Methods for Parabolic Equations

Transcription

1 Maximum Norm Regularity of Implicit Difference Metods for Parabolic Equations by Micael Pruitt Department of Matematics Duke University Date: Approved: J. Tomas Beale, Supervisor William K. Allard Anita T. Layton Micael C. Reed Dissertation submitted in partial fulfillment of te requirements for te degree of Doctor of Pilosopy in te Department of Matematics in te Graduate Scool of Duke University 2011

2 Abstract (Matematics) Maximum Norm Regularity of Implicit Difference Metods for Parabolic Equations by Micael Pruitt Department of Matematics Duke University Date: Approved: J. Tomas Beale, Supervisor William K. Allard Anita T. Layton Micael C. Reed An abstract of a dissertation submitted in partial fulfillment of te requirements for te degree of Doctor of Pilosopy in te Department of Matematics in te Graduate Scool of Duke University 2011

3 Copyrigt c 2011 by Micael Pruitt All rigts reserved except te rigts granted by te Creative Commons Attribution-Noncommercial Licence

4 Abstract We prove maximum norm regularity properties of L-stable finite difference metods for linear-second order parabolic equations wit coefficients independent of time, valid for large time steps. Tese results are almost sarp; te regularity property for first differences of te numerical solution is of te same form as tat of te continuous problem, and te regularity property for second differences is te same as te continuous problem except for logaritmic factors. Tis generalizes results proved by Beale valid for te constant-coefficient diffusion equation, and is in te spirit of work by Aronson, Widlund and Tomeé. To prove maximum norm regularity properties for te omogeneous problem, we introduce a semi-discrete problem (discrete in space, continuous in time). We estimate te semi-discrete evolution operator and its spatial differences on a sector of te complex plan by constructing a fundamental solution. Te semidiscrete fundamental solution is obtained from te fundamental solution to te frozen coefficient problem by adding a correction term found troug an iterative process. From te bounds obtained on te evolution operator and its spatial differences, we find bounds on te resolvent of te discrete elliptic operator and its differences troug te Laplace transform representation of te resolvent. Using te resolvent estimates and te assumed stability properties of te time-stepping metod in te Caucy integral representation of te fully discrete solution operator yields te omogeneous regularity result. iv

5 Maximum norm regularity results for te inomogeneous problem follow from te omogeneous results using Duamel s principle. Te results for te inomogeneous problem imply tat wen te time step is taken proportional to te grid spacing, te rate of convergence of te numerical solution and its differences is controlled by te maximum norm of te local truncation error. As an application of te teory, we prove almost sarp maximum norm resolvent estimates for divergence form elliptic operators on spatially periodic grid functions. Suc operators are invertible, wit inverses and teir first differences bounded in maximum norm, uniformly in te grid spacing. Second differences of te inverse operator are bounded except for logaritmic factors. v

6 Contents Abstract List of Figures List of Symbols Acknowledgements iv viii ix x 1 Introduction 1 2 Preliminaries and Parabolic Regularity Results Preliminaries Parabolic Regularity Results Parabolic Convergence Result Te Semi-Discrete Problem Te Semi-Discrete Problem and its Fundamental Solution Te Frozen Coefficient Fundamental Solution Two Lemmas Constructing Φ Bounding te Fundamental Solution Resolvent Estimates and Fully Discrete Parabolic Regularity Resolvent Estimates for A Fully Discrete Regularity vi

7 5 Divergence Form Elliptic Operators on Spatially Periodic Grid Functions 68 6 Numerical Results Homogeneous Results Inomogeneous Results A Proof of Teorem B Decay in te Semidiscrete Fundamental Solution 103 Bibliograpy 105 Biograpy 107 vii

8 List of Figures 6.1 Implicit Euler: omogeneous first difference TGA: omogeneous first difference Crank-Nicolson: omogeneous first difference Implicit Euler: omogeneous second difference TGA: omogeneous second difference Crank-Nicolson: omogeneous second difference Implicit Euler: omogeneous first difference, one time step comparison TGA: omogeneous first difference, one time step comparison Implicit Euler: omogeneous second difference, one time step comparison TGA: omogeneous second difference, one time step comparison Implicit Euler: inomogeneous first difference TGA: inomogeneous first difference Crank-Nicolson: inomogeneous first difference Implicit Euler: inomogeneous second difference TGA: inomogeneous second difference Crank-Nicolson: inomogeneous second difference viii

9 List of Symbols Symbols R d Te set of all grid points wit grid spacing. L (R d ) Te set of bounded (possibly complex-valued) grid functions. e j Te jt standard basis vector for R d. ˆf Te discrete Fourier transform of f. S + j S j Te forward sift operator in te x j direction. Te backward sift operator in te x j direction. S γ Te composition of sift operators ( S + 1 ) γ1 (S + d ) γd. D + j D j Te forward (divided) difference in te x j direction. Te backward (divided) difference in te x j direction. D γ Te composition of difference operators ( ) D 1 + γ1 (D ) + γd d. n Te floor function; te greatest integer less tan or equal to n. ix

10 Acknowledgements I would like to tank my supervisor, J. Tomas Beale, for te valuable suggestions and encouragement trougout te writing of tis dissertation. Tis researc was supported in part by te National Science Foundation Grant DMS x

11 1 Introduction Parabolic partial differential equations possess te caracteristic property tat solutions to te Caucy problem exibit greater spatial regularity tan te initial data in L p or Sobolev norms. For instance, solutions to te linear parabolic problem u t = Au (1.1) u(x, 0) = u 0 (x) (1.2) on R d [0, T ) satisfy regularity estimates suc as D γ u(x, ) L (R d ) C T t γ /2 u 0 L (R d ), γ 2 (1.3) wen A is a second-order uniformly elliptic operator wit smoot bounded coefficients and continuous u 0. Likewise, for te inomogeneous problem u t = Au + f (1.4) u(x, 0) = 0 (1.5) wit Hölder continuous f, we ave D γ u L (R d ) C T sup f(x, t) L (R d ) γ 2. (1.6) t T 1

12 It is desirable tat numerical metods for solving parabolic problems reflect suc qualitative beavior, particularly in strong norms suc as te maximum norm, wic give control over te exact solution by truncation errors. Unfortunately, explicit finite difference metods require tat te time step k and spatial grid spacing satisfy k = c 2 for some fixed constant c, a significant constraint for practical computation. Wit implicit metods, it is often possible to allow large time steps, suc as k = c, and it is profitable to know if any regularity properties are retained. Early work on maximum norm estimates was done by Aronson (1963) and Widlund (1966), wo proved stability and regularity properties under te small time step restriction k = c 2. In tis dissertation, we sow tat certain implicit L-stable metods possess almost sarp maximum norm discrete regularity properties similar to (1.3) and (1.6), valid for large time steps. We derive discrete regularity estimates analogous to te continuous case for first spatial differences, and for second spatial differences wit te exception of a log factor. Our results extend tose tat Beale (2009) proved for te constant-coefficient diffusion equation to a more general class of parabolic equations wit time-independent coefficients. From te inomogeneous regularity teorem, we derive a convergence result for te inomogeneous problem. For a sufficiently smoot solution to te exact problem, te rate of convergence of te solution and its first differences is controlled by te maximum norm of te local truncation error. Te rate of convergence of second differences is also controlled by te maximum norm of te local truncation error, but contains logaritmic factors in te rate of convergence. As a consequence, if te local truncation error of te sceme is O( 2 +k 2 ), and k = c, ten te rate of convergence of te numerical solution and its first difference to te true solution and its derivative is O( 2 ), and te rate of convergence of second differences is O( 2 log 2 ). Beale (2009) sowed tat te class of time-stepping metods tat satisfy te 2

13 ypoteses of our regularity teorems include second-order metods suc as te modified form of Crank-Nicolson due to Twizell et al. (1996), known as TGA, a second-order singly diagonally implicit Runge-Kutta metod (SDIRK2), as well as te well-known first-order implicit Euler metod. Our results also apply to multi-step metods, including te second-order backward difference formula (BDF2). We examine te problem of obtaining maximum norm discrete regularity properties from te perspective of analytic semigroup teory, influenced by te approac to semi-discrete finite element problems taken by Tomée (2006). Our argument proceeds in tree stages. We begin by introducing te semi-discrete problem (discrete in space, continuous in time) u t = A u (1.7) by replacing te elliptic spatial operator by an elliptic difference operator tat depends upon te grid spacing as a parameter. We construct a solution to te semi-discrete problem to obtain maximum norm bounds on D γ eat, valid for complex t in a wedge about te real axis. In te second stage of our argument, we use analytic semigroup teory to transform our maximum norm bounds on te evolution operator into maximum norm bounds on spatial differences of te resolvent operator (z A ) 1. In te final stage, we use te resolvent estimates in conjunction wit te L-stability assumption on te time-stepping metod to obtain maximum norm regularity bounds on te fully discrete solution. To find maximum norm bounds on D γ eat for complex t in a sector about te real axis, we solve te semi-discrete problem by constructing a semi-discrete fundamental solution. Te fundamental solution is obtained troug a parametrix construction. Aronson (1963) and Widlund (1966) used a fully discrete parametrix construction for proving stability of difference metods for parabolic systems wit small time steps. Friedman (1964) provides a valuable introduction to te tecnique for te 3

14 exact problem. Our semi-discrete version of te parametrix requires some additional subtlety due to its incorporation of complex time. In te parametrix construction, te fundamental solution Γ of te semi-discrete problem is expressed as te sum of te fundamental solution G of te frozen coefficient problem and a correction term Φ. Because G solves a constant-coefficient problem, we can estimate G by examining its Fourier transform. Deforming te contour of integration for te inverse transform into te complex plane allows us to find pointwise bounds on G and its spatial differences. Tis tecnique can be used to sow tat G exibits exponential decay like exp( x / t ). (We migt expect exponential decay like e x 2 /t in analogy wit te eat kernel, but in Appendix B we explain wy tis cannot be obtained.) Te correction term Φ must satisfy an integral equation involving G. Te integral equation for Φ can be solved by an infinite series expansion. Eac term of te series as a bound less singular for small time tan its predecessor. Te first finitely many terms of te series may ave bounds tat are singular in time, altoug eac is less singular tan te bound for G. Te remainder of te terms in te series exibit increasing temporal regularity, and possess rapidly decaying coefficients. Every term in te expansion exibits a uniform rate of decay in exp( x / t ). Tese facts enable us to sow tat te series for Φ converges, and as better temporal regularity tan G. Te pointwise bounds on G and Φ lead directly to pointwise bounds on D γ Γ. Tese are easily leveraged to find maximum norm bounds on D γ eat. Adapting Beale s approac in Beale (2009) for te second part of our argument, a tecnique from analytic semigroup teory now allows us to obtain maximum resolvent estimates troug te Laplace transform representation of te resolvent: D γ (z A ) 1 = 4 0 e zt D γ ea t dt.

15 For tis step, it is critical tat our estimates on D γ eat be valid on a wedge containing te positive real axis so tat we can extend our resolvent estimates to a large enoug portion of te complex plane. Wit maximum norm estimates on D γ (z A ) 1 in and, we are finally able to examine te regularity of fully discrete scemes. Te simplest time-stepping metod for wic our regularity result olds for te omogeneous equation is te familiar L-stable implicit Euler metod. For te implicit Euler metod wit time step k, we approximate te solution of te ODE y t = λy at time nk by y n = (1 kλ) 1 y n 1 = s(kλ)y n 1, were s(kλ) = (1 kλ) 1 is te time-stepping function. In a similar fasion, we approximate te solution to te exact problem (1.3) by u n = s(ka ) n u 0. More generally, for a multi-step metod, we ave operators s n (ka ) for wic u n = s n (ka )u 0. For L-stable metods we can write te operator s n (ka ) or its spatial differences as a contour integral: D γ s n(ka ) = s n (z)d γ (z ka ) 1 dz Γ for Γ a contour originating at e iθ 0 and ending at e iθ 0, for θ 0 (π/2, π), enclosing te spectrum of A. Using our maximum norm resolvent estimates in tis representation of te fully discrete solution gives regularity results for te fully discrete problem. 5

16 Te maximum norm fully discrete regularity results for te inomogeneous problem and te resolvent estimates on te elliptic operator can ten be used to obtain regularity results for te inomogeneous problem. As an application, we derive improved maximum norm resolvent estimates for discrete divergence form elliptic operators on te space of periodic grid functions of mean value zero. Te results in tis section apply to te popular second-order accurate discretization for mixed derivatives found in Samarskii (2001). By restricting our attention to periodic grid functions of mean value zero, we ensure tat te elliptic operators are invertible. For suc an elliptic operator A, we sow tat (A ) 1 and D (A ) 1 ave maximum norm uniformly bounded in, and second differences of (A ) 1 are uniformly bounded except for logaritmic factors. Te key to discovering tese maximum norm estimates is to write D γ (A ) 1 = = D γ ea t dt D γ ea t dt + 1 D γ ea t dt and estimate eac term separately. Te first integral can be andled by our previous semi-discrete results. Te second integral requires more care. We sow tat te maximum norm of D γ eat is controlled by te H m norm of A m/2 e At, wic decays exponentially. Tis requires adapting te semigroup teory of Renardy and Rogers (2004) and te elliptic regularity of Evans (1998). More sopisticated discrete elliptic regularity results ave been sown by Tomée and Westergren (1968), Sreve (1973) and Bondesson (1973) for operators wit smoot coefficients on bounded domains. Our assumption of periodic data wit mean value zero simplifies te regularity teory ere significantly, allowing us to reduce te regularity of te coefficients. To obtain te elliptic resolvent estimates, we make use of te discrete Poincaré inequality and discrete Sobolev inequality. Te discrete Poincaré inequality (also 6

17 known in te literature as Wirtinger s inequality) first appeared in Scoenberg (1950). Te discrete Sobolev inequality was proved originally by Sobolev (1940), and is stated in Sreve (1973). In Capter 2, we state preliminaries and present our main results. In Capter 3 we discuss te semi-discrete problem. It is ere tat we construct te fundamental solution and prove te maximum norm regularity of te semi-discrete evolution operator. In te first section of Capter 4, we use te results of Capter 3 to obtain maximum norm resolvent estimates on discrete elliptic operator. In te second section of Capter 4, we apply our resolvent estimates to obtain fully discrete maximum norm regularity results. In Capter 5 we apply te teory to obtain almost sarp maximum norm resolvent estimates for divergence form discrete elliptic operators on periodic grids. In Capter 6 we present experimental numerical results confirming te main results of Capter 2, and compare te superior regularity properties of certain L-stable metods wit te weaker regularity of te Crank-Nicolson metod. Appendix A is te proof of te convergence result. In Appendix B, we explain te decay property of te semi-discrete fundamental solution. 7

18 Preliminaries and Parabolic Regularity Results 2 We state maximum norm regularity properties of L-stable implicit finite difference scemes valid for large time steps for second-order parabolic equations wit timeindependent coefficients posed on R d [0, ). We are interested finite difference metods for te Caucy problem u t = Au + f A = jl a jl (x) j l + j b j (x) j + c 0 (x) (2.1) u(x, 0) = u 0 (x). Here te operator A satisfies te uniform ellipticity condition a jl (x)ξ j ξ l c 0 ξ 2 (2.2) jl for all ξ R d, wit c 0 independent of x. Furtermore, te coefficients a jl, b j and c must be uniformly bounded and uniformly Hölder continuous. 8

19 2.1 Preliminaries Te discretized problem will be posed on te spatial grids R d = Z d = { x = j : j Z d}, (2.3) were 0 < 1 is te grid spacing. We define te Banac space L (R d { } ): L (R d ) = u(x) : sup u(x) < x R d (2.4) were u(x) is a complex-valued function defined on R d. Te norm for L (R d ) is given by u L (R d ) = sup u(x). (2.5) x R d We will suppress te subscript wen tere is no ambiguity. For an operator L wit domain and range R d, we define te norm Lx L = sup x R x. (2.6) d x 0 We need a discrete Fourier transform in order to express te ellipticity requirement for discrete operators. Te discrete Fourier transform ˆf of te grid function f is defined as ˆf(ξ) = x R d f(x)e i<x,ξ>/. (2.7) Te inverse transform is ten given by f(x) = (2π) d [ π,π] d ˆf(ξ)e i<x,ξ>/ dξ. (2.8) Te symbol of a difference operator is obtained by replacing te sift operator u(x) u(x + β) by e i<β,ξ>. For instance, te symbol of D + j, te forward difference in te x j direction, is (e iξ j 1)/. For a multi-index γ, we define te difference operator D γ = (D+ 1 ) γ1 (D + d )γ d. We denote te symbol of D γ by D γ. 9

20 2.2 Parabolic Regularity Results We select a consistent discretization A of A given by A = jl,σ a jl,σ (x, )S σ D + j D+ l + j,σ b j,σ (x, )S σ D + j + σ c σ (x, )S σ (2.9) were (0, 1] is te grid spacing, x R d are grid points, σ lies in a finite subset of Z d, and a jl,σ, b j,σ and c σ are real-valued functions defined for all x and 0 1. Te coefficients a jl,σ, b j,σ and c σ must be uniformly bounded and must satisfy a uniform Hölder continuity condition suc as a jl,σ (x, ) a jl,σ (y, ) C x y α, (2.10) wit C independent of. We define te sift operators S σ = (S+ 1 ) σ1 (S + d )σ d, were S + j u = u(x + e j) is te forward sift operator in te x j coordinate direction, and (S + j ) 1 u(x) = S j u(x) = u(x e j) is te backward sift operator in te x j direction. We note tat te coefficients a jl,σ (x, ) appearing in te discretization (2.9) do not ave to be te same as te coefficients a jl (x) appearing in (2.1). For eac fixed y, we define te principal symbol p (y, ξ) to be te symbol associated wit te difference operator a jl,σ (y, )SD σ + j D+ l. jl,σ We require tat p (y, ξ) satisfy te uniform ellipticity condition Re { 2 p (y, ξ) } c ξ 2 (2.11) wit constant c independent of y and. For completeness, we also define te full symbol P (y, ξ) to be te symbol associated wit A (y). Hereafter, A operates on te x variable alone of a function of x and y, regarding te y variable as fixed. Te discretizations permitted by (2.9) range from te simple discretization a jl (x)d + j D l + b j (x)dj 0 + c(x) j jl 10

21 to te divergence form discretization appearing in (5.6), provided te a jl are at least C 1+α. We note tat te ellipticity condition ere is stronger tan te uniform ellipticity condition for te exact problem, as not all consistent discretizations satisfy it. For instance, if we were to replace eac derivative in te exact problem by a centered difference, te resulting sceme would fail to satisfy (2.11) as te principal symbol would be zero for te non-zero vector πe j. However, as we will sow in Capter 5, if we replace j by D + j and l by D l, te resulting sceme does satisfy (2.11). Te time discretization wit time step k and u n = u(, nk) is implemented by u n = s n (ka )u 0 (2.12) were s n (kλ) is te time-stepping function for solving numerically te ordinary differential equation y t = λy. For some constant k 0, wen 0 < k k 0, we ave tat s n (ka ) is well-defined as a bounded operator on L (R d ). For single step metods, we can write s n (ka ) = s(ka ) n. We restrict our attention to te class of A-stable and L-stable time-stepping metods, i.e. s n (z) 1, s n ( ) = 0 (2.13) on te left alf-plane. We require tat tere exist a disk B 0 about te origin on wic s n (z) C 0 (1 + c 0 z ) n. (2.14) Furtermore, for eac δ > 0, we must ave an estimate of te form s n (z) C 1 (1 + c 1 z ) ρn (2.15) for some positive constants p, C 1 and c 1, wit constants depending on δ, for all z Σ δ, were Σ δ = {z = z 1 + iz 2 : z 1 0, z 2 δ z 1 }. (2.16) 11

22 Teorem 1. Under te assumptions (2.13)-(2.15) on s n, tere exist a constant k 0 and constants C 0, C 1, C 2 and C 3, independent of 0 < 1 and 0 < k k 0, for wic s n (ka ) C 1 e C 0nk (2.17) D γ s n(ka ) C 2 (nk) 1/2 e C 0nk, γ = 1 (2.18) D γ s n(ka ) C 3 (nk) 1 (1 + log + log nk )e C 0nk, γ = 2. (2.19) A large portion of tis dissertation is te proof of Teorem 1. From Teorem 1, we can deduce te following result, te proof of wic is a simple modification of te proof of Teorem 1.2 in Beale (2009): Teorem 2. For te coice of single-step metod time-stepping function s satisfying te constraints in (2.13)-(2.15), if te problem u t = A u + f (2.20) u(x, 0) = 0 (2.21) is approximated by u n+1 = s(ka )u n + k m q i (ka )(1 η i ka ) 1 f(, nk + τ i k) (2.22) i=1 were k = c for some c > 0, η i > 0 and τ i are fixed numbers and q i is an analytic function on Σ δ for wic q i (ka ) is bounded in norm on L (R d ) independently of and k for k sufficiently small, ten for 0 < nk T we ave u n C 0 sup f(, t) (2.23) t T D γ un C 1 sup f(, t), γ = 1 (2.24) t T D γ un C 2 (1 + log 2 ) sup f(, t), γ = 2. (2.25) t T for constants C 0, C 1 and C 2 depending on c and T but not on or n. 12

23 Tis result may also be extended to multi-step metods. See Beale (2009) for te extension to BDF Parabolic Convergence Result As an application of Teorem 2, we derive a convergence result for te inomogeneous problem. We suppose tat U is a classical solution to u t = Au + f A = jl a jl (x) j l + j b j (x) j + c(x) (2.26) u(x, 0) = u 0 (x) for continuous u 0. We examine discrete scemes consistent wit (2.26) given by te discretization m u n+1 = s(ka )u n + k q i (ka )(1 η i ka ) 1 f(, nk + τ i k) (2.27) i=1 u 0 = u 0 for a rational time-stepping function s(z) = q(z)/r(z) and rational q i. We require tat q i (ka ) be bounded independently of and k for k sufficiently small. We also require tat s be L-stable, so tat s( ) = 0 implies tat te degree of r must be strictly greater tan tat of q. Te consistency of te sceme enables us to take te polynomials q(z), r(z) = 1 + O(z) as z 0. To state our convergence result, we must define te local truncation error of te sceme. To do tis, we re-express te sceme in a form tat directly approximates te exact equation. Multiplying te sceme by r(ka ), we may rewrite it as m r(ka )u n+1 = q(ka )u n + k q i(ka )f(, nk + τ i k) 13 i=1

24 for rational functions functions q i(z) = r(z)q i (z)(1 η i z) 1. Using te fact tat q(z), r(z) = 1 + O(z) as z 0, we may rewrite te sceme as m u n+1 u n = (1 r(ka ))u n+1 + (q(ka ) 1)u n + k q i(ka )f(, nk + τ i k) for polynomials 1 r(z) and q(z) 1 aving no constant term. Dividing by k yields i=1 a sceme in te classical formulation u n+1 u n = A (u n+1, u n ) + k m q i(ka )f(, nk + τ i k), (2.28) i=1 were A (u n+1, u n ) = k 1 (1 r(ka ))u n+1 + k 1 (q(ka ) 1)u n discretizes A and m i=1 q i(ka )f(, nk + τ i k) discretizes f. For conditions on te functions q i tat guarantee consistency and a procedure for generating q i for practical computation, see Capter 8 of Tomée (2006). Having re-expressed te sceme in (2.28), te local truncation error T is defined in te standard way as te quantity satisfying U n+1 U n m = A (U n+1, U n ) + q k i(ka )f(, nk + τ i k) + T n. (2.29) Te total error E is defined by i=1 E n = U n u n. (2.30) Starting from (2.29) and reversing te steps used to obtain (2.28) from (2.27) enables us to write U n+1 = s(ka )U n + k m q i (ka )(1 η i ka ) 1 f(, nk + τ i k) (2.31) i=1 + kq(ka )(1 ηka ) 1 T n for Q(z) = (1 ηz)/r(z) wit η a positive constant. Q(kA ) is uniformly bounded in and k for k sufficiently small as te degree of r is at least one and r as no roots in te left alf-plane. 14

25 We can now state our convergence result, te proof of wic appears in Appendix A. Altoug tis result requires te more restrictive ypotesis tat s be a rational function, te teorem still applies to implicit Euler, TGA, and SDIRK2. Teorem 3. Suppose U is a classical solution to (2.26) and u n is te numerical solution at time step n given by te L-stable sceme in (2.27) for rational s wit k = c. If E is te total error and T is te local truncation error, ten, on any finite interval [0, T ], we ave: E C 0 sup T (, t) (2.32) t T D γ E C 1 sup T (, t), γ = 1 (2.33) t T D γ E C ( ) log 2 sup T (, t) γ = 2 (2.34) t T wit constants depending on c and T, but not on or n for k sufficiently small. 15

26 3 Te Semi-Discrete Problem 3.1 Te Semi-Discrete Problem and its Fundamental Solution To bound D γ eat as an operator on L (R d ), we introduce te semi-discrete problem. Te semi-discrete initial value problem for a grid function u is given by ( L u = A ) u = 0 (3.1) t u(x, 0) = u 0 (x) for A in (2.9). For simplicity of exposition, in tis capter we suppose tat A = jl a jl (x)d 2 jl + j b j (x)d 1 j + c(x), (3.2) were D 2 jl is a consistent discretization of j l and D 1 j is a consistent discretization of j. We may express D 2 jl and D1 j as finite sums of sift operators: D 2 jlu = 2 σ w σ u(x + σ) (3.3) D 1 j u = 1 σ w σ u(x + σ) (3.4) 16

27 for σ in a finite subset of Z d. Te proofs for general A as defined in (2.9) are straigtforward modifications of tose presented in tis capter. For eac, A is a bounded operator on L (R d ) (wose bound depends on ), so tat e A t is well-defined as a bounded operator for t C. We find tat e A t is more well-beaved and, along wit its spatial differences, can be bounded uniformly in. Teorem 4. Tere exists a constant M > 0 and constants C and µ, depending on M but not on, for wic D γ ea t L (R d ) C t γ /2 e µt, γ 2 (3.5) for all t in te wedge T M = {t = t 1 + it 2 : t 2 M t 1, t 1 > 0}. (3.6) Furtermore, if te principal symbol of te difference sceme is real wenever ξ is real, M may be taken to be any positive real number. To prove Teorem 4, we construct a fundamental solution Γ (x, t; y) for (3.1) satisfying L Γ = Γ (x, 0; y) = δ xy (3.7) ( A ) Γ = 0 (t T M ) t were x, y R d are grid points, and δ xy = { 1 x = y 0 x y. (3.8) For suc a fundamental solution, te solution of (3.1) may be written u(x, t) = x Γ (x, t; x )u 0 (x ). (3.9) 17

28 Te construction of Γ (x, t; y) is in te spirit of Levi s parametrix. In te parametrix construction, te fundamental solution is realized as a correction applied to te fundamental solution of te constant-coefficient problem obtained by freezing te coefficients of L at te grid point y. Te frozen coefficient problem wit coefficients eld constant at y is u t = A (y)u (3.10) u(x, 0) = u 0 (x). Associated wit te frozen coefficient problem is its fundamental solution G (x, t; y) satisfying L (y)g = G (x, 0; y) = δ x,0 (3.11) ( A (y) ) G = 0 (t T M ) (3.12) t for wic te solution u(x, t) to (3.10) can be expressed by u(x, t) = x G (x x, t; y)u 0 (x ). (3.13) We now build Γ (x, t; y) as a perturbation of G, writing [ ] t Γ (x, t; y) = G (x y, t; y) + G (x x, t s; x )Φ (x, s; y) ds, (3.14) 0 x wic expresses Γ as te sum of te solution of te frozen coefficient problem and a corrective term depending on a function Φ (x, t; y), to be determined. From te requirement tat L Γ = 0 in (3.7) we can derive an integral equation for Φ (x, t; y). 18

29 We ave tat 0 = L Γ = L G (x y, t; y) + = L G (x y, t; y) + ( A ) t t t 0 0 G (x x, t s; x )Φ (x, s; y) ds x A G (x x, t s; x )Φ (x, s; y) ds x x G (x x, 0; x )Φ (x, t; y) t 0 x = L G (x y, t; y) Φ (x, t; y) + t 0 t G (x x, t s; x )Φ (x, s; y) ds L G (x x, t s; x )Φ (x, s; y) ds. x In te final step of te computation we ave used te fact tat G (x x, 0; x ) = δ x,x. Tis yields te integral equation for Φ : [ ] t Φ (x, t; y) = L G (x y, t; y) + L G (x x, t s; x )Φ (x, s; y) ds. 0 x (3.15) We can solve (3.15) by seeking Φ (x, t; y) in te form Φ (x, t; y) = Φ (m) (x, t; y), (3.16) m=0 were and Φ (0) (x, t; y) = L G (x y, t; y) (3.17) [ ] t Φ (m) (x, t; y) = Φ (0) (x, t s; x )Φ (m 1) (x, s; y) ds. (3.18) 0 x To establis te existence of Φ, we require pointwise bounds on Φ (m). 19

30 3.2 Te Frozen Coefficient Fundamental Solution As te construction and estimation of Φ depends eavily on te frozen coefficient fundamental solution, we turn our attention to G next. It is profitable to examine G troug te Fourier transform, were we can exploit te analyticity properties of te symbol to find spatial decay. We begin wit a bound on te symbol of A (y). Teorem 5. Let 0 < B < 1. Tere exists a positive constant M and constants c, κ and ω depending on M and B, but not on or y, for wic Re {P (y, ξ + iβ)t} c ξ 2 t t + κ β 2 + ω t (3.19) 2 2 for all (ξ + iβ) S B, all t T M and all grid points y, were S B = { (ξ + iβ) C d : ξ j π, β j B }. (3.20) Furtermore, if p (y, ξ) is real wenever ξ is real, M > 0 may be taken arbitrarily large. Proof. We first consider te principal symbol p (y, ξ + iβ) of A (y). Multiplying by 2, te function 2 p (y, ξ + iβ) is analytic on S B and bounded in magnitude independent of 0 < 1. Te real part of 2 p (y, ξ + iβ)t is given by Re { 2 p (y, ξ + iβ)t } = = ( Re { 2 p (y, ξ + iβ) }) t 1 ( Im { 2 p (y, ξ + iβ) }) t 2. (3.21) We denote te symbol of D 2 jl by D 2 jl. As 2 D 2 jl is a polynomial in e±iξ j, e ±iξ l, we can extend it to an entire function of ξ j + iβ j and ξ l + iβ l. We write te Taylor expansion for 2 D 2 jl : 2 D 2 jl (ξ j + iβ j, ξ l + iβ l ) = µ,ν N µ+ν 2 20 c µν (ξ j + iβ j ) µ (ξ l + iβ l ) ν. (3.22)

31 As D 2 jl is consistent wit j l, te first coefficient, c 1,1, in te Taylor expansion must be 1. If we expand te products, we may separate te resulting terms into tree categories: terms containing only powers of ξ j and ξ l, terms containing only powers of β j and β l, and te remaining cross terms. As we are restricting our attention to te compact set S B and eac cross term contains at least one component of ξ and one component of β, te cross terms ave sum of magnitude at most O( ξ β ). Likewise, as te terms containing only powers of β ave at minimum two factors of β, teir sum as magnitude at most O( β 2 ) on S B. Tus, we may write 2 D jl 2 (ξ j + iβ j, ξ l + iβ l ) = µ,ν N µ+ν 2 c µν (ξ j ) µ (ξ l ) ν + O( ξ β ) + O( β 2 ) = 2 D 2 jl (ξ j, ξ l ) + O( ξ β ) + O( β 2 ). Substituting tis into te formula for p (y, ξ + iβ), we ave ) 2 p (y, ξ + iβ) = 2 ( jl a jl (y) D 2 jl (ξ j, ξ l ) + O( ξ β ) + O( β 2 ) (3.23) = 2 p (y, ξ) + O( ξ β ) + O( β 2 ). Tis bound may be taken independent of y as te a jl are uniformly bounded. Using te uniform ellipticity ypotesis on te first term, we ave Re { 2 p (y, ξ + iβ) } c ξ 2 + O( ξ β ) + O( β 2 ). For eac ɛ > 0 we may bound te O( ξ β ) term by ɛ ξ 2 + C ɛ β 2. By coosing ɛ sufficiently small we find for some positive constants c and κ. Re { 2 p (y, ξ + iβ) } c ξ 2 + κ β 2 For te imaginary part of p (y, ξ + iβ), we use te elementary bound Im { 2 p (y, ξ + iβ) } C ( ξ 2 + β 2). 21

32 Terefore, for t T M, we ave tat Re { 2 p (y, ξ + iβ)t } = ( Re { 2 p (y, ξ + iβ) }) t 1 ( Im { 2 p (y, ξ + iβ) }) t 2 ( c ξ 2 + κ β 2) t 1 + C ( ξ 2 + β 2) t 2 c ξ 2 t 1 + κ β 2 t 1 + CM ξ 2 t 1 + CM β 2 t 1. (3.24) If we take M so tat CM < c, ten we ave, for some constants c and κ : Re { 2 p (y, ξ + iβ)t } c ξ 2 t 1 + κ β 2 t 1. Dividing troug by 2 and using te fact tat (1 + M 2 ) 1/2 t t 1 t on T M, we discover ξ 2 β 2 Re {p (y, ξ + iβ)t} c t + κ t. (3.25) 2 2 We now turn to te full symbol. From te elementary bound ( ξ b j (y) D j 1 + c(y) C + β ) + 1, j we ave for eac ɛ > 0: b j (y) D j 1 + c(y) ɛ ξ 2 2 j + ɛ β C ɛ. Consequently, {( ) } Re b j (y) D j 1 + c(y) t j ɛ ξ 2 t + ɛ β 2 2 t + C ɛ t. (3.26) 2 Adding te bounds in (3.25) and (3.26) gives te result. In te case were p (y, ξ) is real wenever ξ is real, (3.23) yields te improved bound Im { 2 p (y, ξ + iβ) } ɛ ξ 2 + C ɛ β 2, 22

33 as Im { 2 p (y, ξ) } = 0. Tis allows us to take M to be as large as we wis in (3.24) provided we first coose ɛ sufficiently small. Teorem 6. For te box S B in Teorem 5, tere exists a constant C independent of for wic P (x, ξ + iβ) P (y, ξ + iβ) C x y α ( ξ 2 2 ) + β (3.27) for all x, y R n and all (ξ + iβ) S. Proof. We estimate: P (x, ξ + iβ) P (y, ξ + iβ) = = [a jl (x) a jl (y)] D jl 2 + j jl C x y α ( ξ 2 2 C x y α ( ξ 2 2 [b j (x) b j (y)] D 1 j + [c(x) c(y)] ) ( + β 2 ξ + C x y α 2 + β ) + C x y α ) + β using te uniform Hölder continuity of te coefficients and elementary bounds on D 2 jl and D 1 j. In order to prove several pointwise bounds on te fundamental solution to te frozen coefficient problem, we will need a sort lemma. 23

34 Lemma 7. Suppose { } xj y j β j = sign{x j y j } min, B 2κ t (3.28) for some positive constants B and κ. Ten for some constants c and C depending on B and κ, we ave e <x y,β>/+κ β 2 t 2 Ce c x y /, t 2 2 (3.29) e <x y,β>/+κ β 2 t 2 Ce c x y / t, t 2. (3.30) Proof. Witout loss of generality, suppose tat te first k components of β satisfy and te last are given by β j = x j y j, j = 1,, k 2κ t β j = B, j = k + 1,, d. Ten we ave < x y, β > = k x j y j 2 j=1 2κ t + d j=k+1 B x j y j (3.31) and κ β 2 t 2 = k ( ) xj y j 2 2 t κ 4κ 2 t j=1 d j=k+1 κb 2 t 2 = k x j y j 2 j=1 4κ t + d j=k+1 κb 2 t 2. (3.32) However, in te second sum of te last line, we ave tat B x j y j 2κ t so tat B 2 B x j y j 2κ t and tus κb 2 t B x j y j. (3.33)

35 Using (3.33) in (3.32) gives us κ β 2 t 2 k x j y j 2 j=1 4κ t + d j=k+1 B x j y j. 2 Combining (3.31) wit (3.32) yields te estimate < x y, β > + κ β 2 t 2 k x j y j 2 4κ t j=1 d j=k+1 B x j y j. 2 Exponentiating, we find e <x y,β>/+κ β 2 t 2 e k j=1 x j y j 2 /4κ t e d j=k+1 B x j y j /2. Applying te fact tat e r2 Ce r for r 0 to te first factor on te rigt, we ave e <x y,β>/+κ β 2 t 2 Ce k j=1 x j y j /(2 κ t ) e d j=k+1 B x j y j /2, and te result follows immediately. We now examine te frozen coefficient fundamental solution G (x, t; y). Our primary strategy will be to consider G in te Fourier Transform. Analyticity of te transform of G enables us to employ contour deformation tecniques from complex analysis to exibit exponential spatial decay in G. Teorem 8. Wit M as in Teorem 5, for any multi-index γ, tere exist constants C 1, C 2 and ω for wic D γ G (x y, t; y) C 1 γ e C2 x y / e ω t, t 2 2 (3.34) D γ G (x y, t; y) C 1 d t d/2 γ /2 e C 2 x y / t e ω t, t 2 (3.35) for all x, y R d, 0 < 1 and t T M. 25

36 Proof. Writing G in te transform, we ave D γ G (x y, t; y) = (2π) d γ ˆD (ξ)ep (y,ξ)t+i<x y,ξ>/ dξ. (3.36) [ π,π] d Using Caucy s integral formula and te periodicity of P (y, ξ + iβ), in eac integral we may deform te contour of integration from te real axis to te segment joining π + iβ and π + iβ were β B as in (5). Tis allows us to express D γ G (x y, t; y) = (2π) d γ ˆD (ξ + iβ)ep (y,ξ+iβ)t+i<x y,ξ>/ <x y,β>/ dξ. [ π,π] d We first assume tat t 2 2. We bound te difference operator by C/ γ and use (3.19) to obtain D γ G (x y, t; y) C γ e κ β 2 t / 2 <x y,β>/+ω t e c ξ 2 t /2 dξ. [ π,π] d Te integral is bounded by a constant, so tat we ave D γ G (x y, t; y) C γ e κ β 2 t / 2 <x y,β>/+ω t. Te result follows by coosing β j as in Lemma 7. We proceed to te case t 2. From te bound ˆD γ (ξ + iβ) C ( ξ γ γ ) + β γ γ (3.37) on te symbol of te difference operator and (3.19) applied in (3.36) we find tat D γ G (x y, t; y) Ce κ β 2 t / 2 <x y,β>/+ω t [ π,π] d ( ξ γ γ ) + β γ e c ξ 2 t / 2 dξ γ (3.38) 26

37 on te sector T M guaranteed by (3.19). Making te cange of variables ξ j = ξ j t / and extending te integral over all of R d we ave D γ G (x y, t; y) Ce κ β 2 t / 2 <x y,β>/+ω t d t d/2 Wit te coice of β j in Lemma 7, we find R d ( ) ξ γ β γ + e c ξ 2 dξ t γ /2 γ ( ) 1 C d t d/2 β γ + e κ β 2 t / 2 <x y,β>/+ω t. t γ /2 γ ( ) 1 D γ G (x y, t; y) C d t d/2 β γ + e c x y / t +ω t. t γ /2 γ As our coice of β as eac component bounded by x y, we are guaranteed tat 2κ t β γ γ x y γ C. t γ Using tis fact, factoring and absorbing constants leads us to find ( ) γ D γ G (x y, t; y) C d t d/2 t γ /2 x y 1 + e c x y / t +ω t. t Employing te fact tat r k e cr C ɛ,k e (c ɛ)r, r 0 (3.39) for real-valued functions and exploiting te scaling in x y / t in te exponential enables us to conclude D γ G (x y, t; y) C d t d/2 γ /2 e c x y / t e ω t. 27

38 Teorem 9. Let γ be any multi-index for wic γ 2. Wit M as in Teorem 5, tere exist constants C 1, C 2 and ω for wic D γ G (x y, t; y 1 ) D γ G (x y, t; y 2 ) C 1 γ y 1 y 2 α e C 2 x y / e ω t, t 2 2 (3.40) D γ G (x y, t; y 1 ) D γ G (x y, t; y 2 ) C 1 d t d/2 γ /2 y 1 y 2 α for all x, y, y 1, y 2 R d, 0 < 1 and t T M. e C 2 x y / t e ω t, t 2 Proof. We again turn to te transform to obtain te estimate. We ave tat D γ G (x y, t; y 1 ) D γ G (x y, t; y 2 ) = = (2π) d = (2π) d [ π,π] d [ π,π] d (3.41) γ ˆD (ξ + iβ) ( e P (y 1,ξ+iβ)t e ) P (y 2,ξ+iβ)t e i<x y,ξ>/ <x y,β>/ dξ γ ˆD (ξ + iβ) ( e P (y 1,ξ+iβ)t P (y 2,ξ+iβ)t 1 ) e P (y 2,ξ+iβ) e i<x y,ξ>/ <x y,β>/ dξ. We use (3.19), (3.37) and te fact tat e z 1 z e z (3.42) to find D γ G (x y, t; y 1 ) D γ G (x y, t; y 2 ) Ce <x y,β>/+κ β 2 t / 2 +ω t [ π,π] d ( ξ γ γ ) + β γ γ P (y 1, ξ + iβ) P (y 2, ξ + iβ) t e P (y 1,ξ+iβ) P (y 2,ξ+iβ) t e c ξ 2 t / 2 dξ. 28

39 We use te bound in Teorem 6 to obtain D γ G (x y, t; y 1 ) D γ G (x y, t; y 2 ) Ce <x y,β>/+κ β 2 t / 2 +ω t [ π,π] d ( ξ γ γ ) + β γ y γ 1 y 2 α ( ) ξ 2 β 2 t + t + t e C y 1 y 2 α ( ξ 2 t / 2 + β 2 t / 2 + t ) 2 2 e c ξ 2 t / 2 dξ. For small enoug δ, wit y 1 y 2 δ, we ten find D γ G (x y, t; y 1 ) D γ G (x y, t; y 2 ) C y 1 y 2 α e <x y,β>/+κ β 2 t / 2 +ω t [ π,π] d ( ξ γ +2 γ +2 ) + β γ t e c ξ 2 t / 2 dξ. γ +2 Te remainder of te proof is similar to te proof of Teorem 8, continuing from (3.38). Te factor of t is absorbed into te exponential, causing an increase in ω. For y 1 y 2 > δ, te result is a straigtforward consequence of Teorem 8. Teorem 10. Let γ be any multi-index for wic γ 2. Wit M as in Teorem 5, tere exist constants C 1, C 2 and ω for wic D γ G (x 1 y, t; y) D γ G (x 2 y, t; y) C 1 ( γ +1) x 1 x 2 e C 2 x 2 y / e ω t, t 2 2 (3.43) D γ G (x 1 y, t; y) D γ G (x 2 y, t; y) C 1 d t d/2 γ /2 1/2 x 1 x 2 e C 2 x 2 y / t e ω t, t 2 (3.44) for all x 1, x 2, y R d wit x 1 x 2 t, 0 < 1 and t T M. 29

40 Proof. We write te difference in te transform, deform te contour of integration, and factor te integrand to obtain D γ G (x 1 y, t; y) D γ G (x 2 y, t; y) = = (2π) d [ π,π] d ˆD γ (ξ + iβ) ( e P (y,ξ+iβ)t+i<x 1 y,ξ+iβ>/ e ) P (y,ξ+iβ)t+i<x 2 y,ξ+iβ>/ dξ = (2π) d [ π,π] d ˆD γ (ξ + iβ) e ( P (y,ξ+iβ)t+i<x 2 y,ξ>/ <x 2 y,β>/ e i<x 1 x 2,ξ+iβ>/ 1 ) dξ. Using (3.42), we bound te factor e i<x 1 x 2,ξ+iβ>/ 1 x1 x 2 ξ + iβ ( ξ x 1 x 2 + β e x 1 x 2 ξ+iβ ) e ɛ x 1 x 2 2 ξ 2 / 2 +ɛ x 1 x 2 2 β 2 / 2 +1/ɛ ( ξ C x 1 x 2 + β ) e ɛ x 1 x 2 2 ξ 2 / 2 +ɛ x 1 x 2 2 β 2 / 2. As x 1 x 2 t by ypotesis, e i<x 1 x 2,ξ+iβ>/ 1 ( ξ C x 1 x 2 + β ) e ɛ ξ 2 t / 2 +ɛ β 2 t / 2. Combining tis wit (3.19) gives us D γ G (x 1 y, t; y) D γ G (x 2 y, t; y) C x 1 x 2 e <x 2 y,β>/+κ β 2 t / 2 +ω t [ π,π] d ( ξ γ +1 γ ) + β γ +1 e c ξ 2 t / 2 dξ. γ +1

41 Te remainder of te proof is similar to te proof of Teorem 8, continuing from (3.38). Teorem 11. Wit M as in Teorem 5, tere exist constants C 1, C 2 and ω for wic L G (x y, t; y) C 1 2+α e C2 x y / e ω t, t 2 2 (3.45) L G (x y, t; y) C 1 d t d/2 1+α/2 e C 2 x y / t e ω t, t 2, (3.46) for all x, y R d, 0 < 1 and t T M. Proof. Writing L G (x y, t; y) in te transform, we ave L G (x y, t; y) = (2π) d (P (x, ξ + iβ) P (y, ξ + iβ)) [ π,π] d e P(y,ξ+iβ)t+i<x y,ξ>/ <x y,β>/ dξ. By (3.19) and Teorem 6, we find L G (x y, t; y) C x y α e <x y,β>/+κ β 2 t 2 +ω t [ π,π] d ( ) ξ 2 + β e c ξ 2 t 2 2 dξ. (3.38). Te remainder of te proof is similar to te proof of Teorem 8, continuing from Teorem 12. Wit M as in Teorem 5, tere exist constants C 1, C 2 and ω for 31

42 + jl wic (L G )(x 1 y, t; y) (L G )(x 2 y, t; y) C 1 2+α/2 x 1 x 2 α/2 ( e C 2 x 1 y / + e C 2 x 2 y / ) e ω t, t 2 2 (3.47) (L G )(x 1 y, t; y) (L G )(x 2 y, t; y) ( ) C 1 d t d/2 1+α/4 x 1 x 2 α/2 e C 2 x 1 y / t + e C 2 x 2 y / t e ω t, t 2 (3.48) for all x 1, x 2, y R d, 0 < 1 and t T M. Proof. We treat te case t 2 explicitly. Te case t 2 2 is andled similarly. We begin by supposing x 1 x 2 t. We write out L G (x 1 y, t; y) L G (x 2 y, t; y) = B 2 + B 1 + B 0 were B 2 = jl (a jl (x 1 ) a jl (y)) D 2 jlg (x 1 y, t; y) jl (a jl (x 2 ) a jl (y)) D 2 jlg (x 2 y, t; y) = jl (a jl (x 1 ) a jl (x 2 )) D 2 jlg (x 1 y, t; y) ( D 2 jl G (x 1 y, t; y) D 2 jlg (x 2 y, t; y) ) (a jl (x 2 ) a jl (y)) = F 1 + F 2 and B 1 and B 0 are defined similarly for te lower-order discretizations. Here we ave re-arranged te terms so tat our previous estimates are directly applicable. We bound F 1 using te uniform Hölder continuity of te a jl and (3.35), giving us F 1 C x 1 x 2 α d t d/2 1 e C 2 x 1 y / t e ω t. 32

43 As x 1 x 2 t by ypotesis, we ave tat x 1 x 2 α x 1 x 2 α/2 t α/4, wic gives us F 1 C x 1 x 2 α/2 d t d/2 1+α/4 e C 2 x 1 y / t e ω t. We may bound F 2 using te uniform Hölder continuity of te a jl and (3.44), yielding F 2 C x 2 y α x 1 x 2 d t d/2 3/2 e C 2 x 2 y / t e ω t. We can rewrite te above as ( ) α x 2 y F 2 C t α/2 x 1 x 2 d t d/2 3/2 e C 2 x 2 y / t e ω t. t Te scaling in te exponent now gives us F 2 C x 1 x 2 d t d/2 3/2+α/2 e C 3 x 2 y / t e ω t. We split up x 1 x 2 = x 1 x 2 α/2 x 1 x 2 1 α/2 and use te ypotesis x 1 x 2 t on te second factor to find F 2 C x 1 x 2 α/2 t 1/2 α/4 d t d/2 3/2+α/2 e C 3 x 2 y / t e ω t C x 1 x 2 α/2 d t d/2 1+α/4 e C 3 x 2 y / t e ω t. Summing te bounds for F 1 and F 2 (along wit similar bounds for B 1 and B 0 ) yields te result in te case x 1 x 2 t. Te result for x 1 x 2 t is an immediate consequence of Teorem 11. Teorem 13. Let M be as in Teorem 5. Suppose γ is a multi-index wit γ = 2. Ten tere exists a constant C depending on M but not on or x for wic: D γ (x x, t; x ) C 2+α e ω t, t 2 2 (3.49) x D γ (x x, t; x ) C t 1+α/2 e ω t, x t 2, t 2 (3.50) 33

44 for t T M. Proof. We prove te case were t 2 explicitly. Te case were t 2 2 is similar. Let χ : R R be a C 0 function wit χ(r) 1 for r 1 and χ(r) 0 for r 2. For an arbitrary grid point q, we express x [D γ G (x x, t; x )] = = x x 2 + [(D γ G (x x, t; x ) D γ G (x x, t; q)) χ( x x )] x x 2 + x x 2 [D γ G (x x, t; q)χ( x x )] [D γ G (x x, t; x )(1 χ( x x ))] = J 1 + J 2 + J 3 To andle J 1, we use Teorem 9 and te fact tat χ is bounded to find tat J 1 C d t d/2 1 x q α e C 2 x x / t e ω t. x x 2 For te particular coice q = x, we ave J 1 q=x C d t d/2 1 x q α e C 2 x x / t e ω t x x 2 C d t d/2 1 x x α e C 2 x x / t e ω t x x 2 C d t d/2 1+α/2 e C 3 x x / t e ω t, x x 2 were in te last step we ave exploited te scaling in te exponential to trade Hölder regularity for temporal regularity. Te final inequality can be rewritten as J 1 q=x C t 1+α/2 e ω t d t d/2 e C 3 x x / t. x x 2 34

45 Te sum on te rigt can be regarded as a Riemann sum. If we select δz = / t, ten δz 1, and te sum is bounded by j Z d e C3 jδz δz d. Tis is a Riemann sum for te exponentially decaying function e C 3 x and is uniformly bounded in 0 < δz 1 by a constant. Terefore, J 1 q=x C t 1+α/2 e ω t. To andle J 2, we interpret te difference as acting on x and difference by parts, so tat J 2 = x x 2 [D γ 1 G (x x, t; q)d γ 2 χ( x x )] were γ 1 = γ 2 = 1 are multi-indices. As χ is C 0, D e 2 χ( x x ) C for C independent of and x. We use tis fact and Teorem 8 for γ = 1 to find J 2 C d t d/2 1/2 e C 2 x x / t e ω t x x 2 Te sum on te rigt is C t 1/2 e ω t multiplied by te discretization of an integral, so tat J 2 C t 1/2 e ω t. For J 3, we use te fact tat χ is bounded and apply te estimate in Teorem 8 wit γ = 2 to find J 3 C d t d/2 1 e C 2 x x / t e ω t. x x 2 As x x 2, we can multiply te sum on te rigt by x x, and exploit te scaling in te exponential to find J 3 C d x x t d/2 1 e C 2 x x / t e ω t x x 2 d t d/2 1/2 e C 4 x x / t e ω t. 35

46 Te final step can be expressed as te rescaling of an integral, enabling us to conclude tat J 3 C t 1/2 e ω t. Adding te bounds on J 1, J 2 and J 3 (for te coice q = x) gives te result. 3.3 Two Lemmas Two lemmas simplify te proof tat te series for Φ converges. We define ( ) Ψ (m) x y (x, τ, y) = exp (C 2 mɛ) τ (3.51) were C 2 is given in Teorem 11. Lemma 14. Suppose ɛ is a positive constant and m is a positive integer for wic (m + 1)ɛ < C 2. Let x and y be generic grid points, and 0 < s < t. Tere exists a constant d(ɛ) (independent of x, y, C 2 and m) for wic: (i) Wit no oter restriction on t or s: Ψ (0) (x,, x )Ψ (m) (x,, y) d(ɛ)ψ (m) (x,, y). (3.52) x R d (ii) If t 2 2 : x R d Ψ (0) (x,, x )Ψ (m) (x, s, y) d(ɛ)ψ (m) (x, t, y). (3.53) (iii) If t 2 2 : Ψ (0) (x, t s, x )Ψ (m) (x,, y) d(ɛ)ψ (m+1) (x, t, y) (3.54) x R d 36

47 (iv) If t 2 2 and (t s) 2 : d (t s) d/2 Ψ (0) (x, t s, x )Ψ (m) (x,, y) d(ɛ)ψ (m) (x, t, y). x R d (3.55) (v) If t 2 2 and (t s) 2 : d (t s) d/2 Ψ (0) (x, t s, x )Ψ (m) (x, s, y) d(ɛ)ψ (m) (x, t, y). x R d (3.56) (vi) If t 2 2 and s 2 : d s d/2 Ψ (0) (x, t s, x )Ψ (m) (x, s, y) d(ɛ)ψ (m+1) (x, t, y). (3.57) x R n Furtermore, a similar result olds wen m is replaced by 0 on te left and side and m is replaced by 1 on te rigt and side of (i),(ii), (iv) and (v), and (iii) and (vi) old for m = 0 witout modification. Proof. (i) Applying te triangle inequality to te grid points x, y and x and multiplying by te negative constant (C 2 mɛ)/ results in te relationsip: (C 2 mɛ) x x / (C 2 mɛ) x y / (C 2 mɛ) x y /. From tis we subtract mɛ x x / from bot sides to obtain tat C 2 x x / (C 2 mɛ) x y / (C 2 mɛ) x y / mɛ x x /. Tis inequality and te monotonicity of te exponential allow us to reason Ψ (0) (x,, x )Ψ (m) (x,, y) = e C 2 x x / e (C 2 mɛ) x y / x x R d x e (C 2 mɛ) x y / e mɛ x x / e (C 2 mɛ) x y / x e mɛ x x / d(ɛ)e (C2 mɛ) x y / = d(ɛ)ψ (m) (x,, y). Te sum is independent of, and as m is a positive integer, we may take te constant d(ɛ) to be te sum wen m = 1, so tat d(ɛ) is independent of m. 37

48 (ii) Starting wit an easy consequence of te triangle inequality, (C 2 mɛ) x x / t (C 2 mɛ) x y / t (C 2 mɛ) x y / t, we replace t in te denominator by positive real numbers of smaller magnitude, leading to: (C 2 mɛ) x x / (C 2 mɛ) x y / s (C 2 mɛ) x y / t. By subtracting mɛ x x / from eac side, we ave tat C 2 x x / (C 2 mɛ) x y / s (C 2 mɛ) x y / t mɛ x x /. Te remainder of te proof is similar in spirit to te proof of (i). (iii) We apply te triangle inequality to te grid points x, x and y and multiply by (C 2 (m + 1)ɛ) to obtain (C 2 (m + 1)ɛ) x x / t (C 2 (m + 1)ɛ) x y / t (C 2 (m + 1)ɛ) x y / t. We replace t on te left and side by smaller constants, so tat (C 2 (m + 1)ɛ) x x / t s (C 2 (m + 1)ɛ) x y / (C 2 (m + 1)ɛ) x y / t. Subtracting ɛ x y / from eac side and subtracting (m + 1)ɛ x x / t s from te left gives C 2 x x / t s (C 2 mɛ) x y / (C 2 (m + 1)ɛ) x y / t ɛ x y /. Te remainder of te proof is similar in spirit to te proof of (i). (iv) Te proof is very similar to te proof of (v), presented below. 38

49 (v) We begin by applying te triangle inequality to te grid points x, x and y, and multiply by te negative constant (C 2 mɛ) to find (C 2 mɛ) x x / t (C 2 mɛ) x y / t (C 2 mɛ) x y / t. We replace bot occurrences of t in te denominators on te left and side by smaller constants, so tat (C 2 mɛ) x x / t s (C 2 mɛ) x y / s (C 2 mɛ) x y / t. Subtracting mɛ x x t s from bot sides gives C 2 x x / t s (C 2 mɛ) x y / s (C 2 mɛ) x y / t mɛ x x / t s. We can now bound te sum on te left in (3.56) by d (t s) d/2 Ψ (0) (x, t s, x )Ψ (m) (x, s, y) = x R d = d (t s) d/2 x e C 2 x x / t s e (C 2 mɛ) x y / s d (t s) d/2 x e (C 2 mɛ) x y / t e mɛ x x / t s. We define η = / t s. Ten η 1, and d (t s) d/2 e C 2 x x / t s e (C mɛ) x y / s x R d e (C 2 mɛ) x y / t e mɛjη η d j Z d d(ɛ)e (C 2 mɛ) x y / t = d(ɛ)ψ (m) (x, t, y). In te second to last step, we ave recognized tat te expression involving η is te discretization of an integral and is uniformly bounded for 0 < η 1. 39

50 (vi) Te proof is similar in spirit to te proofs of (iii) and (v). We note tat te only bounds among (i)-(vi) tat require giving up some exponential decay are (iii) and (vi). Wen m = 0, eac bound requires giving up some exponential decay, and te proofs begin similarly to tat of (iii). Lemma 15. Let 0 < γ 1 < 1 < γ 2 and t > 0. Ten t 0 (t s) 1+γ 1 s 1+γ 2 ds = Γ(γ 1)Γ(γ 2 ) Γ(γ 1 + γ 2 ) t 1+γ 1+γ 2, (3.58) were Γ wit one argument is te standard Γ function. Proof. We make te cange of variables s = s/t, so tat te integral becomes 1 1 (t ts ) 1+γ 1 (ts ) 1+γ 2 (t) ds = t 1+γ 1+γ 2 (1 s ) 1+γ 1 (s ) 1+γ 2 ds. (3.59) 0 0 We recognize te integral on te rigt as a standard form of te Beta function, and, by a well-known property of te Beta function, we ave 1 0 (1 s ) 1+γ 1 (s ) 1+γ 2 ds = B(γ 1, γ 2 ) = Γ(γ 1)Γ(γ 2 ) Γ(γ 1 + γ 2 ), from wic te result follows immediately. 3.4 Constructing Φ To sow tat Φ converges, we seek pointwise bounds on eac term Φ (m). Te estimate on te first term, Φ (0) (x, t; y) = L G (x y, t; y), was andled specially by way of a contour deformation in Teorem 11. Te subsequent terms Φ (m) 1 m ˆm for ˆm defined below contain singularities of diminising order in t, and for tese ˆm terms, te bound on eac successive term in Φ requires abandoning some 40 for