NUMERICAL METHODS FOR PARABOLIC EQUATIONS

Transcription

1 NUMERICAL METHODS FOR PARABOLIC EQUATIONS LONG CHEN As a model problem of general parabolic equations, we sall mainly consider te following eat equation and study corresponding finite difference metods and finite element metods () u t u = f in Ω (, T ), u = on Ω (, T ), u(, ) = u in Ω. Here u = u(x, t) is a function of spatial variable x Ω R n and time variable t (, T ). Te ending time T could be +. Te Laplace operator is taking wit respect to te spatial variable. For te simplicity of exposition, we consider only omogenous Diriclet boundary condition and comment on te adaptation to Neumann and oter type of boundary conditions. Besides te boundary condition on Ω, we also need to assign te function value at time t = wic is called initial condition. For parabolic equations, te boundary Ω (, T ) Ω {t = } is called te parabolic boundary. Terefore te initial condition can be also tougt as a boundary condition.. BACKGROUND ON HEAT EQUATION For te omogenous Diriclet boundary condition witout source term, in te steady state, i.e., u t =, we obtain te Laplace equation u = in Ω and u Ω =. So u = no matter wat te initial condition is. Indeed te solution will decay to zero exponentially. Let us consider te simplest -D problem Apply Fourier transfer in space û(k, t) = u t = u xx in R (, T ), u(, ) = u. R u(x, t)e ikx dx. Ten û x = ( ik)û, û xx = k 2 û, and û t = û t. So we get te following ODE for eac Fourier coefficient û(k, t) û t = k 2 û, û(, ) = û Te solution in te frequency domain is û(k, t) = û e k2t. We apply te inverse Fourier transform back to (x, t) coordinate and get u(x, t) = 4πt R e (x y)2 4t u (y) dy. For a general bounded domain Ω, we cannot apply Fourier transform. Instead we can use te eigenfunctions of A = (Laplace operator wit zero Diriclet boundary condition). Since A is SPD, we know tat tere exists an ortogonormal basis formed by

2 2 LONG CHEN eigenfunctions of A, i.e., L 2 = span{φ, φ 2,..., }. We expand te function in suc bases u(x, t) = k uk (t)φ k (x). Te eat equation ten becomes and te solution is u k t (t) = λ k u k, u k () = u k u k = u k e λ kt, for k =, 2,.... Again eac component will exponentially decay to zero since te eigenvalue λ k of A is positive. And te larger te eigenvalue, te faster te decay rate. Tis spectral analysis is mainly for teoretical purpose and te numerical application is restricted to special domains. In practice, for domains of complex geometry, it is muc arder to finding out all eigenvalue and eigenfunctions tan solving te eat equation numerically. In te following sections we will talk about finite difference and finite element metods. (2) (3) 2. FINITE DIFFERENCE METHODS FOR -D HEAT EQUATION In tis section, we consider a simple -D eat equation u t = u xx + f in (, ) (, T ), u() = u() =, u(x, ) = u (x). to illustrate te main issues in te numerical metods for solving parabolic equations. Let Ω = (, ) be decomposed into a uniform grid { = x < x <... < x N+ = } wit x i = i, = /N, and time interval (, T ) be decomposed into { = t < t <... < t M = T wit t n = n, = T/M. Te tensor product of tese two grids give a two dimensional rectangular grid for te domain Ω (, T ). We now introduce tree finite difference metods by discretizing te equation (2) on grid points. 2.. Forward Euler metod. We sall approximate te function value u(x i, t n ) by U n i and u xx by second order central difference u xx (x i, t n ) U n i + U n i+ 2U n i 2. For te time derivative, we use te forward Euler sceme (4) u t (x i, t n ) U n+ i Ui n. Togeter wit te initial condition and te source Fi n = f(x i, t n ), we ten end wit te system (5) (6) U n+ i Ui n = U i n + U i+ n 2U i n 2 + Fi n, i N, n M Ui = u (x i ), i N, n =. To write (5) in a compact form, we introduce te parameter λ = / 2 and te vector U n = (U n, U n 2,..., U n N )t. Ten (5) can be written as, for n =,, M were A = I + λ = U n+ = AU n + F n, 2λ λ λ 2λ λ λ 2λ λ λ 2λ.

3 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 3 Starting from t =, we can evaluate point values at grid points from te initial condition and tus obtain U. After tat, te unknown at next time step is computed by one matrixvector multiplication and vector addition wic can be done very efficiently witout storing te matrix. Terefore it is also called time marcing. Remark 2.. Because of te omogenous Diriclet boundary condition, te boundary index i =, N + is not included. For Neumann boundary condition, we do need to impose equations on tese two boundary nodes and introduce gost points for accurately discretize te Neumann boundary condition; See Capter: Finite difference metods for elliptic equations. Te first issue is te stability in time. Wen f =, i.e., te eat equation witout te source, in te continuous level, te solution sould be exponential decay. In te discrete level, we ave U n+ = AU n and want to control te magnitude of U in certain norms. Teorem 2.2. Wen te time step 2 /2, te forward Euler metod is stable in te maximum norm in te sense tat if U n+ = AU n ten U n U u. Proof. By te definition of te norm of a matrix N A = a ij = 2λ + 2λ. max i=,,n j= If 2 /2, ten A = and consequently U n A U n U n. Exercise 2.3. Construct an example to sow, numerically or teoretically, tat if > 2 /2, ten U n > U. Teorem 2.2 and Exercise 2.3 imply tat to ensure te stability of te forward Euler metod, we ave to coose te time step in te size of 2 wic is very restrictive, say, = 3 ten t = 6 /2. Altoug in one step, it is efficient, it will be very expensive to reac te solution at te ending time T by moving forward wit suc tiny time step. For time dependent equations, we sall consider not only te computational cost for one single time step but also te total time to arrive a certain stopping time. Te second issue is te accuracy. We first consider te consistency error. Define u n I = (u(x, t n ), u(x 2, t n ),, u(x N, t n )) t. We denote a general differential operator as L and its discritization as L n. Note tat te action Lu is te operator L acting on a continuous function u wile L n is applied to vectors suc as U n or u n I. Te consistent error or so-called truncation error is to pretend we know te exact function values and see wat te error from te approximation of te differential operator. More precisely, for te eat equation we define Lu = u t u and L n V = (V n+ V n )/ / 2 V n and te truncation error τ n = L n u n I (Lu) n I. Te error is denoted by E n = U n u n I. Estimate of te truncation error is straigtforward using te Taylor expansion.

4 4 LONG CHEN Lemma 2.4. Suppose u is smoot enoug. For te forward Euler metod, we ave τ n C + C 2 2. Te convergence or te estimate of te error E n is a consequence of te stability and consistency. Te metod can be written as We ten obtain te error equation L n U n = F n = f n I = (Lu) n I. (7) L n (U n u n I ) = (Lu) n I L n u n I, wic can be simply written as L n En = τ n. Teorem 2.5. For te forward Euler metod, wen 2 /2 and te solution u is smoot enoug, we ave U n u n I Ct n ( + 2 ). Proof. We write out te specific error equation for te forward Euler metod Consequently E n+ = AE n τ n. n E n = A n E A n l τ l. l= Since E =, by te stability and consistency, we obtain n E n τ l Cn( + 2 ) = Ct n ( + 2 ). l= 2.2. Backward Euler metod. Next we introduce te backward Euler metod to remove te strong constraint of te time step-size for te stability. Te metod is simply using te backward difference to approximate te time derivative. We list te resulting linear systems: (8) (9) U n i U n i = U i n + U i+ n 2U i n 2 + Fi n, i N, n M Ui = u (x i ), i N, n =. In te matrix form (8) reads as () (I λ )U n = U n + F n. Starting from U, to compute te value at te next time step, we need to solve an algebraic equation to obtain U n = (I λ ) (U n + F n ). Te inverse of te matrix, wic involves te stiffness matrix of Laplacian operator, is not easy in ig dimensions. For -D problem, te matrix is tri-diagonal and can be solved very efficiently by te Tomas algoritm. Te gain is te unconditional stability. Teorem 2.6. For te backward Euler metod witout source term, i.e., (I λ )U n = U n, we always ave te stability U n U n u.

5 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 5 Proof. We sall rewrite te backward Euler sceme as Terefore for any i N, wic implies and te desired result ten follows. ( + 2λ)U n i = U n i + λu n i + λu n i+. ( + 2λ) U n i U n + 2λ U n, ( + 2λ) U n U n + 2λ U n, Te truncation error of te backward Euler metod can be obtained similarly. Lemma 2.7. Suppose u is smoot enoug. For te backward Euler metod, we ave τ n C + C 2 2. We ten use stability and consistency to give error analysis of te backward Euler metod. Teorem 2.8. For te backward Euler metod, wen te solution u is smoot enoug, we ave U n u n I Ct n ( + 2 ). Proof. We write te error equation for te backward Euler metod as ( + 2λ)E n i = E n i + λe n i + λe n i+ τ n i. Similar to te proof of te stability, we obtain Consequently, we obtain E n E n + τ n n E n τ l Ct n ( + 2 ). l= Exercise 2.9. Apply te backward Euler metod to te example you constructed in Exercise 2.3 to sow tat numerically te sceme is stable. From te error analysis, to ave optimal convergent order, we also need 2 wic is still restrictive. Next we sall give a unconditional stable sceme wit second order truncation error O( ) Crank-Nicolson metod. To improve te truncation error, we need to use central difference for time discritization. We keep te forward discretization as (U n+ i Ui n)/ but now treat it is an approximation of u t (x i, t n+/2 ). Tat is we discretize te equation at (x i, t n+/2 ). Te value U n+/2 i is taken as average of Ui n and U n+ i. We ten end wit te sceme: for i N, n M Ui n () and for i N, n = U n+ i = Ui n + U i+ n 2U i n U i = u (x i ). U n+ i + U n+ n+ i+ 2Ui 2 + F n+/2 i,

6 6 LONG CHEN In matrix form, (47) can be written as (I 2 λ )U n+ = (I + 2 λ )U n + F n+/2. It can be easily verify tat te truncation error is improved to second order in time Lemma 2.. Suppose u is smoot enoug. For Crank-Nicolson metod, we ave τ n C 2 + C 2 2. Exercise 2.. Prove te maximum norm stability of Crank-Nicolson metod wit assumptions on λ. Wat is te weakest assumption on λ you can impose? In te next section, we sall prove Crank-Nicolson metod is unconditionally stable in te l 2 norm and tus obtain te following second order convergence; See Exercise 3. Teorem 2.2. For Crank-Nicolson metod, we ave E n Ct n (C 2 + C 2 2 ). 3. VON NEUMANN ANALYSIS A popular way to study te L 2 stability of te eat equation is troug Fourier analysis and its discrete version. Wen Ω = R, we can use Fourier analysis. For Ω = (, ), we can use eigenfunctions of Laplacian operator. To study te matrix equation, we establis te discrete counter part. Lemma 3.. If A = diag(b, a, b) be a N N matrix, ten te eigenvalue of A is and its corresponding eigenvector is were λ k = a + 2b cos θ k, k =,, N φ k = 2 (sin θ k, sin 2θ k,, sin Nθ k ), θ k = kθ = kπ N +. Proof. It is can be easily verified by te direct calculation. Remark 3.2. Te eigenvectors do not depend on te values a and b! We ten define a scaling of l 2 inner product of two vectors as N (U, V ) = U t V = U i V i, wic is a mimic of L 2 inner product of two functions if we tougt U i and V i represent te values of corresponding functions at grid points x i. It is straigtforward to verify te eigenvectors {φ k } N k= forms an ortonormal basis of R N wit respect to (, ). Indeed te scaling 2 is introduced for te normalization. For any vector V R N, we expand it in tis new basis N V = ˆv k φ k, wit ˆv k = (V, φ k ). Te discrete Parseval identity is k= V = ˆv, i=

7 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 7 were ˆv = (ˆv,, ˆv N ) t is te vector formed by te coefficients. Teˆin te coefficient indicates tis is a mimic of Fourier transform in te discrete level. Suppose U n+ = AU n, wit A = diag(b, a, b). Ten te stability in is related to te spectrum radius of A. Tat is U n+ A U n, and since A is symmetric wit respect to (, ), A = ρ(a) = max k N λ k(a). Note tat = /2. Te stability in te scaled l 2 norm is equivalent to tat in norm. Here te scaling is cosen to be consistent wit te scaling of discrete Laplace operator. Now we analyze te l 2 -stability of te tree numerical metods we ave discussed. Recall tat λ = δ/ 2. Forward Euler Metod. U n+ = AU n, A = diag(λ, 2λ, λ). Tus It is easy to verify tat ρ(a) = max k N 2λ + 2λ cos θ k. λ /2 = ρ(a). Let us write A = A N, te uniform stability (wit respect to N) implies λ /2, i.e., sup ρ(a N ) = λ /2. N Terefore we obtain te same condition as tat for te maximum norm stability. Backward Euler Metod. U n+ = A U n, A = diag( λ, + 2λ, λ). Tus and ρ(a ) = ρ(a) = max k N + 2λ 2λ cos θ k, ρ(a ) for any λ >. Terefore backward Euler is also unconditionally stable in l 2 norm.

8 8 LONG CHEN Crank-Nicolson Metod. U n+ = A BU n, were A = diag( λ, 2 + 2λ, λ), 2 B = diag(λ, 2 2λ, λ) 2 Since A and B ave te same eigenvectors, we ave and ρ(a λ k (B) B) = max k N λ k (A) = max λ + λ cos θ k k N + λ λ cos θ k, ρ(a B) for any λ >. Terefore we proved Crank-Nicolson metod is unconditionally stable in l 2 norm. Question 3.3. Is Crank-Nicolson metod unconditional stable in te maximum norm? You can google Crank Nicolson metod maxnorm stability to read more. For general scemes, one can write out te matrix form and plug in te coefficients to do te stability analysis. Tis metodology is known as von Neumann analysis. Formally we replace (2) U n j ρ n e iθj in te sceme and obtain a formula of te amplification factor ρ = ρ(θ). Te uniform stability is obtained by considering inequality max ρ(θ). θ π To fascinate te calculation, one can factor out ρ n e iθj and consider only te difference of indices. Te eigen-function is canged to e iθ = cos(θ) + i sin(θ) to account for possible different boundary conditions. Te te frequency of discrete eigenvectors φ k = e iθ k(:n) t is canged to a continuous variable θ in (2). We skip te index k and take te maximum of θ over [, π] wile te range of te discrete angle θ k satisfying π θ k ( )π. As an example, te readers are encouraged to do te following exercise. Use one exercise as an example. Exercise 3.4 (Te θ metod). For θ [, ], we use te sceme: for i N, n M U n+ i Ui n (3) = ( θ) U i n + U i+ n 2U i n 2 + θ U n+ i + U n+ n+ i+ 2Ui 2, and for i N, n = Ui = u (x i ). Give a complete error analysis (stability, consistency, and convergence) of te θ metod. Note tat θ = is forward Euler, θ = is backward Euler, and θ = /2 is Crank-Nicolson metod. Exercise 3.5 (Leap-frog metod). Te sceme is an explicit version of Crank-Nicolson. For i N, n M (4) U n+ i U n i 2 = U n i + U n i+ 2U n i 2.

9 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 9 Te computation of U n+ can be formulated as linear combination of U n and U n. Namely it is an explicit sceme. To start te computation, we need two initial values. One is te real initial condition U i = u (x i ), i N. Te oter U, so-called computational initial condition, can be computed using one step forward or backward Euler metod. Give a complete error analysis (stability, consistency, and convergence) of te leap-frog metod. 4. VARIATIONAL FORMULATION AND ENERGY ESTIMATE We multiply a test function v H (Ω) and apply te integration by part to obtain a variational formulation of te eat equation (): given an f L 2 (Ω) (, T ], for any t >, find u(, t) H (Ω), u t L 2 (Ω) suc tat (5) (u t, v) + a(u, v) = (f, v), for all v H (Ω). were a(u, v) = ( u, v) and (, ) denotes te L 2 -inner product. We ten refine te weak formulation (5). Te rigt and side could be generalized to f H (Ω). Since map H (Ω) to H (Ω), we can treat u t (, t) H (Ω) for a fixed t. We introduce te Sobolev space for te time dependent functions ( ) /q T L q (, T ; W k,p (Ω)) := {u(x, t) u Lq (,T ;W k,p (Ω)) := u(, t) q k,p dt < }. Our refined weak formulation will be: given f L 2 (, T ; H (Ω)) and u H (Ω), find u L 2 (, T ; H (Ω)) and u t L 2 (, T ; H (Ω)) suc tat { ut, v + a(u, v) = f, v, v H (6) (Ω), and t (, T )a.e. u(, ) = u were, is te duality pair of H and H. We assume te equation (6) is well posed. For te existence and uniqueness of te solution [u, u t ], we refer to []. In most places, we sall still use te formulation (5) and assume f, u t L 2. Remark 4.. Te topology for te time variable sould be also treat in L 2 sense. But in (5) and (6) we still pose te equation point-wise (almost everywere) in time. In particular, one as to justify te point value u(, ) does make sense for a L 2 type function wic can be proved by te regularity teory of te eat equation. To easy te stability analysis, we treat t as a parameter and te function u = u(x, t) as a mapping u : [, T ] H (Ω), defined as u(t)(x) := u(x, t) (x Ω, t T ). Wit a sligt abuse of te notation, ere we still use u(t) to denote te map. Te norm u(t) or u(t) is taken wit respect to te spatial variable. We ten introduce te operator L : L 2 (, T ; H (Ω)) L 2 (, T ; H (Ω)) L 2 (Ω)

10 LONG CHEN as (Lu)(, t) = t u u in H (Ω), for t (, T ] a.e. (Lu)(, ) = u(, ). Ten te equation (6) can be written as Lu = [f, u ]. Here we explicitly include te initial condition. Te spatial boundary condition is build into te space H (Ω). In most places, wen it is clear from te context, we also use L for te differential operator only. We sall prove several stability results of L wic are known as energy estimates in [3]. Teorem 4.2. Suppose [u, u t ] is te solution of (5) and u t L 2 (, T ; L 2 (Ω)), ten for t (, T ] a.e. (7) u(t) u + (8) u(t) 2 + (9) u(t) 2 + u(s) 2 ds u 2 + u t (s) 2 ds u 2 + f(s) ds f(s) 2 ds, f(s) 2 ds. Proof. Te solution is defined via te action of all test functions. Te art of te energy estimate is to coose an appropriate test function to extract desirable information. We first coose v = u to obtain (u t, u) + a(u, u) = (f, u). We manipulate tese tree terms as: (u t, u) = 2 (u2 ) t = d 2 dt u 2 = u d dt u ; Ω a(u, u) = u 2 ; (f, u) f u or (f, u) f u 2 ( f 2 + u 2 ). Te inequality (7) is an easy consequence of te following inequality u d u f u. dt From d 2 dt u 2 + u 2 2 ( f 2 + u 2 ), we get d dt u 2 + u 2 f 2. Integrating over (, t), we obtain (8). Te inequality (9) can be proved similarly by coosing v = u t and left as an exercise.

11 NUMERICAL METHODS FOR PARABOLIC EQUATIONS as From (8), we can obtain a better stability for te operator L : L 2 (, T ; H (Ω)) L 2 (, T ; H (Ω)) L 2 (Ω) u L2 (,T ;H (Ω)) u + f L 2 (,T ;H (Ω)). Since te equation is posed a.e for t, we could also obtain maximum-norm estimate in time. For example, (7) can be formulated as and (9) implies u L (,T ;L 2 (Ω)) u + f L (,T ;L 2 (Ω)), u L (,T ;H (Ω)) u + f L2 (,T ;L 2 (Ω)). Exercise 4.3. Prove te stability result (9). Exercise 4.4. Prove te energy estimate (2) u 2 e λt u 2 + e λ(t s)) f 2 ds, were λ = λ min ( ) >. Te estimate (2) sows tat te effect of te initial data is exponential decay. 5. FINITE ELEMENT METHODS: SEMI-DISCRETIZATION IN SPACE Let {T, } be a quasi-uniform family of triangulations of Ω. Te semi-discretized finite element metod is: given f V (, T ], u, V H (Ω), find u L 2 (, T ; V ) suc tat (2) { ( t u, v ) + a(u, v ) = f, v, u (, ) = u, v V, t R +. Te sceme (2) is called semi-discretization since u is still a continuous (indeed differential) function of t. Te initial condition u is approximated by u, V and coices of u, is not unique. We can expand u = N i= u i(t)ϕ i (x), were ϕ i is te standard at basis at te vertex x i for i =,, N, te number of interior nodes, and te corresponding coefficient u i (t) now is a function of time t. Te solution u can be computed by solving an ODE system (22) M u + Au = f, were u = (u,, u N ) t is te coefficient vector, M, A are te mass matrix and te stiffness matrix, respectively, and f = (f,, f N ) t. For te linear finite element, one can cose a numerical quadrature to make M diagonal, wic is known as te mass lumping, so tat te ODE system (22) can be solved efficiently by mature ODE solvers. We sall apply our abstract error analysis developed in Capter: Unified Error Analysis to estimate te error u u in certain norms. Te setting is ( ) X = L 2 (, T ; H T /2 (Ω)), and u X = u(t) 2 dt ( ) Y = L 2 (, T ; H T /2 (Ω), and f Y = f(t) 2 dt ( ) X = L 2 T /2 (, T ; V ), and u X = u (t) 2 dt

12 2 LONG CHEN ( ) Y = L 2 (, T ; V ), and f T /2. Y = f (t) 2, dt Recall te dual norm f, v f, = sup, for f V v V v. I = R (t) : H (Ω) V is te Ritz-Galerkin projection, i.e., R u V suc tat a(r u, v ) = a(u, v ), v V. Π = Q (t) : H (Ω) V is te projection Q f, v = f, v, v V. P : X X is te natural inclusion L : X Y H (Ω) is Lu = t u u, Lu(, ) = u(, ), and L = L X : X Y V. Te equation we are solving is L u = Q f, u (, ) = u,. in V, t (, T ] a.e. 5.. Stability. Adapt te proof of te energy estimate for L, we can obtain similar stability results for L. Te proof is almost identical and tus skipped ere. Teorem 5.. Suppose u satisfy L u = f, u (, ) = u,, ten (23) (24) (25) u (t) 2 + u (t) 2 + u (t) u, + u (s) 2 ds u, 2 + t u (s) 2 ds u, 2 + f (s) ds f (s) 2, ds, f (s) 2 ds. Note tat in te energy estimate (24), te is replaced by a weaker norm, since we can apply te inequality f, u f, u. Te weaker norm, can be estimated by f, f C f Consistency. Recall tat te consistency error is defined as L I u L u = L I u Π Lu. Te coice of I = R simplifies te consistency error analysis. Lemma 5.2. For te semi-discretization, we ave te error equation (26) (27) L (R u u ) = t (R u Q u), t >, in V, (R u u )(, ) = R u u,. Proof. Let A =. By our definition of consistency, te error equation is: for t > L (R u u ) = L R u Q Lu = t (R u Q u) + (AR u Q Au). Te desired result ten follows by noting tat AR = Q A in V. One can also verify (26) by writing out te variational formulation.

13 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 3 Te error equation (26) olds in V wic is a weak topology. Te motivation to coose I = R is tat in tis weak topology AR u, v = ( R u, v ) = ( u, v) = Au, v = Q Au, v, or simply in operator form AR = Q A. Tis tecnique is firstly proposed by Weeler [4]. Apply te stability to te error equation, we obtain te following estimate on te discrete error R u u. Teorem 5.3. Te solution u of (2) satisfy te following error estimate (28) R u u u, R u + (29) R u u 2 + (3) R u u 2 + (u R u) 2 ds u, R u 2 + t Q u t R u ds t (R u u ) 2 ds u, R u 2 + t Q u t R u 2, ds, t Q u t R u 2 ds. We ten estimate te two terms involved in tese error estimates. Te first issue is on te coice of u,. An optimal one is obviously u, = R u so tat no error coming from approximation of te initial condition. However, tis coice requires te inversion of a stiffness matrix wic is not ceap. A simple coice would be te nodal interpolation, i.e. u, (x i ) = u (x i ) or any oter coice wit optimal approximation property (3) u, R u u u, + u R u 2 u 2, and similarly u, R u u 2. According to (2), te affect of te initial boundary error will be exponentially decay to zero as t goes to infinity. So in practice, we can coose te simple nodal interpolation. On te estimate of te second term, we assume u t H 2 (Ω) and assume H 2 -regularity result old for Poisson equation (for example, te domain is smoot or convex), ten (32) t Q u t R u = Q (I R )u t (I R )u t 2 u t 2. Te negative norm can be bounded by te L 2 -norm as t Q u t R u, t Q u t R u C t Q u t R u 2 u t 2. Wen using quadratic and above polynomial, we can prove a stronger estimate for te negative norm (33) u R u C r+ u r were r is te order of te element, ten we could obtain a superconvergence in L 2 -norm ( /2 R u u C r+ u t 2 r ds).

14 4 LONG CHEN 5.3. Convergence. Te convergence of te discrete error comes from te stability and consistency. For functions in finite element space, te H semi-norm can be estimate by a simple inverse inequality R u u C R u u. Teorem 5.4. Suppose te solution u to (6) satisfying u t L 2 (, T ; H 2 (Ω)) and te H 2 regularity olds. Let u be te solution of (2) wit u, satisfying (3). We ten ave ) (34) R u u + R u u C ( u u t 2 ds. To estimate te true error u u, we need te approximation error estimate of te projection R u R u + u R u C u 2. Teorem 5.5. Suppose te solution u to (6) satisfying u t L 2 (, T ; H 2 (Ω)). Ten te solution u of (2) wit u, aving optimal approximation property (3) satisfy te following optimal order error estimate: ) (35) u u + u u C ( u u t 2 ds *Superconvergence and maximum norm estimate. Te error e = R u u satisfies te evolution equation (26) wit e () = wit te cose u, = R u, i.e., (36) t e + A e = Q t (R u u). Terefore e (t) = exp( A (t s))τ ds, wit τ = Q t (R u u). Due to te smooting effect of te semi-group e A t, we ave te following estimate. Here we follow te work by Garcia-Arcilla and Titi [2]. Lemma 5.6 (Smooting property of te eat kernel). For τ V, we ave (37) max t T exp( A (t s))a τ ds C log max τ. t T Proof. Let λ m and λ M be te minimal and maximal eigenvalue of A. Ten it is easy to ceck λ M e λ M (t s) if (t s) λ e A(t s) M, A (t s) if λ M (t s) λ m, λ m e λm(t s) if (t s) λ Note tat λ m = O() and λ M C 2. We get e A(t s) A τ ds max t T m. C log max t T τ. Teorem 5.7 (Superconvergence in H -norm). Suppose te solution u to (6) satisfying u t L (, T ; H 2 (Ω)). Let u be te solution of (2) wit u, = R u. Ten (38) max R u u C log 2 max u t 2. t T t T

15 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 5 Wen u t L 2 (, T ; H 2 (Ω)), ten ( /2 (39) R u u C 2 u t 2 2 ds). Proof. We multiply A /2 to (36) and apply Lemma 5.6 to get T e (T ) = A /2 e (T ) = e A (T s) A /2 τ ds T e A (T s) A ds max C log 2 max u t 2. t T In te last step, we ave used te fact A /2 τ = τ τ. To get (39), we use te energy estimate (3). t T A /2 τ Since te optimal convergent rate for u R u or u u is only first order, te estimate (38) and (39)) are called superconvergence. To control te maximum norm, we use te discrete embedding result (for 2-D only) R u u C log R u u, and te error estimate of R in te maximum norm to obtain te following result. u R u C log 2 u 2,, Teorem 5.8 (Maximum norm estimate for linear element in two dimensions). Suppose te solution u to (6) satisfying u L (, T ; W 2, ) and u t L 2 (, T ; W 2, ) or u t L (, T ; W 2, (Ω)). Let u be te solution of (2) wit u, = R u. Ten in two dimensions ( ) /2 ] (4) (u u )(t) C log [ u 2 2, + u t 2 2 ds, (4) max (u u )(t) C log 2 2 max [ u(t) 2, + u t (t) 2 ]. t T t T For ig order elements, we could get superconvergence in L 2 norm. Let us define te order of te polynomial as te degree plus, wic is te optimal order wen measuring te approximation property in L p norm. For example, te order of te linear polynomial is 2. Wen te order of polynomial r is bigger tan 3 (,i.e., quadratic and above polynomial), we can prove a stronger estimate for te negative norm (42) u R u C r+ u r. Using te tecnique in Lemma 5.6 and Teorem 5.7, we ave te following estimate on te L 2 norm. Teorem 5.9 (Superconvergence in L 2 -norm). Suppose te solution u to (6) satisfying u t L (, T ; H r (Ω)). Let u be te solution of (2) wit u, = R u. Ten (43) max (R u u )(t) C log r+ max (u t)(t) 2 r. t T t T

16 6 LONG CHEN Wen u t L 2 (, T ; H r (Ω)), ten ( /2 (44) R u u C r+ u t 2 r ds). Proof. From (36), we ave e (T ) = T T e A (T s) τ ds e A (T s) A /2 C log τ. ds max t T A /2 In te second step, we ave used te estimate T e A (T s) A /2 ds C log, wic can be proved by te estimate e A (T s) A /2 (t s) To prove (44), we simply use te energy estimate (29) and (42). In -D, using te inverse inequality R u u /2 R u u, we could obtain te superconvergence in te maximum norm ( /2 (45) R u u C r+/2 u t 2 r ds). Again using te inverse inequality u R u C u R u, te superconvergence of L 2 norm, and te maximum norm estimate of Ritz-Galerkin projection, for r 3, u R u C r u r,, we can improve te maximum norm error estimate. Teorem 5. (Maximum norm estimate for ig order elements in two dimensions). Suppose te solution u to (6) satisfying u L (, T ; W 2, ) and u t L 2 (, T ; W 2, ) or u t L (, T ; W 2, (Ω)). Let u be te solution of (2) wit u, = R u. Ten in two dimensions and for r 3 ( ) /2 ] (46) (u u )(t) C [ u r r, + u t 2 r ds. τ 6. FINITE ELEMENT METHODS: SEMI-DISCRETIZATION IN TIME In tis section, we consider te semi-discretization in time. We first discretize te time interval (, T ) into a uniform grid wit size = T/N and denoted by t n = n for n =,... N. A continuous function in time will be interpolated into a vector by (I n f)(, t n ) = f(, t n ). Recall tat A = : H H. We list tree scemes in operator form. Forward Euler Metod. u = u Backward Euler Metod. u = u u n u n + Au n = f n. u n u n + Au n = f n.

17 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 7 Crank-Nicolson Metod. u = u u n u n + A(u n + u n )/2 = f n /2. Note tat tese equations old in H sense. Taking Crank-Nicolson as an example, te equation reads as (47) (un u n, v) + 2 ( un + u n, v) = (f n /2, v) for all v H. We now study te stability of tese scemes. We rewrite te Backward Euler metod as (I + A)u n = u n + f n. Since A is SPD, λ min (I + A) and consequently, λ max ((I + A) ). Tis implies te L 2 stability n (48) u n u n + f n u + f k. Te stability (49) is te discrete counter part of (7): discretize te integral n f ds by a Riemann sum. Similarly one can derive te L 2 stability for te C-N sceme n (49) u n u + f k /2. k= k= Te integral n f ds is approximated by te middle point rule. Remark 6.. For C-N metod, te rigt and side can be also cosen as (f n + f n )/2. It corresponds to te trapezoid rule of te integral. For nonlinear problem A(u), it can be A((u n + u n )/2) or (A(u n ) + A(u n ))/2. Wic one to cose is problem dependent. Te energy metod can be adapted to te semi-discretization in time easily. For example, we cose v = (u n + u n )/2 in (47) to get 2 un 2 2 un 2 + u n /2 2 = (f n /2, u n /2 ), wic implies te counter part of (8) n n (5) u n 2 + u k /2 2 u 2 + f k /2 2. k= Exercise 6.2. Study te stability of te forward and backward Euler metod. We ten study te convergence. We use C-N as a typical example. We apply te discrete operator L n to te error I n u u n L n (I n u u n ) = L n I n u I n Lu = u(, tn ) u(, t n ) t u(, t n /2 ). Te consistency error is (5) u(, t n ) u(, t n ) t u(, t n /2 ) C()2. By te stability result, we ten get k= I n u u n Ct n 2.

18 8 LONG CHEN From te consistency error estimate (5), one can easily see te backward and forward Euler metods are only first order in time. Hig order scemes, e.g., Runge-Kutta metod, can be constructed by using a ig order approximation of te integral. We first rewrite te differential equation (in time) as an integral equation: n t n u t (, t) dt + n t n Au(, t) dt = n t n f(, t) dt. Te first term is simply (u(, t n ) u(, t n )/. Te rigt and side can be approximated as accurately as we want since f is known. Te subtle part is te middle one. We can apply ig order quadrature using more points but u is only known at t n. So approximation at quadrature points sould be computed first. For details, we refer to [?]. Add more on ig order (explicit and implicit) discretization in time. 7. FINITE ELEMENT METHOD OF PARABOLIC EQUATION: FULL DISCRETIZATION Te full discretization is simply a combination of discretization in space and time. We consider te following tree scemes: Forward Euler Metod. u = u, (52) (un u n, v ) + a(u n, v ) = f n, v, v V, n N Backward Euler Metod. u = u, (53) (un u n, v ) + a(u n, v ) = f n, v, v V, n N Crank-Nicolson Metod. u = u, (54) (un u n, v ) + a((u n + u n )/2, v ) = f n /2, v, v V, n N We write te semi-discretization and te full discretization as L L L n. Te discrete error is R u(t n ) u n. Te setting is ( X = L 2 (, T ; H T (Ω)), and u X = u(t) 2 dt ( Y = L 2 (, T ; H T (Ω), and f Y = f(t) 2 dt X ) /2 ) /2 = V V V, and u X = ( N k= uk 2 ) /2 ) /2. ( Y = V V V, and f = N Y k= f k 2 I n R : L 2 (, T ; H (Ω)) V V V is te composition of te nodal interpolation in time and Ritz projection in space I n R u = R I n u = (R u)(t n ). I n Q : L 2 (, T ; H (Ω)) V V V is te composition of te nodal interpolation in time and L 2 projection in space I n Q f = Q I n f = (Q f)(t n ).

19 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 9 L n : X Y V is u = u, and for n D t + u n un Forward Euler, L n u n = Dt u n un Backward Euler, Dt u n (un + un )/2 Crank-Nicolson. were D t + is te forward difference and Dt te backward difference in time. Te solution u n solves te full discrete equation: L n u n = I n+s Q f, in V, for n, u = u,, were s = for backward Euler and forward Euler, and s = /2 C-N metod. 7.. Stability. We write te discretization using operator form and present te stability in L 2 norm. Again denoted by A = V. Forward Euler Metod. (55) u n = (I A )u n Backward Euler Metod. + f n. (56) u n = (I + A ) (u n + f n ). Crank-Nicolson Metod. (57) u n = (I + [ 2 A ) (I ] 2 A )u n + f n /2. Since A = is symmetric in te L 2 inner product, to obtain te stability in L 2 norm, we only need to study te spectral radius of tese operators. Forward Euler Metod. provided ρ(i A ) = λ max (A ), 2 λ max (A ). Note tat λ max (A ) = O( 2 ). We need te time step is in te size of 2 to make te forward Euler stable. Backward Euler Metod. For any >, since A is SPD, i.e., λ min (A ) > Crank-Nicolson Metod. For any >, ρ((i + A ) ) = ( + λ min (A )). ρ((i + 2 A ) (I 2 A 2 )) = max λ k(a ) k N + 2 λ k(a ), We summarize te stability as te following teorem wic is a discrete version of (7).

20 2 LONG CHEN Teorem 7.. For forward Euler metod, wen < /λ max (A ) n (58) u n u + f k. For backward Euler k= n (59) u n u + f k+. For Crank-Nicolson metod k= n (6) u n u + f k+/2. It is interesting to note tat te last term in te stability result (58), (59), and (6) are different Riemann sums of te integral in te continuous level. k= Exercise 7.2. Derive te discrete version of (8) and (9) Consistency. Te error equation will be L n I n R u I n Q Lu = D n t I n R u I n t u = (D n t I n u I n t u) + D n t I n (R u u). Forward Euler. and (D n t I n u I n t u) = u(t n+) u(t n ) D n t I n (R u u) = n+ Backward Euler. and t u(t n ) C tt u, t n (R u t u t ) ds 2 (D n t I n u I n t u) = u(t n) u(t n ) D n t I n (R u u) = n n+ t n t u(t n ) C tt u t n (R u t u t ) ds 2 n u t 2 ds. t n u t 2 ds. Crank-Nicolson. For Crank-Nicolson, it is important to note tat we are solving So te error equation is L n u n = I n /2 Q f. L n I n R u I n /2 Q Lu = R u(t n ) R u(t n ) t u(t n /2 ) + AR 2 (u(t n) + u(t n )) Q Au(t n /2 ) = D t I n (R u u) + D t I n u t u(t n /2 ) [ ] + Q A 2 (u(t n) + u(t n )) u(t n /2 ) = I + I 2 + I 3.

21 NUMERICAL METHODS FOR PARABOLIC EQUATIONS 2 We ten estimate eac term as I = 2 n I 2 C ttt u 2, I 3 C tt u 2. t n u t 2 ds, 7.3. Convergence. Te convergence is a consequence of stability, consistency, and te approximation. For simplicity, we list te result for Crank-Nicolson and assume u, = R u. Teorem 7.3. Let u, u n be te solution of (6) and (54), respectively wit u, = R u. Ten for n, n n u(t n ) u n C 2 u t 2 ds + C 2 ( ttt u + tt u ). REFERENCES [] L. C. Evans. Partial Differential Equations. American Matematical Society, 998. [2] B. GarcÍa-Arcilla and E. S. Titi. Postprocessing te galerkin metod: Te finite-element case. SIAM J. Numer. Anal., 37(2):47 499, 2. [3] V. Tomée. Galerkin finite element metods for parabolic problems, volume 25 of Springer Series in Computational Matematics. Springer-Verlag, Berlin, second edition, 26. [4] M. F. Weeler. A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal., : , 973.