is a bounded and connected domain and that a ij C(Ω)
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More information