CONDITION-DEPENDENT HILBERT SPACES FOR STEEPEST DESCENT AND APPLICATION TO THE TRICOMI EQUATION Jason W. Montgomery

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1 CONDITION-DEPENDENT HILBERT SPACES FOR STEEPEST DESCENT AND APPLICATION TO THE TRICOMI EQUATION Jason W. Montgomery Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS August 2014 APPROVED: John W. Neuberger, Major Professor Joseph Iaia, Committee Co-chair W. Ted Mahaiver, Committee Member Robert J. Renka, Committee Member Su Gao, Chair of the Department of Mathematics Mark Wardell, Dean of the Toulouse Graduate School

2 Montgomery, Jason W. Condition-Dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation. Doctor of Philosophy (Mathematics), August 2014, 58 pp., 3 tables, 11 figures, bibliography list, 21 titles. A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain.

3 Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomitype equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.

4 Copyright 2014 by Jason W. Montgomery ii

5 TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES v vi CHAPTER 1 INTRODUCTION 1 CHAPTER 2 BACKGROUND Orthogonal Projections Sobolev Spaces Lax-Richtmeyer Equivalence Theorem 9 CHAPTER 3 CONDITION-DEPENDENT HILBERT SPACES FOR STEEPEST DESCENT Steepest Descent Method Convergence Theorems Hilbert Space Example Method for a Nonlinear Equation 20 CHAPTER 4 APPLICATION TO FINITE PROBLEMS Steepest Descent Method Singular Problem Laplace-Dirichlet Problem Overdetermined Problem Nonlinear Problem 28 CHAPTER 5 THE TRICOMI EQUATION Finite-difference Method The Tricomi-Dirichlet Problem The Tricomi-ladder Problem 39 iii

6 5.4. Continuity Properties of Ladder Conditions Summary of Ladder Conditions 46 APPENDIX MATLAB CODE 48 BIBLIOGRAPHY 57 iv

7 LIST OF TABLES Page Table 4.1. Accuracy of the SDM when applied to a singular ODE with endpoint conditions. 25 Table 4.2. Accuracy of the SDM when applied to an overdetermined problem. 28 Table 4.3. Accuracy of the SDM when applied to a nonlinear initial value problem. 29 v

8 LIST OF FIGURES Page Figure 1.1. An illustration of ladder conditions used to deform Dirichlet conditions Figure 5.1. into initial-boundary value conditions on (0, 1) ( 1/2, 1/2). Solid lines indicate the portion of the domain where conditions are specified. The arrows in the rightmost set of conditions indicate derivative values. 4 Solutions to the finite Tricomi-Dirichlet problem with boundary conditions given by v(x, y) = y 2, i.e., Q (1) m v for m = 49 and m = Figure 5.2. Q (1) m v for m = 17, 19, and 21 and v(x, y) = y ( ) Figure 5.3. D m,2,2 Q (1) m P m v for m = 49 and m = 149 and v(x, y) = y Figure 5.4. Boxplots for m vs. β m (1) and m vs. ω m (1) for m = 15, 17,..., 125. Note the log 10 scaling of the vertical axes. This illustrates the instability of the Tricomi-Dirichlet method. 38 Figure 5.5. Tricomi-ladder solutions: Q (2) m v for m = 49 and m = 149 and v(x, y) = y Figure 5.6. Tricomi-ladder solutions: Q (2) m v for m = 17, 19, and 21 and v(x, y) = y ( ) Figure 5.7. D m,2 Q (2) m P m v for m = 49 and m = 149 and v(x, y) = y ( ) Figure 5.8. D m,2,2 Q (2) m P m v for m = 49 and m = 149 and v(x, y) = y Figure 5.9. Boxplots for m vs. β m (2) and m vs. ω m (2) for m = 15, 17,..., 125. This illustrates the stability of the Tricomi-ladder method. 42 Figure Plot of mesh size m vs. m 2 E m (v) for v(x, y) = y 2. This provides a strong indication that E m (v) is proportional to 1/m. 46 vi

9 CHAPTER 1 INTRODUCTION Consider the Tricomi equation, which is given by (1) Ju 11 + u 22 = 0 where J(x, y) = y, (x, y) R 2. On a domain Ω R 2 the Tricomi equation is Elliptic if J > 0 Parabolic if J = 0 Hyperbolic if J < 0 and it is mixed if J changes signs in Ω. A long-standing problem in partial differential equations (PDEs) is that of defining boundary conditions for a given domain Ω so that there exists a unique solution to (1). Francesco Tricomi produced one of the first main results on (1) in his seminal work [20]. Tricomi considered a mixed domain with boundary defined by a curve Σ in the upper half-plane with distinct endpoints on the parabolic line and the unique pair of intersecting characteristic curves which meet the respective endpoints of Σ. Tricomi used open Dirichlet conditions for which values are prescribed on only part of the boundary and showed that if given values on Σ and one of the two characteristic edges there exists a unique solution to (1) on Ω. F. I. Frankl [6] examined less restrictive characteristic-dependent domains than those studied by Tricomi. Frankl showed a type of open Dirichlet conditions to be necessary and sufficient for existence and uniqueness of solutions to (1). C. S. Morawetz [14] provided a maximum principle and a uniqueness theorem for strong solutions to (1) which satisfy open Dirichlet conditions defined with respect to characteristic curves. Morawetz [15] addressed the closed Dirichlet problem on lenslike domains with quite extreme restrictions on the boundary geometry and produced a well-posedness result. A recent result by D. Lupo et. al [10] demonstrates well-posedness on a large class of domains 1

10 for two types of closed boundary conditions, Dirichlet conditions and mixed conditions which prescribe Dirichlet conditions on the boundary in the upper half-plane and Neumann conditions on the remaining boundary. The problem in [10] is varied slightly from (1) in that the inhomogenous equation Ju 11 + u 22 = f is considered and homogeneous closed boundary conditions are imposed. The problem is shown to be well-posed for solutions in a weighted Sobolev space W 1,2 (Ω; J) with the norm [ ] 1/2 u W 1,2 (Ω;J) = J u u u 2, Ω u W 1,2 (Ω; J). Results on local regularity of such solutions assume J 1/2 f L 2 (Ω) and J 1/2 f 1 L 2 (Ω). The problem of characterizing the set of solutions to the Tricomi equation on a rectangular domain centered on the x-axis remains essentially unknown, despite many decades of effort. What efforts there have been in this direction seem focused on solving this problem using only boundary conditions. We strongly expect that the almost exclusive focus on boundary conditions has kept the characterization problem unsolved. The direction of the present work is to present a new numerical approach which strongly supports a conjecture about what supplementary conditions as opposed to boundary conditions might solve the characterization problem. It is noted that the Tricomi equation is a principal example for all those studying the important problem of transonic flow (see [2], [4], [6], and [16]) and hence a solution of the characterization problem for the Tricomi equation would have far reaching implications for transonic problems. A broader aim is to generally promote a new paradigm for PDEs which endorses the use of non-boundary data when prescribing supplementary conditions. The following problem acts as a motivator for this approach. P: Let a = 1/2, I a = [ a, a], and Ω = (0, 1) ( a, a) R 2. For t I a let D t : C 2 (Ω) C 0 (Ω) be the operator defined by D t u = (J t)u 11 + u 22, u C 2 (Ω). The problem is to find a Banach space W C 0 (Ω), a Banach space Y, and a set of non-trivial operators {S t : t I a } so that the following properties, (i)-(iv), hold true. (i) For each t I a, C (Ω) dom(s t ) and { S t u : u C (Ω) } Y. 2

11 (ii) For each v C (Ω), (2) α v = {(t, S t v) I a Y } is a continuous function and dom(α v ) = I a. (iii) For each t I a, the problem (3) D t u = 0 L 2 (Ω), S t u = S t v Y, has a unique solution u W for each v W. (iv) For each v W, (4) β v = {(t, u) I a W : u satisfies (3)} is a continuous function and dom(β v ) = I a. Problem P is analogous to finding necessary and sufficient domain-dependent conditions for the Tricomi equation so that both the conditions and the solutions vary continuously as the domain is shifted continuously in the plane from an elliptic region to a hyperbolic region. Prior results on the Tricomi equation address property (iii) of problem P while no results on property (iv) are currently offered. Conditions satisfying property (iv) are essential for a complete understanding of the Tricomi equation, especially in any role the Tricomi equation might play in a unification of elliptic and hyperbolic equations. Direction for solving problem P is offered by the following operators. Leave a, I a, Ω, and {D t : t I a } defined as in problem P. Let I = [0, 1]. For t I a, define B t : C (Ω) C 0 (I) and S t : C (Ω) C 0 (I) 4 so that for each u C (Ω) and each s I, (5) [B t u] (s) = u(s, a) u(s, t) a t if a t < a, u 2 (s, a) if t = a, 3

12 and (6) [S t u] (s) = u(0, s a) u(1, s a) u(s, a) [B t u] (s). Let (7) K t = { (x, y) Ω : x {0, 1} or y {t, a} }, t I a. It is easily seen that S t u = S t v if and only if u = v on K t, a t < a. The conditions given by specifying values on K t, a < t < a, will be referred to as ladder conditions. It is shown that ladder conditions satisfy properties (i) and (ii) stated in problem P for Y = (C 0 (I) 4, sup ). The continuity property in (ii) is equivalent to a continuous deformation from Dirichlet conditions to initial-boundary value (IBV) conditions when using ladder conditions. See Figure 1.1 for an illustration. It follows that problem (3) transitions from an elliptic-dirichlet problem to a hyperbolic-ibv problem as t varies from a to a. Moreover, for each t ( a, a), problem (3) may be reduced to an elliptic-dirichlet problem on (0, 1) (t, a) and a resulting hyperbolic-ibv problem on (0, 1) ( a, t). Numerical experiments support ladder conditions as a candidate for satisfying the remaining properties in problem P and are presented in this work. Figure 1.1. An illustration of ladder conditions used to deform Dirichlet conditions into initial-boundary value conditions on (0, 1) ( 1/2, 1/2). Solid lines indicate the portion of the domain where conditions are specified. The arrows in the rightmost set of conditions indicate derivative values. 4

13 The unconventional aspect of ladder conditions is that data is prescribed on the interior of a mixed domain. The following example illustrates the value of using non-boundary data. Consider the ordinary differential equation (ODE) (8) u (t) + u(t) 2 = 0, t [0, 1]. An applied perspective may view (8) as a time-dependent equation and impose an initial condition u(0) = c R at the starting time t = 0 in order to determine the solution on [0, 1]. The immediate obstacle is to determine if c allows for a solution on the entirety of [0, 1]. It is eventually found that there exists a function u C 1 ([0, 1]) satisfying (8) and u(0) = c if and only if c ( 1, ). A greater complication is that the set of appropriate c values is domain-dependent. A pure approach may use the condition (9) 1 0 u(t)dt = c in order to resolve this issue. For any c R there exists a unique function u C 1 ([0, 1]) satisfying (8) and (9). The solution u is given by Moreover, the function u(t) = e c 1, t [0, 1]. 1 + (e c 1)t f = { (t, u) [0, 1] C 1 ([0, 1]) : u satisfies (8) and (9) } is a continuous bijection between the linear space R and the set of solutions to (8). Similar results occur when imposing the integral conditions on any interval [0, T ], T > 0. A steepest descent method is applied to discretizations of the Tricomi equation in order to investigate conditions imposed on a given domain. Prior application of steepest descent to mixed PDEs is given in [7], [8], and Chapter 13 of [17]. Application of steepest descent to solve differential equations was first introduced by Cauchy [3]. The value of Cauchy s method, in contrast with many other methods, is that it does not require boundary conditions. However, the method is extremely inefficient in a numerical setting. 5

14 A far more efficient method of steepest descent in Sobolev spaces was introduced by J. W. Neuberger [17]. The increased efficiency is largely due to the use of derivative values for defining the norm and the resulting gradient. W. T. Mahavier [11] extended this technique to singular differential equations by utilizing weighted Sobolev spaces. Only two examples exist for a Sobolev space in which convergence occurs in one iteration, [18] and [12]. Such a space X is defined for application to a large class of wellposed problems of the form (10) Du = 0, Su = Sv, v X fixed, where D is a linear differential operator and S is a linear operator used to impose supplementary conditions on the given domain. The key aspect of the descent method is the use of supplementary conditions to define the inner product, and thus the Hilbert space, whereas traditional Sobolev descent methods impose conditions during later stages of the descent method. Section 3.3 provides an example in which the Hilbert space resulting from the S-dependent inner product is shown to generate a standard Sobolev space. The case where (10) is overdetermined for most v is also considered. Given v as the initial guess, repeated iterations of steepest descent produce a limit u ker(d) satisfying Su Sv K < Sw Sv K, w ker(d), w u, where K is the Hilbert space containing the range of S. Chapter 4 deals with finite problems. The descent method is applied to discretizations of a singular ODE and the Laplace-Dirichlet problem in Sections 4.2 and 4.3, respectively. An overdetermined problem and a nonlinear problem are examined in Section 4.4 and Section 4.5, respectively. The Tricomi equation is discussed in Chapter 5. Instability of Tricomi-Dirichlet solutions is discussed in Section 5.2. Section 5.3 provides evidence supporting the claim that ladder conditions satisfy property (iii) of problem P for the solution space W = C 0 (Ω) W 1,2 (Ω). Section 5.4 contains a theorem stating that ladder conditions satisfy properties (i) 6

15 and (ii). Evidence supporting the claim that ladder conditions satisfy property (iv) is also presented. 7

16 CHAPTER 2 BACKGROUND 2.1. Orthogonal Projections Let Q be a closed linear subspace of a Hilbert space (X,, ). The orthogonal complement of Q in X is defined by Q = {x X : x, q = 0 q Q}. Theorem 2.1. Q is a closed linear subspace of X. Any vector x X can be decomposed uniquely in the form (11) x = q + r, where q Q and r Q. Proof of Theorem 2.1 is found in [21], pp The element q in (11) is called the orthogonal projection of x upon Q and is denoted by P Q x. The function P Q is called the projection operator upon Q. Theorem 2.2. The projection operator P = P Q is a continuous linear operator such that P = P 2 and P x, y = x, P y for all x, y X. Proof of Theorem 2.2 is found in [21], pp Theorem 2.3. (Riesz representation theorem). Let (X,, ) be a Hilbert space and f a continuous linear functional on X. Then there exists a uniquely determined vector y f X such that f(x) = y f, x, x X. Proof of Theorem 2.3 is found in [21], pp Sobolev Spaces Let Ω R n, n a positive integer. The collection of infinitely differentiable functions from Ω to R with compact support in Ω is denoted by C c (Ω). Let L 1 loc (Ω) be the collection of locally integrable functions on Ω with respect to Lebesgue measure. 8

17 A function u : Ω R is said to have a weak α-th partial derivative if there exists v L 1 loc (Ω) satisfying ud α φ = ( 1) α vφ, φ Cc (Ω). Ω Ω The function v is denoted by D α u. defined by Let k be a positive integer and 1 p <. The Sobolev space W k,p (Ω) may be W k,p (Ω) = {u L p (Ω) : D α u L p (Ω)}. An alternate definition of W k,p (Ω) is given by H k,p (Ω) which is defined as the closure of u Ck (Ω) : D α u p L p (Ω) < with respect to the norm (12) u k,p = 0 α k 0 α k D α u p L p (Ω) 1/p, u C k (Ω). Equivalence of W k,p (Ω) and H k,p (Ω) was shown by N. Meyers and J. Serrin [13]. By definition, W k,p (Ω) is a Banach space. In the special case of p = 2, W k,p (Ω) is a Hilbert space under the inner product (13) u, v k,p = 0 α k D α u, D α v L p (Ω), u, v W k,p (Ω). R. Adams [1] gives an extensive treatment of Sobolev spaces Lax-Richtmeyer Equivalence Theorem The original Lax-Richtmeyer equivalence theorem is found in [9]. The more generalized theorem by J. Sanz-Serna and C. Palencia [19] is used here. Let X and Y be Banach spaces. Suppose A : X Y is a linear operator so that the problem (14) Au = f 9

18 is well-posed. That is, the range of A is dense in Y and there exists a continuous linear operator E : Y X so that the composition EA is the identity in dom(a). The approximating problems are given as follows. Suppose H is a set of positive numbers such that 0 is the unique limit point of H. For each h H, let X h and Y h be normed vector spaces and consider the problem (15) A h u h = f h, f h Y h, where A h : X h Y h is a linear operator. Assume that for each h H, problem (15) is well-posed, with solution operator E h = A 1 h. Let r h : X X h and s h : Y Y h denote discretization operators, and suppose { r h : h H} { s h : h H} is uniformly bounded. The family defines a method for the solution of (14). M = (X h, Y h, A h, r h, s h ) M is said to be a convergent method for problem (14) if for each f Y, lim h r h Ef E h s h f Xh = 0. The method M is said to be consistent at u dom(a) if lim h A h r h u s h Au Yh = 0. The method M is consistent if it is consistent at each u in a set D 0 such that the image A(D 0 ) is dense in Y. Lastly, the method M is stable if the collection of operator norms is uniformly bounded. { E h : h H} Theorem 2.4. Let X, Y, A, X h, Y h, A h, r h, s h be as above. If the method is consistent and stable then it is convergent. If the method is convergent, then it is stable provided that the following conditions holds: There exists a constant K such that, for each h H and each g Y h with g Yh 1, there exists an element f Y such that f Y K and s h f = g. Proof of Theorem 2.4 is given in [19]. 10

19 CHAPTER 3 CONDITION-DEPENDENT HILBERT SPACES FOR STEEPEST DESCENT A problem-specific steepest descent method (SDM) is constructed for application to a linear differential equation paired with uniqueness-inducing conditions which might yield an overdetermined problem. The SDM converges to the unique solution to the differential equation for which change in the assumed condition values is minimal. If the conditions yield a well-posed problem, the solution is given by the first iteration of steepest descent. Throughout the chapter j will denote the identity map on R Steepest Descent Method Let Ω be a bounded open set in R m, m N, with a piecewise smooth boundary Ω. Consider a linear differential operator D which maps C n (Ω), n N, into H def = L 2 (Ω). Suppose S is a linear operator from C n (Ω) into a Hilbert space K so that the following property holds: There exists M > 0 such that (16) u 2 L 2 (Ω) M ( Du 2 H + K) Su 2, u C n (Ω). ( ) D The pairing of D and S is denoted by A = which maps C n (Ω) into H K. Define the S bilinear form, A by (17) u, v A = Au, Av H K, u, v C n (Ω). Symmetry and linearity of, A is clear. The bound in (16) implies that, A is positivedefinite, and thus, A is an inner product on C n (Ω). Let X denote the completion of ( C n (Ω),, A ) under the induced metric. The fact X L 2 (Ω) follows from (16). By design, each of D, S, and A is continuous on a dense subset of X, and thus each may be extended continuously to have domain X. The inner product, A is now defined on X and (X,, A ) is seen to be a Hilbert space. A solution to Du = 0 is found by minimizing the function { φ = (u, a) X R : a = 1 } 2 Du 2 H. 11

20 The directional derivative of φ, at u X and in the direction v X, is given by φ (u)v = Du, Dv H. By way of the Cauchy-Schwarz inequality and the trivial bound Dv H v A, v X, the inequality (18) φ (u)v Du H v A holds for all u, v X. Hence, for each u X, φ (u) : X R is a continuous linear functional. Applying Riesz representation theorem, for each u X there exists a unique element g X so that (19) φ (u)v = g, v A, v X. Define G : X X to map u X to the element g X that satisfies (19). Let I denote the identity map on X. The SDM for minimizing φ is given by selecting v X as an initial guess for a minimum of φ and defining the sequence of iterations u 0 = v and u k+1 = (I G)u k, k 0. The transformation (20) T = yields the limit of the SDM, when it exists. { } (u, v) X 2 : v = lim (I G) k u k 3.2. Convergence Theorems Let Q = {Au : u X} and note that Q is closed in H K since A : X H K is norm-preserving. Denote by P the orthogonal projection of H K onto Q. Lemma 3.1. (I G)u = A 1 P 0 H for all u X. Su 12

21 ( ) Du Proof. Fix u X. It is first shown that Gu = A 1 P. For an arbitrary v X, 0 K φ (u)v = Du, Dv H (21) and = Du, Dv H + 0 K, Sv K ( ) ( ) Du Dv =, 0 K Sv H K ( ) Du =, Av 0 K 0 K H K ( ) Du =, P Av = = P ( Du 0 K A 1 P ), Av ( Du 0 K H K H K ), v A (22) φ (u)v = Gu, v A. By equations (21) and (22), ( ) Du (23) Gu A 1 P, v 0 K A = 0, v X. ( ) Du Therefore, Gu A 1 P is the zero element of X, and thus 0 K ( ) Du Gu = A 1 P. ( ) Du Since u = A 1 Au = A 1 P Au = A 1 P then Su (( ) ( )) Du Du (I G)u = A 1 P = A 1 P Su 0 K 0 K (( )) 0H. Su The following theorem states that if conditions given by S allow for a solution to Du = 0 then a single iteration of steepest descent is sufficient for convergence to the solution. 13

22 Theorem 3.2. Fix v X. If there exists a solution to (24) Du = 0 H, Su = Sv, in X then the solution is given by u = (I G)v. Proof. Fix v X and let u = (I G)v. ( ) 0H Q, and thus Sv Using Lemma 3.1, Therefore, u satisfies (24). P ( ) 0H = Sv Au = A(I G)v = AA 1 P Since there exists a solution to (24) then ( ) 0H. Sv ( ) 0H = Sv ( ) 0H. Sv The following theorem and corollary provide a characterization of the limit of the SDM in the case where (24) has no solution. Theorem 3.3. T is the orthogonal projection of (X,, A ) onto ker(d). Proof. It is first shown that T has domain X and range ker(d). Fix u X. For each k N define u k = (I G) k u, a k = Du k H, b k = Su k K. An inequality is derived in order to show that (u k ) converges in X. Consider a fixed k N. Since u k+1 = (I G)u k then (25) u k+1 A = A(I G)u k H K. ( ) 0H Recall from Lemma 3.1 that A(I G)u k = P. Then equation (25) implies Su k ( (26) u k+1 A = P 0H. Su k) H K 14

23 ( Since P is a projection then P 0H Su k) H K inequality, equation (26) yields the inequality ( 0H Su k) H K = Su k K. Using the previous (27) u k+1 A Su k K. Square both sides of (27) and then expand the left-hand side to arrive at the desired inequality (28) Du k+1 2 H + Su k+1 2 K Su k 2 K. Inequality (28) implies b 2 k+1 b 2 k a 2 k+1 b 2 k, k N. We now have that (b k : k N) is a decreasing sequence bounded below by 0, and thus b def = lim k b k exists. Moreover, (b k ) is Cauchy in R and inequality (28) implies a def = lim a k lim b 2 k 1 b2 k = 0. k k Since a = 0 then Du k 0 H, and thus Gu k 0 X. Fix ɛ > 0. Choose N > 0 so that Gu k A < ɛ for k > N. Then u k+1 u k A = (I G)u k u k A = Gu k A < ɛ, k > N. This shows (u k ) to be Cauchy in X. Therefore, T has domain X. Moreover, DT u H = Since X is complete then T v def = lim k u k is in X. D lim u k = lim Du k H = a = 0, k H k and thus T has range in ker(d). A consequence of Theorem 3.2 is that (I G) k v = v for all k N whenever v ker(d), and thus (29) T v = v, v ker(d). Therefore, the range of T is ker(d). 15

24 For the remainder of the proof it is sufficient to show that T 2 = T and T op 1. The former follows immediately from (29). The latter is shown by recalling inequality (28) and noting that Sv K v A, v X. This implies that u k+1 A u k A, k N. Therefore, T u A = lim u k = lim u k A lim u A = u A. k A k k Since the previous inequality holds for the arbitrary u fixed at the beginning of the proof then T op 1 is shown. Therefore, T is an orthogonal projection. Since T had domain X and range ker(d) and is an orthogonal projection then T is the orthogonal projection of X onto ker(d). The following corollary is an immediate consequence of Theorem 3.3. Corollary 3.4. For each v X, T v is the unique function ker(d) so that (30) ST v Sv K < Sw Sv K, w ker(d), w T v. Proof. Fix v X. Since T is the projection of X onto ker(d) then T v is the closest element in ker(d) to v, with respect to the norm A. Note that for any w ker(d), w v 2 A = Dw Dv 2 H + Sw Sv 2 K = Dv 2 H + Sw Sv 2 K }{{}. constant By the previous identity, minimization of w v A is equivalent to minimization of Sw Sv K for w ker(d). Corollary (3.4) offers a distinct advantage over prior descent methods since the minimal distance property of the orthogonal projection T simplifies to a minimal distance in terms of S. This allows a user to characterize solutions to Du = 0 in terms of a variety of S operators. Such liberty offers a potentially valuable tool for experiments in the search for necessary and sufficient conditions for existence and uniqueness of solutions to Du = 0. Remark 3.5. To solve the inhomogeneous equation Du = f H one need only redefine the function φ by φ(u) = 1 Du 2 f 2 H, u X. The Hilbert space (X,, A ) is maintained and analogous convergence results hold. 16

25 3.3. Hilbert Space Example The L 2 norm in (16) is chosen so that the resulting function space X is well-defined and contained in a familiar function space, the square integrable functions. However, solutions to a differential equation often call for more desirable properties. This section considers an example problem for which the space X is equivalent to a standard Sobolev space. Let Ω = (0, 1) R. Define the first-order linear differential operator D by Du = u + pu, u C 1 (Ω) where p is continuous on Ω. The initial value operator S is defined by Su = u(0), u C 1 (Ω). It is shown that the function space X, which is generated by D and S as outlined in Section 3.1, is well-defined and is equivalent to the Sobolev space W 1,2 (Ω). Define Au = Du, u C 1 (Ω). Since A has a trivial kernel in C 1 (Ω) then the Su bilinear form, A, defined by u, v A = Au, Av L 2 (Ω), u, v C1 (Ω), is an inner product on C 1 (Ω). Let A denote the induced norm on (C 1 (Ω),, A ). One of the several choices for a norm on W 1,2 (Ω) is given by 1/2 u W 1,2 (Ω) ( u = 2 L 2 (Ω) + u 2 L (Ω)), u W 1,2 (Ω). 2 Lemma 3.6 allows one to define the problem-specific function space X. Lemma 3.6. A bounds W 1,2 (Ω) on C1 (Ω). Proof. Fix a nonzero u C 1 (Ω). Let f = Du, and note that where j u = (Su)e w + e w e w(s) f(s)ds w = j 0 0 p(s)ds. 17

26 Choose M > 1 as an upper bound of both e w and e w. The first bound is shown for u L 2 (Ω). For t Ω, t u(t) 2 = ((Su)e w(t) + e w(t) ( Su R M + M 2 f L 2 (Ω) ) 2M ( Su 4 2 R + Du 2 L 2 (Ω) = 2M 4 u 2 A. 0 ) 2 e w(s) f(s)ds ) 2 Therefore, 1 (31) u 2 L 2 (Ω) = u(t) 2 dt M 4 u 2 A dt = 2M 4 u 2 A. A bound on u L 2 (Ω) is now derived. Since u (t) 2 = (f(t) + u(t)) 2 2(f(t) 2 + u(t) 2 ) for all t Ω then 1 u 2 L 2 (Ω) = u (t) 2 dt f(t) 2 dt u(t) 2 dt 2 Du 2 L 2 (Ω) + 4M 4 u 2 A = 2 ( 1 + 2M 4) u 2 A. It follows from the previous bound and the bound in (31) that (32) u W 1,2 (Ω) 2(1 + 3M 4 ) u A. Let X denote the closure of C 1 (Ω) with respect to A. Since C 1 (Ω) is dense in W 1,2 (Ω) then Lemma 3.6 implies (33) X W 1,2 (Ω). 18

27 Proposition 3.7 completes this section by showing that X is in fact W 1,2 (Ω), and thus A generates W 1,2 (Ω). Proposition 3.7. X = W 1,2 (Ω). Proof. By way of (33) it is sufficient to show that W 1,2 (Ω) bounds A on C1 (Ω). Fix a nonzero u C 1 (Ω) and let M > 1 be an upper bound of p. Since Du L 2 (Ω) = u + pu L 2 (Ω) M ( u L 2 (Ω) + u L 2 (Ω) ) then Du 2 L 2 (Ω) M 2 ( u L 2 (Ω) + u L 2 (Ω) 2M ( u 2 2 L 2 (Ω) + u 2 L 2 (Ω) = 2M 2 u 2 W 1,2 (Ω). ) 2 ) Using the previous bound and the fact that Su 2 R u 2 sup 2 u 2 W 1,2 (Ω) we have that u A 2(1 + M 2 ) u W 1,2 (Ω). Since C 1 (Ω) is dense in X then W 1,2 (Ω) X. Therefore, X = W 1,2 (Ω). Remark 3.8. A deeper investigation is needed in order to understand condition-dependent Sobolev spaces. For example, if D is an ordinary differential operator with a leading coefficient p that vanishes on the domain of interest, is the Sobolev space for the SDM equivalent to one of the more common weighted Sobolev spaces defined in terms of p? Secondly, how might the resulting Sobolev spaces change with respect to the conditions operator S? 19

28 3.4. Method for a Nonlinear Equation An example of a nonlinear differential equation is considered in order to illustrate a variable metric analogue of the descent method. Let Ω = (0, 1) R and let W = W 1,2 (Ω). Define F (u) = u + u 2, u W, and Su = u(0), u W. Consider the problem of finding u W so that (34) F (u) = 0, Su = Sv, v W fixed. Note that (34) has a solution in W if and only if v(0) > 1. Define A(u) = F (u), u W. Note that the directional derivative of A, at Su u W and in the direction v W, is given by A (u)v = F (u)v = v + 2uv. Sv v(0) For each u W define the symmetric bilinear form, u by v, w u = A (u)v, A (u)w L 2 (Ω) R, v, w W. Since ker(a (u)) = {0}, u W, then, u is an inner product on W. Let u denote the induced norm on (W,, u ), u W. Lemma 3.9. For each u W, u bounds W on W. Proof of Lemma 3.9 follows from Lemma 3.6. By Lemma 3.9, one may define X u, u W, to be the completion of (W,, u ) and have the inclusion X u W. Lemma X u = W for each u W. 20

29 Proof of Lemma 3.10 follows from Proposition 3.7. Let X = W. Reference to X u X, u X, is made only to indicate the local inner product, u. Define φ(u) = 1 2 Du 2 L 2 (Ω), u X. Lemma For each u X, φ (u) is a continuous linear functional on X u. Proof. Fix u X. Note that φ (u)v = F (u), F (u)v L 2 (Ω), v X. Let M = F (u) L 2 (Ω) 0. For v X, φ (u)v 2 = F (u), F (u)v L 2 (Ω) 2 F (u) 2 L 2 (Ω) F (u)v 2 L 2 (Ω) ( ) F (u) 2 L 2 (Ω) F (u)v 2 L 2 (Ω) + Sv 2 R = M 2 A (u)v 2 L 2 (Ω) R = M 2 v 2 u. Therefore, φ (u)v M v u for all v X, and continuity is shown. Applying the Riesz representation theorem, the gradient operator G is defined to map u X to the unique Gu X satisfying φ (u)v = Gu, v u v X. The SDM for solving (34) is given by setting u 0 = v and defining u k+1 = u k h k Gu k, k = 0, 1,..., where h k is chosen to minimize φ(u k+1 ). One may also define h k = 1 for all k = 0, 1,... Convergence of the SDM is not shown, nor is it claimed. In a numerical setting, convergence to an approximate solution to (34) is observed if and only if v(0) > 1. 21

30 CHAPTER 4 APPLICATION TO FINITE PROBLEMS An analogue of the SDM is constructed for operators on a finite-dimensional space. The SDM is applied to discretizations of a singular ODE, the Laplace-Dirichlet problem, an overdetermined ODE with respect to the given conditions, and a nonlinear ODE. The dagger notation,, denotes the adjoint (resp. transpose) of a given operator (resp. matrix) Steepest Descent Method Define X = R m for some m N. Suppose both D : X R p, p N, and S : X R q, q N, are linear. Define A : X R p+q by Au = Du, u X. Consider the problem Su (35) Du = 0 and Su = Sv, v X. Assume that (36) ker(a) = {0 X} so that any solution to (35) is unique. For each d N the standard Euclidean inner product on R d is denoted by, d, i.e., u, v d = d u k v k, u = k=1 u 1. u d, v = v 1. v d Rd. The symmetric bilinear form, A is defined by u, v A = Au, Av p+q for every u, v X. By statement (36),, A defines an inner product on X. Since X is finite-dimensional then (X,, A ) is a Hilbert space. Define φ : X R by φ(u) = 1 2 Du 2 p, u X. Lemma 4.1. φ (u) is a continuous linear functional on X for each u X. 22

31 Proof of Lemma 4.1 follows from the Cauchy-Schwarz inequality. Lemma 4.1 allows one to define the descent gradient Gu, u X, to be the unique element in X so that φ (u)v = Gu, v A, v X. Theorem 4.2. Let Then Gu, u X, is found by solving B = D D + S S. BGu = D Du. Proof. Let B = D D + S S. Fix u X. Since φ (u)v = Du, Dv p = D Du, v m and φ (u)v = Gu, v A = DGu, Dv p + SGu, Sv q = D DGu, v m + S SGu, v m = ( D D + S S ) Gu, v m = BGu, v m for all u, v X, then BGu = D Du. Analogues of Theorem 3.2, Theorem 3.3, and Corollary 3.4 follow by similar arguments. Let I denote the identity on X. Define T = { } (u, v) X 2 : v = lim (I G) k u. k 23

32 The SDM allows one to solve a finite problem by way of the symmetric positivedefinite operator B. Immediate value of the SDM is seen in experimentation. Any solution to (35) is unique if and only if B is invertible. Equation (35) has a solution if and only if the first iteration satisfies (35). Moreover, an analysis of numerical stability offers insight into the limiting behavior of solutions to (35) as m. This can offer valuable insight concerning solutions to a linear differential equation and supplementary conditions which are approximated by solutions to (35). Application of this approach is given in Chapter 5. Another aspect of the SDM is the case where (35) is overdetermined for a given v X. An analysis of S(v T v) describes the S-change needed to find a solution to Du = 0. The extent in which (35) is overdetermined can be quantified and information on how to resolve the issue may be gained. This approach is illustrated in Section Singular Problem (37) This section concerns application of the SDM to discretizations of the singular ODE ( j 1 ) u u = 0 2 on Ω = [0, 1] where j denotes the identity map on R. If requiring a solution to be continuously differentiable then an initial condition is sufficient to ensure uniqueness of a solution. However, the finite-difference scheme which models the vanishing coefficient will require two conditions. The conditions (38) u(0) = 1 and u(1) = 1 are imposed. Allowing loss of continuity for the derivative at t = 0, the solution to (37) and (38) is given by 2t + 1, 0 t 1 u(t) = 2, 1 2t 1, 2 < t 1 2, Consider an odd integer m 3 for the following definitions concerning the finitedifference method. Let Ω m be a uniform m-point partition of Ω. X m denotes the collection of functions from X m to R. Elements in X m will be considered as elements in R m. Define 24

33 the restriction map P m : C 1 (Ω) X m by P m v = x where x(t) = v(t) for all t Ω m. Let δ = 1. For x = (x m 1 1,..., x m ) X m, define (x 1 1/2) + (x 2 1/2) 2 D m x =. (x m 1 1/2) + (x m 1/2) 2 x 2 x 1 δ x m x m 1 δ x 1 + x 2 2. x m 1 + x m 2, S m x = x 1. x m For a given v C (Ω) the approximation to (37) and (38) is given by Q m v = P m v ( D md m + S ms m ) 1 D m D m P m v. Table 4.1 list the accuracy results of w = Q m v for v(t) = cos(2πt), t Ω, and m = 10 2, 10 3, and The accuracy of D m w and w u are measured by both the supremum norm and the approximating L 2 norm. 1 m sup D m u (Dm u) 2 1 sup w u m 1 m (w u) Table 4.1. Accuracy of the SDM when applied to a singular ODE with endpoint conditions Laplace-Dirichlet Problem Define Du = u 11 + u 22 and Su = u Ω for u C 2 (Ω), Ω = (0, 1) 2. The Laplace- Dirichlet problem is given by Du = 0, Su = Sv, v C 2 (Ω). The SDM is applied to the following finite-difference scheme. Fix an integer m 3. Define the following. 25

34 Ω m is a uniform m-by-m grid of Ω and δ = 1 m 1. X m = R m R m R m2 is the collection of functions from Ω m to R. P m : C 2 (Ω) X m is the restriction map. For x = (x i,j ) i,j=1,...,m X m, define ( xi 1,j 2x i,j + x i+1,j D m x = δ 2 + x ) i,j 1 2x i,j + x i,j+1 R m 2 R m 2 δ 2 i,j=2,...,m 1 S m x = x Ω Ωm. Representing an element of x X m as a vector in R m2 and defining the corresponding matrix that approximates the Laplacian is tedious. Simply suppose a reshaping is done so that D m and S m may be considered as matrices and the transpose operator is applicable. For v C 2 (Ω), define Q m v = P m v ( ) D md m + S ms 1 m D m D m P m v. Define v(x, y) = cos(2πx) cos(2πy), (x, y) Ω. Accuracy measurements of w = Q m v follows for m = 50, 100, and 150. D 50 w sup = D 100 w sup = D 150 w sup = Overdetermined Problem Define Du = u u and S = u(0) u(1) While the problem for u C 1 (Ω), Ω = [0, 1]. Let A = D S. Du = 0, Su = Sv, v C 1 (Ω), is overdetermined for most v it was shown in Section 3.1 that gradient descent under the inner product u, v A = Au, Av L 2 (Ω) R 2 converges to a solution to Du = 0. 26

35 Consider a fixed integer m 3. Recall the following objects from Section 4.2: Ω m is a uniform m-point partition of Ω and δ = 1 m 1. X m = R m is the collection of functions from Ω m to R. P m : C 1 (Ω) X m is the restriction map. For x = (x 1,..., x m ) X m, define D m x = x 2 x 1 δ. x m x m 1 δ x 1 + x 2 2. x m 1 + x m 2, S m x = x 1. x m Fix v C 1 (Ω). The SDM is given by defining u 0 = P m v and for k = 1, 2,..., u k = u k 1 h k g k 1 where g k 1 = ( D md m + S ms m ) 1 D m D m u k 1 and h k > 0. The optimal choice for h k, i.e., the value h k which minimizes D m u k m 1, is given by (39) h k = D mu k 1, D m g k 1 m 1 D m g k 1 2 m 1 for each k. Consider v(t) = cos(2πt), t [0, 1]. Let u 0 = v and define the descent iterations u 1, u 2, u 3 by using optimal step sizes h 1, h 2, h 3 as defined in (39). The accuracy results of solving D m u = 0, measured by sup D m u k, are listed in Table 4.2 for m = 10 2, 10 3, 10 4, and k = 0, 1, 2, 3. 27

36 sup D m u k m k = 0 k = 1 k = 2 k = Table 4.2. Accuracy of the SDM when applied to an overdetermined problem Nonlinear Problem ( ) F (u) Define F (u) = u +u 2 and S = u(0) for u C 1 (Ω), Ω = [0, 1]. Define A(u) =, S u C 1 (Ω). The problem (40) F (u) = 0, Su = Sv, v C 1 (Ω), is examined. Consider a fixed integer m 3. Recall the following objects from Section 4.2: Ω m is a uniform m-point partition of Ω and δ = 1 m 1. X m = R m is the collection of functions from Ω m to R. P m : C 1 (Ω) X m is the restriction map. For u = (u 1,..., u m ) X m and x = (x 1,..., x m ) X m, define ( ) 2 x 2 x x1 1 + x 2 2 F m (x) = δ. x m x. ( m 1 xm 1 + x m δ 2 x 2 x 1 u 1 + u 2 Fmx u = δ. x m x 2 2. m 1 S m x = x 1 x m δ. u m 1 + u m 2 ) 2, x 1 + x 2 2 x m 1 + x m 2, 28

37 Fix v C 1 (Ω). The SDM is given by defining u 0 = P m v and for k = 1, 2,..., where u k = u k 1 g k 1 g k 1 = ( F u m F u m + S ms m ) 1 F u m F m (u k 1 ). Note that unit step size is used for each iteration. Let v(t) = cos(2πt), t Ω. The solution to (40) is u(t) = t, t Ω. Accuracy measurements of D m u k sup are given in Table 4.3 for m = 10 2, 10 3, 10 4 and k = 0, 1, 2, 3, 4. sup D m u k m k = 0 k = 1 k = 2 k = 3 k = Table 4.3. Accuracy of the SDM when applied to a nonlinear initial value problem. 29

38 CHAPTER 5 THE TRICOMI EQUATION Define J(x, y) = y, (x, y) R 2. The Tricomi equation is given by (41) Ju 11 + u 22 = 0. A finite-difference method is constructed in Section 5.1 to approximate solutions to (41) on Ω = (0, 1) ( 1/2, 1/2). Section 5.2 examines the common approach of imposing boundary conditions on (41) by investigating numerical stability for Tricomi-Dirichlet solutions, and evidence indicates that the method is unstable. The remainder of the chapter focuses on ladder conditions in the context of problem P of the introduction. Section 5.3 examines ladder conditions with respect to property (iii) of problem P. Numerical evidence indicates that the Tricomi-ladder method is stable and supports the claim that solutions lie in C 0 (Ω) W 1,2 (Ω). A numerical method for testing ladder conditions against property (iv) of problem P is defined in Section 5.4. Experiments are presented and support ladder conditions as satisfying property (iv). Section 5.4 also contains a theorem stating that ladder conditions satisfy properties (i) and (ii) of problem P Finite-difference Method Let Ω = (0, 1) ( 1/2, 1/2) R 2. A finite-difference method is defined on uniform partitions of Ω in order to approximate solutions to (41) which satisfy ladder conditions and Dirichlet conditions, respectively. Define Consider a fixed odd integer m 3 for the following definitions. Let δ m = 1/(m 1). x k = (k 1)δ m and y k = (k 1)δ m 1, k = 1,..., m. 2 The uniform m-by-m grid of Ω is given by G m = {(x i, y j ) : i, j = 1,..., m}. 30

39 C(G m ) will denote the collection of functions with domain G m and range R. For u C(G m ), u i,j will denote the value u(x i, y j ) for i, j = 1,..., m. The first-order differences D m,1, D m,2, the second-order differences D m,1,1, D m,2,2 and the finite Tricomi operator D m are defined to map from C(G m ) to R m 2 R m 2 as follows: For each u C(G m ) and each i, j {2,..., m 1}, (42) (43) (44) (45) (46) [D m,1 u] i,j = u i 1,j + u i+1,j 2δ m, [D m,2 u] i,j = u i,j 1 + u i,j+1 2δ m, [D m,1,1 u] i,j = u i 1,j 2u i,j + u i+1,j, δm 2 [D m,2,2 u] i,j = u i,j 1 2u i,j + u i,j+1, δm 2 [D m u] i,j = y j [D m,1,1 u] i,j + [D m,2,2 u] i,j. Note that m is chosen to be odd so that the coefficient y j in (46) vanishes for some j; specifically, y j = 0 for j = (m + 1)/2. The following operators are used to impose supplementary conditions. The Dirichlet operator S (1) m : C(G m ) R 4(m 1) is defined so that S (1) m u = S (1) m v if and only if u = v on G m Ω. Ladder conditions on Ω specify function values on K = { (x, y) Ω : x {0, 1} or y {0, 1/2} }, and thus the finite ladder operator S (2) m : C(G m ) R 4(m 1) is defined so that S (2) m u = S (2) m v if and only if u = v on G m K. Define P m : C 0 (R 2 ) C(G m ) so that for u C 0 (R 2 ), [P m u] i,j = u(x i, y j ) for i, j = 1,..., m. The Tricomi-Dirichlet and Tricomi-ladder problems are approximated by solving, respectively, (47) D m u = 0, S (1) m u = S (1) m P m v, v C (R 2 ), 31

40 and (48) D m u = 0, S (2) m u = S (2) m P m v, v C (R 2 ). For k = 1, 2, define A (k) m By way of Matlab, A (1) m by A (k) m u = D mu S (k) m u, u C(G m ). is known to be invertible for m = 3, 5,..., 201. The following lemma provides a maximum principle which is used to prove that each A (2) m, m 3 odd, is invertible. Note that Ω + def = {(x, y) Ω : y > 0} and Ω def = {(x, y) Ω : y < 0}. Lemma 5.1. Fix an odd m 3. If u C(G m ) satisfies D m u = 0 on G m Ω +, then max { u(x i, y j ) : (x i, y j ) G m Ω +} = max { u(x i, y j ) : (x i, y j ) G m Ω +}. Proof. Arguing by contradiction, suppose there exists indices a, b so that (x a, y b ) G m Ω + yields the maximum value of u and u(x a, y b ) is strictly greater than values of u on G m Ω +. Without loss of generality, suppose u(x a, y b ) > 0. Recall that u i,j denotes u(x i, y j ) for all i, j. It follows from D m u = 0 that u a,b = 1 2(1 + y b ) (y b(u a 1,b + u a+1,b ) + (u a,b 1 + u a,b+1 )). Now, if one of the neighboring values of u a,b is strictly less than u a,b then u a,b = < = 1 2(1 + y b ) (y b(u a 1,b + u a+1,b ) + (u a,b 1 + u a,b+1 )) 1 2(1 + y b ) (y b(2u a,b ) + (2u a,b )) 1 2(1 + y b ) 2(1 + y b)u a,b = u a,b Since u a,b < u a,b cannot hold then each neighboring value of u a,b must be as large as u a,b. Since u a,b is the global max then the neighboring value must be equal to u a,b. 32

41 By an inductive argument, this can be extended to the entirety G m Ω +, i.e., u is constant on G m Ω +. However, this contradicts the assumption that u a,b is strictly greater than all boundary values. Therefore, the global maximum must be given by a boundary value. Theorem 5.2. For each odd m 3, A (2) m is invertible. Proof. Fix m 3 to be odd. A (2) m u = 0. It is first shown that A (2) m has a trivial nullspace. Suppose u C(G m ) satisfies Then D m u = 0 and S (2) m u = 0. The latter implies that u = 0 on G m Ω +. By Lemma 5.1, u = 0 on G m Ω +. The zero values on G m Ω + are now extended uniquely to the remainder of G m. For 1 i 1 < i 2 m and j = 1,..., m, let R i1 :i 2,j = (u i1,j, u i1 +1,j,..., u i2 1,j, u i2,j). R i1 :i 2,j is simply a vector of the function values of u over the j-th row of G m and the i 1 -th through i 2 -th columns. It follows from D m u = 0 that R 2:m 1,j = 2R 2:m 1,j+1 R 2:m 1,j+2 y j (R 1:m 2,j 2R 2:m 1,j + R 3:m,j ) for every j = 1, 2,..., m 2. Consequently, if R 1:m,j+1 = 0 = R 1:m,j+2 then R 2:m 1,j = 0, j = 1, 2,..., m 2. Since u = 0 on G m Ω + then R 1:m,m 1 = 0 = R 1:m,m. Combining this with that fact that u i,j = 0 whenever j = 1 or j = m, which follows from S (2) m u = 0, an argument by induction shows that R 1:m,j = 0 for j = m 2, m 3,..., 2, 1. This shows that u vanishes on each row of G m. Therefore u is the zero function in C(G m ), showing A (2) m to have a trivial nullspace. 33

42 Invertibility now relies on showing dim ( Domain ( A (2) m )) ( ( )) = dim Range A (2) m. Since A (2) m has a trivial nullspace then (49) dim ( Domain ( A (2) m Since dim (Range (D m )) = (m 2) 2 and dim (50) dim ( Range ( A (2) m By inequalities (49) and (50), )) ( ( )) dim Range A (2) m. )) ( ( Range S (2) m = 4(m 1) then )) (m 2) 2 + 4(m 1) = m 2 = dim ( Domain ( )) A (2) m. dim ( Domain ( A (2) m )) ( ( )) = dim Range A (2) m, and thus A (2) m is invertible. For k = 1, 2, define Q (k) m (51) Q (k) m v = ( ( I : C (R 2 ) C(G m ) by D T md m + S (k) m T S (k) m ) ) 1 D T md m P m v, v C (R 2 ). Q (1) m v satisfies (47), m = 3, 5,..., 201, and Q (2) m v satisfies (48), m 3 odd. It is noted that for conditions derived from known solutions to (41) each condition type produces finite solutions that converge to the known solution. That is, for v satisfying (41) and k {1, 2}, P m v Q (k) m v sup eventually reaches machine precision in experiments. Examples of known solutions used in convergence experiments include v(x, y) = x 3 xy 3 and v(x, y) = 6x 2 y y The Tricomi-Dirichlet Problem Consider the Tricomi-Dirichlet problem (47) with boundary conditions derived from the function v defined by v(x, y) = y 2. Figure 5.1 shows Q (1) m v for m = 49 and m = 149. The two solutions share a hill-valley-hill shape in the hyperbolic region. However, there is increased variation between adjacent function values as the grid becomes finer. Most mesh sizes produce a similar shape in the hyperbolic region, but that is the extent of any noticeable 34

43 pattern in the finite solutions. On a local level, the behavior is unpredictable in that some solutions exhibit extreme disturbance when compared to the solutions on neighboring mesh sizes. For example, see Figure 5.2 which shows Q (1) m v for m = 17, 19, and 21. Second derivative approximations offer a clear illustration of disturbance in the hyperbolic region. Figure 5.3 which shows D m,2,2 Q (1) m v for m = 49 and m = 149. Note the extreme difference in the range of values, i.e., ( ) ( ) max D 149,2,2 Q (1) 149v > 10 max D 49,2,2 Q (1) 49 v. Figure 5.1. Solutions to the finite Tricomi-Dirichlet problem with boundary conditions given by v(x, y) = y 2, i.e., Q (1) m v for m = 49 and m = 149. Figure 5.2. Q (1) m v for m = 17, 19, and 21 and v(x, y) = y 2. Quantification of the stability or lack thereof, follows. Let E (1) m denote the inverse of A (1) m, m 3 odd. The method is said to be stable if there exists M > 0 which bounds the operator norm of E (1) m for all odd m 3. The standard Euclidean norm m 2 is used as the norm for both the domain and range of E (1) m since each is of dimension m 2. For each odd 35

44 ( ) Figure 5.3. D m,2,2 Q (1) m P m v for m = 49 and m = 149 and v(x, y) = y 2. m 3, the operator norm of E (1) m E (1) m = sup is defined by { E m (1) u m 2 u m 2 : u R m2, u 0 In order to test stability, define w i,j C (R 2 ), i, j Z, by w i,j (x, y) = sin(πix) cos(πjy) for all (x, y) R 2. Let W = {w i,j : i, j = 1, 2,..., 10}. Using W as a sample of functions to investigate the norm ratios, define the sample distribution β m (1) } β (1) m = { E (1) m P m w m 2 P m w m 2 : w W Figure 5.4 displays a standard boxplot for each distribution β (1) m, m = 15, 17,..., 125. The method is also tested by using a sample of random vectors. Denote by W m, m = 15, 17,..., 125, a collection of 100 m 2 -dimensional vectors randomly generated by Matlab from a uniform distribution of [ 1, 1], and let ω (1) m = { E (1) m w m 2 w m 2 : w W m } It is emphasized that W i and W j are independent for i j since the collections are not derived from a fixed collection of smooth functions. Boxplots of ω m, m = 15, 17,..., 125, are shown in Figure 5.9. Note that the vertical axes in Figure 5.9 are scaled by log 10 so that the ebb and flow trend in the data is noticeable. The behavior of β (1) 19 is extreme relative to β (1) 17 and β (1) 21 ; the.. by }. 36

45 same holds for the ω-distributions. Similar disturbances are present for larger m values. For { } example, estimates on the operator norms indicate that E 121 (1) 100 max E 119, (1) E 123 (1). 37

46 Figure 5.4. Boxplots for m vs. β (1) m and m vs. ω (1) m for m = 15, 17,..., 125. Note the log 10 scaling of the vertical axes. This illustrates the instability of the Tricomi-Dirichlet method. 38

47 Stability has been tested with various samples and the results consistently indicate extreme disturbances at the same mesh sizes, e.g., m = 19 and m = 121. These mesh sizes also produce finite solutions with extreme variation between neighboring values. Experimental results repeatedly indicate that the unstable behavior of the finite solutions depends on mesh width rather than boundary values. This is easily seen when viewing a plot of the norm ratios E m (1) v m 2 v m 2 for a fixed v R m2. The plot mimics those in Figure 5.9, and there is no indication of an eventual bound on the ratios. A subcollection of odd m values may produce a stable method, but finding such a collection is a number theory problem beyond the scope of this work The Tricomi-ladder Problem The following conjecture is tested in this section. Conjecture 5.3. For each v C (Ω) there exists a unique function u C 0 (Ω) W 1,2 (Ω) so that u = v on K = { { (x, y) Ω : x {0, 1} or y 0, 1 }} 2 and u satisfies (41). Consider the Tricomi-ladder problem with conditions again given by v(x, y) = y 2. Figure 5.5 shows Q (2) m v, m = 49 and m = 149. Figure 5.6 shows Q (2) m v, m = 17, 19, and 21, illustrating the behavior on successive mesh sizes. The poor behavior of the solutions under Dirichlet conditions is in no way present in solutions that follow from ladder conditions. The solutions appear to converge uniformly. Define the characteristic curves { η 0 = (x, y) Ω : y = ( } 3 x) 2 3, 2 η 1 = { (1 x, y) Ω : (x, y) η 0 }. Let C denote the open characteristic region with four edges given by the two lines y = 0 and y = 1/2 and the two characteristic curves η 0 and η 1. The plots in Figures 5.5 and 5.6 show an upward bend of the surface along η 1. The bend sharpens with a finer mesh, and 39

48 Figure 5.5. Tricomi-ladder solutions: Q (2) m v for m = 49 and m = 149 and v(x, y) = y 2. Figure 5.6. Tricomi-ladder solutions: Q (2) m v for m = 17, 19, and 21 and v(x, y) = y 2. one might expect the first derivative approximations D m,2 Q (2) m v to converge to a function that loses continuity along η 0 and η 1. Figure 5.7 provides an illustration of this when noting the steepening slope of D m,2 Q (2) m v, m = 49 and m = 149, along η 0 and η 1. However, the derivative approximations D m,2 Q (2) m v, m 3 odd, do appear to converge uniformly on C with continuity maintained across the parabolic line. Examining the second derivative approximations, see Figure 5.8, uniform convergence on C and continuity across the parabolic line is again a likely result. However, the behavior of D m,2,2 Q (2) m v, m 3 odd, shows an ever-growing oscillatory effect on the hyperbolic region outside of C. The amplitudes increase and the frequency decreases as m grows, with the maximum amplitude on each mesh observed near η 0 and η 1. Hence solutions in Conjecture 5.3 are limited to W 1,2 (Ω). The stability of the Tricomi-ladder method is tested by using the same samples W 40

49 ( ) Figure 5.7. D m,2 Q (2) m P m v for m = 49 and m = 149 and v(x, y) = y 2. ( ) Figure 5.8. D m,2,2 Q (2) m P m v for m = 49 and m = 149 and v(x, y) = y 2. ( and {W m : m = 15, 17,..., 125}. For each odd m 3, let E m (2) = β (2) m = { E (2) m P m w m 2 P m w m 2 : w W }. A (2) m ) 1 and define The boxplots of each β (2) m, m = 15, 17,..., 125, are shown in Figure 5.9. ω (2) m, m = 15, 17, 125, is defined by ω (2) m = { E (2) m w m 2 w m 2 : w W m }, and the boxplots of each ω (2) m are shown in Figure