3 Parabolic Differential Equations

Transcription

1 3 Parabolic Differential Equations 3.1 Classical solutions We consider existence and uniqueness results for initial-boundary value problems for te linear eat equation in Q := Ω (, T ), were Ω is a bounded domain in R d wit boundary Γ := Ω and T R +. For given functions (3.1.1) f C(Q), u D C(Γ (, T )), u C(Ω), satisfying te compatibility condition (3.1.2) u D (x, ) = u (x), x Γ, we look for a function u : Q R suc tat (3.1.3a) (3.1.3b) (3.1.3c) u t (x, t) u(x, t) = f(x, t), (x, t) Q, u(x, t) = u D (x, t), (x, t) Γ (, T ), u(x, ) = u (x), x Ω. (3.1.4) Definition: Under te assumptions (3.1.1),(3.1.2) a function u C(Q) suc tat u(x, ) C 1 ((, T )) for x Ω, and u(, t) C 2 (Ω) for t (, T ), is called a classical solution of te initial-boundary value problem (3.1.3a)-(3.1.3c), if u satisfies (3.1.3a)-(3.1.3c) pointwise. Te initial-boundary value problem (3.1.3a)-(3.1.3c) is said to be well-posed, if its solution u depends continuously on te data f, u D, u wit respect to te topology defined by some norm on C(Q). In te following, we summarize te most important existence and uniqueness results for classical solutions of te initial-boundary value problem (3.1.3a)-(3.1.3c). For proofs and furter details we refer to te literature (cf., e.g., Jost (22)). Te uniqueness of a classical solution follows from te weak maximum principle. (3.1.5) Teorem: Assume tat (3.1.1) and (3.1.2) old true and denote by R Q te reduced boundary R Q := (Ω {}) (Γ [, T ]). Moreover, suppose

2 76 3 Parabolic Differential Equations tat te function u C(Q) suc tat u(x, ) C 1 ((, T )) for x Ω and u(, t) C 2 (Ω) for t (, T ) satisfies te inequality u (x, t) u(x, t) t, (x, t) Q. Ten, tere olds max u(x, t) = (x,t) Q max u(x, t). (x,t) R Q If u 1, u 2 are two classical solutions, te application of Teorem (3.1.5) to u 1 u 2 and u 2 u 1 implies u 1 = u 2, i.e., we ave: Corollary ( ) Under te assumptions (3.1.1),(3.1.2) te initial-boundary value problem (3.1.3a)-(3.1.3c) admits a unique classical solution. Remark ( ) In case Ω = R d, te boundary condition (3.1.3b) as to be replaced by te growt condition u(x, t) = O(exp(λ x 2 )) for x (λ > ). Besides te weak maximum principle tere olds te following strong maximum principle: (3.1.6) Teorem: Under te assumptions of Teorem (3.1.5) suppose tat te function u C(Q) suc tat u(x, ) C 1 ((, T )) for x Ω and u(, t) C 2 (Ω) for t (, T ) satisfies te inequality u (x, t) u(x, t), (x, t) Q. t If tere exists a point (x, t ) Q suc tat tere olds u(x, t ) = max u(x, t), (x,t) Q u(x, t) = const., (x, t) Ω [, t ]. Te existence of a classical solution of te initial-boundary value problem (3.1.3a)-(3.1.3c) follows from an explicit representation wit respect to te data f, u D, u wic can be given by means of a kernel function of te linear eat equation. (3.1.7) Definition: Let Ω be a domain in R d. A function K : Ω Ω R + R is called a kernel function of te linear eat equation, if tere olds (i) t K(x, y, t) xk(x, y, t) =, (x, y, t) Ω Ω R +, (ii) K(x, y, t) =, (x, y, t) Γ Ω R +, (iii) lim K(x, y, t)v(x) dx = v(y), y Ω, v C(Ω). t Ω

3 3.1 Classical solutions 77 Te existence of a kernel function can be sown by means of te fundamental solution Λ(x, y, t) := (4πt) d/2 exp( x y 2 ), (x, y, t) R d R d R + 4t of te linear eat equation: Suppose tat u C(R d ) is bounded and let u : R d R + R be te function defined by te convolution u(x, t) = Λ(x,, t) u ( ) = Λ(x, y, t)u (y) dy. R d Ten u C (R d R + ), and u is a classical solution of te linear eat equation in R d R + satisfying te initial condition u(x, ) = u (x), x Ω. Indeed, computing te partial derivatives Λ/ t and 2 Λ/ x i x j, 1 i, j d, it is easy to sow tat Λ, considered as a function of x, satisfies te linear eat equation in (x, t) R d R +. It follows tat t u(x, t) = ( t Λ(x,, t)) u ( ) = ( x Λ(x,, t)) u ( ) = x u(x, t). Since Λ C (R d R d R + ), differentiation implies tat u C (R d R + ). For a bounded C 2 domain Ω R d, one can also use te fundamental solution to obtain an explicit representation of te solution of te initial-boundary value problem (3.1.3a)-(3.1.3c) under a omogeneous initial condition, i.e., u, and tus deduce te existence of a kernel function: (3.1.8) Teorem: Let Ω R d be a bounded C 2 domain. Ten, te linear eat equation as a kernel function K C (Ω Ω R + ) suc tat K(x, y, t) > and K(x, y, t) = K(y, x, t), (x, y, t) Ω Ω R +. Using te kernel function K from Teorem (3.1.8) we obtain te following representation of te unique solution of te initial-boundary value problem (3.1.3a)-(3.1.3c): (3.1.9) Teorem: Let Ω R d be a bounded C 2 domain and suppose tat te assumptions (3.1.1),(3.1.2) old true. Ten, te initial-boundary value problem (3.1.3a)-(3.1.3c) admits a unique solution u C(Q) C (Q) wic as te representation + Ω t u(x, t) = K(x, y, t s)f(y, s) dy ds + Ω t K(x, y, t)u (y) dy ν Γ y K(x, y, t s)u D (y, s) dσ(y) ds. o Γ

4 78 3 Parabolic Differential Equations Remark ( ) If te compatibility condition (3.1.2) does not old true, one can establis te existence of a unique solutionu C((Ω R + ) \ (Γ {})) C (Q) wit te same representation as in Teorem (3.1.9). Teorem (3.1.9) sows te well-posedness of te initial-boundary value problem (3.1.3a)-(3.1.3c). It also reflects te smooting property of parabolic differential operators. Te solution of te initial-boundary value problem is C in Q even if te initial-boundary data are only continuous. 3.2 Finite difference metods We consider te following initial-boundary value problem for a parabolic differential equation (3.2.1a) (3.2.1b) (3.2.1c) Lu := u t + A(t)u = f in Q := Ω [, T ], u = u D on Γ (, T ), u(, ) = u in Ω. Here, Ω R d stands for a bounded Lipscitz domain wit boundary Γ = Ω and T >. Furter, for any t (, T ) we suppose tat A(t) is a linear second order elliptic differential operator. Moreover, we assume f C(Q), u D C(Γ [, T ]) and u C(Ω) and tat (3.2.1a)-(3.2.1c) is well-posed. Te numerical solution can be based on a semi-discretization eiter in space or in time. Te first approac is referred to as te metod of lines. We assume a grid point set (3.2.2) Ω := {x = (x ν1,, x νd ) T }. and define Ω and Γ as te sets of interior and boundary grid points according to Ω := Ω Ω and Γ := Ω Γ. Moreover, we assume A (t), t (, T ), to be a compact finite difference approximation of A(t) as in capter 2.2 and suppose tat f (t) C(Ω ), t (, T ], u D C(Γ ) and u C(Ω ). Ten, te metod of lines amounts to te solution of te initial-value problem (3.2.3a) (3.2.3b) (u ) t + A (t)u = f (t), t (, T ), u () = u for a system of first-order ordinary differential equations in te unknowns u (x), x Ω. Tis system is typically stiff so tat te numerical integration of (3.2.3a),(3.2.3b) as to be based on implicit or semi-implicit scemes. Te second approac is called Rote s metod. Discretizing (3.2.1a) implicitly in time, e.g. by te backward Euler sceme, wit respect to a uniform partition (3.2.4) I k := {t m := mk, k := T/(M + 1), M N}

5 3.2 Finite difference metods 79 of te time interval [, T ], at eac time step we ave to solve an elliptic boundary value problem of te form (3.2.5a) (3.2.5b) (I + ka(t k ))u(t k ) = u(t k 1 ) + kf(t k ) in Ω, u(, t k ) = u D (t k ) on Γ. Consequently, te tecniques of capter 2.2 and 2.4 can be applied for te numerical solution of (3.2.5a),(3.2.5b). In te sequel, we will focus on finite difference metods in time and space. We set Q,k := Ω I k. For D,k Q,k we refer to C(D,k ) as te linear space of grid functions on D,k. For grid functions u,k C(Q,k ) we define te forward difference quotient (3.2.6) D + k u,k(x, t) := k 1( ) u,k (x, t + k) u,k (x, t), were x Ω, t I k \ {T }, and te backward difference quotient (3.2.7) D k u,k(x, t) := k 1( ) u,k (x, t) u,k (x, t k), were x Ω, t I k \ {}. Moreover, let A (t), t Īk, be a difference approximation of A(t) as in capter 2.2 and suppose tat f,k C(Q,k ), u D C(Γ Īk) and u C(Ω ). Using te forward difference quotient, we obtain te difference operator L,k := D + k + A (t) and tus te explicit finite difference metod (3.2.8a) (3.2.8b) (3.2.8c) L,k u,k = f,k in Ω (I k \ {T }), u,k = u D auf Γ I k, u,k = u in Ω. Te use of te backward difference quotient leads to te difference operator L,k := D k + A (t) and te implicit finite difference metod (3.2.9a) (3.2.9b) (3.2.9c) L,k u,k = f,k in Ω (I k \ {}), u,k = u D auf Γ I k, u,k = u in Ω. If we know u,k (x, t) in x Ω, t = t m, m M, in case of te explicit finite difference metod we obtain te approximation u,k (x, t) in x Ω, t = t m+1, simply by solving (3.2.8a) for u,k (x, t m+1 ). On te oter and, for te implicit finite difference metod te computation of u,k (x, t m ), x Ω, requires te solution of a linear algebraic system. Te multiplication of (3.2.9a) by some Θ [, 1] (for t+k instead of t) and of (3.2.8a) by 1 Θ and te subsequent addition of te resulting difference equations results in te so-called Θ-metod:

6 8 3 Parabolic Differential Equations (3.2.1a) (3.2.1b) (3.2.1c) L Θ,ku,k = f Θ,k, x Ω, t I k \ {T }, u,k (x, t) = u D (x, t), x Γ, t I k, u,k (x, ) = u (x), x Ω, were L Θ,k stands for te difference operator L Θ,ku,k := k 1 (u,k (, t + k) u,k (, t)) + + ΘA (t + k)u,k (, t + k) + (1 Θ)A (t)u,k (, t) and f,k Θ for te grid function f Θ,k := Θf,k (, t + k) + (1 Θ)f,k (, t). For Θ =, we obtain te explicit finite difference metod (3.2.8a)-(3.2.8c), wereas Θ = 1 results in te implicit finite difference metod (3.2.9a)- (3.2.9c). In case Θ = 1/2, we refer to (3.2.1a)-(3.2.1c) as te Crank-Nicolsen metod. We study te convergence of te Θ-sceme witin te framework of a more general finite difference approximation of (3.2.1a)-(3.2.1c) using a two-level (one-step) finite difference metod, i.e., a metod involving only two subsequent time levels t j 1 and t j. To tis end let M (t j ), N (t j 1 : C(Ω ) C(Ω ) be two finite difference approximations of A(t j ) and A(t j 1 ), respectively, and assume g (, t j 1, t j ) C(Ω ). Moreover, let L,k : C(Q,k ) C(Q,k ) be te finite difference operator given by (3.2.11) L,k u,k (, t j ) := (I,k + km (t j ))u,k (, t j ) (I,k )kn (t j 1 )u,k (, t j 1 ) We consider te two-level sceme (3.2.12a) (3.2.12b) (3.2.12c) L,k u,k (x, t) = g,k (x, t), x Ω, t Īk \ {}, u,k (x, t) = u D (x, t), x Γ, t Īk \ {}, u,k (x, ) = u (x), x Ω. For te specification of te convergence we measure te global discretization error u Q,k u,k in a suitable discrete norm,k of te linear space C(Q,k ) of grid functions on Q,k. For instance, we may coose te discrete maximum norm u,k,k := max (x,t) Q,k u,k (x, t) or te discrete analogue of te L 2 ([, T ]; L 2 (Ω))-norm (cf. capter 3.3) u,k,k := ( k M 1/2, u,k (, t j ) 2,) 2 j= were 2, stands for te discrete L 2 -norm on C(Ω ).

7 3.2 Finite difference metods 81 (3.2.13) Definition: Assume tat u is te classical solution of te initialboundary value problem (3.2.1a)-(3.2.1c) and tat u,k C(Q,k ) is te solution of te finite difference metod (3.2.12a)-(3.2.12c). Te finite difference metod is called convergent, if tere olds u u,k,k for, k. It is said to be convergent of order p 1 in and p 2 in k (p i N, 1 i 2), if tere exists a constant C >=, independent of and k, suc tat u u,k,k C ( p1 + k p2 ). As in te numerical solution of initial-value problems for ordinary differential equations, sufficient conditions for convergence are given by te consistency and te stability. (3.2.14) Definition: Under te assumptions of Definition (3.2.13), te grid function L,k u(x, t) g,k (x, t), x Ω, t Ī k τ,k (x, t) := u(x, t) u D,k (x, t), x Γ, t Īk u(x, ) u (x), x Ω is called te local discretization error. Te finite difference metod (3.2.12a)- (3.2.12c) is called consistent wit te initial-boundary value problem (3.2.1a)- (3.2.1c), if tere olds τ,k,k for, k. Te finite difference metod is said to be consistent of order p 1 in and p 2 in k (p i N, 1 i 2), if tere exists a positive constant C, independent of and k, suc tat τ,k,k C ( p1 + k p2 ). Assuming a sufficiently smoot classical solution u of (3.2.1a)-(3.2.1c), consistency and order of consistency can be sown by Taylor expansion. Example. We consider te Θ-metod (3.2.1a)-(3.2.1c) applied to te linear eat equation, i.e., A(t) =, t [, T ], and A (t) =, t [, T ], were stands for te discrete Laplace operator, and u D = ud Γ Īk, u = u Ω. We assume tat te classical solution satisfies u C 2 ([, T ]; C 4 (Ω)). Ten, Taylor expansion reveals tat for Θ 1/2 te Θ-metod is consistent of order p 1 = 2 in and p 2 = 1 in k. We note tat te expansion point is (x, t) for te explicit metod and (x, t + k) for te implicit metods. If te classical solution u satisfies u C 3 ([, T ]; C 4 (Ω)), te Crank-Nicolsen metod (Θ = 1/2) is consistent of order p 1 = 2 in and p 2 = 2 in k. As te expansion point one as to coose (x, t + k/2). Te stability of te difference approximation (3.2.12a)-(3.2.12c) means continuous dependence of te solution u,k on te data g,k, u D,k, u.

8 82 3 Parabolic Differential Equations (3.2.15) Definition: Let u,k and z,k be te solutions of te finite difference metod (3.2.12a)-(3.2.12c) wit respect to te data g,k, u D,k, u and te perturbed data g,k + δg,k, u D,k + δud,k, u + δu wit grid functions δg,k C(Ω ), δu D,k C(Γ Īk), δu C(Ω ). Ten, te finite difference metod (3.2.12a)-(3.2.12c) is said to be stable, if for any ε > tere exists δ = δ(ε,, k) > suc tat u,k z,k,k < ε for all perturbations δu and (δg,k, δu D,k ) satisfying δu + (δg,k, δu D,k,k < δ. As in te case of finite difference metods for elliptic boundary value problems, te stability of consistent finite difference approximations is a sufficient condition for convergence. (3.2.16) Teorem: Assume tat (3.2.12a)-(3.2.12c) is a finite difference metod being consistent wit te well-posed initial-boundary value problem (3.2.1a)- (3.2.1c). If it is stable, ten it is convergent, and te order of convergence is te same as te order of consistency. Proof. Te solution u of te initial-boundary value problem (3.2.1a)-(3.2.1c) satisfies te equations L,k u(x, t) = g,k (x, t) + τ,k (x, t), x Ω, t Īk \ {}, u(x, t) = u D,k(x, t) + τ,k (x, t), x Γ, t Īk \ {}, u(x, ) = u (x) + τ,k (x, ), x Ω. Setting δg,k (x, t) = τ,k (x, t), x Ω, t Īk \ {}, δu D,k = τ,k(x, t), x Γ, t Īk \{}, and δu (x) = τ,k(x, ), x Ω, and observing τ,k,k for, k, te assertion follows from te stability (3.2.15). Remark ( ) In te literature, Teorem (3.2.16) is sometimes referred to as te Teorem of Lax (cf., e.g., Tomas (1995)). Under te same assumptions, it can be furter sown tat te stability also constitutes a necessary condition for te convergence of a consistent finite difference metod. Te equivalence of te stability and te convergence of consistent finite difference approximations of well-posed initial-boundary value problems is called te Lax-Rictmyer equivalence teorem. For a proof we refer to Strikwerda (24). Te finite difference (3.2.12a)-(3.2.12c) leads to a linear algebraic system. Hence, stability can be verified by arguments from numerical linear algebra. Tis will be illustrated by te subsequent example. Example. We study te stability of te Θ-sceme (3.2.1a)-(3.2.1c) applied to te eat equation (3.1.3a)-(3.1.3c) in Ω = (a, b) 2, a, b R, a < b. For Ω =

9 {x 1,, x n ) T, we denote by u (j) 3.3 Weak solutions 83 te vector u (j) = (u, k(x 1, t j ),, u,k (x n, t j)) T, j M + 1. Ten, (3.2.1a)-(3.2.1c) is equivalent to te linear algebraic system (3.2.17) (I + kb (Θ))u (j) = (I kc (Θ))u (j 1) + b (j). Here, B (Θ) = ΘA, C (Θ) = (1 Θ)A, were A R n n is a block-tridiagonal matrix and b (j) R n. If δu () R n and δb (j) R n, 1 j M + 1, are perturbations in te initial data and te rigt-and side, respectively, and z (j) denotes te solution of (3.2.17) wit respect to te perturbed data, by induction we can sow (3.2.18) u (j) z(j) D (Θ) j δu () + X j 1 + l= D (Θ) j l 1 (I + kb (Θ)) 1 δb (l), were D (Θ) := (I + kb (Θ)) 1 (I kc (Θ)). Note tat te norms stand for te Euclidean vector norm and te associated spectral norm of a matrix, respectively. It follows from (3.2.18) tat for stability it suffices to sow ρ(d (Θ)) < 1 were ρ(d (Θ)) is te spectral radius of D H (Θ). Since we know te eigenvalues of A, te eigenvalues of D (Θ) can be assessed easily (Exercise). Under te constraint k 2 (2 4Θ), Θ < 1/2, we deduce ρ(d (Θ) < 1, wereas ρ(d H (Θ) < 1, Θ 1/2, witout any constraints. Hence, te Θ-sceme is conditionally stable for Θ < 1/2 and unconditionally stable for Θ 1/2. For furter results on te stability of finite difference metods for parabolic initial-boundary value problems we refer to Kreiss and Lorenz (1989), Strikwerda (24) and Tomas (1995). 3.3 Weak solutions We consider te following initial-boundary value problem for a parabolic differential equation (3.3.1a) u + A(t)u = f t in Q := Ω (, T ), (3.3.1b) u = on Γ (, T ), (3.3.1c) u(, ) = u in Ω. Here, Ω R d is a bounded Lipscitz domain wit boundary Γ = Ω and T >. We furter assume A(t), t (, T ), to be te formally self-adjoint second order linear elliptic differential operator ( ) A(t)u(, t) := a(, t) u(, t) + b(, t) u(, t) + c(, t)u(, t).

10 84 3 Parabolic Differential Equations We suppose tat a = (a ij ) d i,j=1, a ij C[, T ], L (Ω)), 1 i, j d, and assume tat for all t (, T ) te matrix-valued function (a ij (t, )) d i,j=1 is uniformly positive definite in Ω, i.e., tere exists a constant α > suc tat d a ij (t, x)ξ i ξ j α ξ 2, ξ R d. i,j=1 Moreover, we suppose tat b = (b 1,, b d ) T suc tat b i C([, T ], L (Ω)), 1 i d, and c C([, T ], L (Ω)) suc tat c(x, t), t [, T ], for almost all x Ω as well as f L 2 ((, T ), H 1 (Ω)) and u L 2 (Ω). Following Renardy and Rogers (1993), we provide a weak solution concept for initial-boundary value problems associated wit parabolic differential operators in a more general framework: Let H and V be Hilbert spaces wit continuous and dense embeddings V H V were H is identified wit its dual H. We denote by, V,V te dual product of V and V. We refer to L 2 ((, T ); H) as te Hilbert space equipped wit te norm T u 2 L 2 ((,T );H) := u(t) 2 H dt and define L 2 ((, T ); V ) and its norm analogously. Moreover, we denote by H 1 ((, T ); V ) te Hilbert space wit te norm T u 2 H 1 ((,T );V ) := ) ( u(t) 2 V + u t(t) 2 V dt. (3.3.2) Lemma: Te space L 2 ((, T ); V ) H 1 ((, T ); V ) is continuously embedded in C([, T ]; H). Proof. We assume u C 1 ([, T ]; H). For t [, T ], we coose t [, T ] suc tat It follows tat u(t ) 2 H = T 1 u(t) 2 H = u(t ) 2 H + 2 u(t) 2 H dt = T 1 u 2 L 2 ((,T );H). t t T T 1 u 2 L 2 ((,T );H) + 2 (u s (s), u(s)) H ds u t V u V dt T 1 u 2 L 2 ((,T );H) + 2 u H 1 (,T );V ) u L 2 ((,T );V ).

11 We conclude using tat C 1 ([, T ]; H) is dense in C([, T ]; H). 3.3 Weak solutions 85 Now, let A(t) : V V, t [, T ], be a bounded linear operator depending continuously on t. We assume tat te bilinear form a(t,, ) : V V R given by (3.3.3) a(t, u, v) := A(t)u, v V,V, t [, T ] satisfies te coercivity condition (3.3.4) a(t, v, v) α v 2 V β v 2 H, v V, t [, T ], were α and β are positive constants independent of t [, T ]. Given an initial condition u H and a rigt-and side f L 2 ((, t), V ), we consider te evolution equation (3.3.5a) (3.3.5b) u t A(t)u = f(t), u() = u. (3.3.6) Teorem: Let H, V, A(t) : V V, t [, T ], and u, f be given as above. Ten, te evolution equation (3.3.5a),(3.3.5b) admits a unique solution u L 2 ((, T ), V ) H 1 ((, T ), V ). Proof. Witout restriction of generality we may assume tat a(t,, ) is uniformly V -elliptic, i.e., β = in (3.3.4). For te proof of te existence of a solution we use te tecnique of te Galerkin approximation, i.e., we approximate (3.3.5a),(3.3.5b) by a sequence {V n } N of finite dimensional subspaces V n V, n N, suc tat V n = span{ϕ 1,, ϕ n }, were ϕ n, n N, are linearly independent elements in V wose linear ull is dense in V. We denote by P n, n N, te ortogonal projection of H onto V n and define u n C 1 ([, T ]; V n ) as te solution of te initial value problem (3.3.7a) (3.3.7b) ( du n dt, ϕ i) H + a(t, u n, ϕ i ) = f(t), ϕ i V,V, 1 i d, u n () = P n u. Te ansatz u n (t) = n i=1 κ i(t)ϕ i sows tat (3.3.7a),(3.3.7b) represents an initial value problem for a system of linear ordinary differential equations of first order in (κ 1,, κ n ) T wic as a unique solution. Observing P n u H u H, n N, we deduce te existence of a constant C >, independent of n N, suc tat ) (3.3.8) u n L 2 ((,T );V ) C ( f L 2 ((,T );V ) + u H, n N. Consequently, te sequence {u n } N is bounded in L 2 ((, T ); V ). Tis implies te existence of a subsequence N N and an element u L 2 ((, T ); V ) suc tat

12 86 3 Parabolic Differential Equations (3.3.9) u n u (n N, n ) in L 2 ((, T ); V ). Now, suppose tat ϕ N C ((, T ); V N ), N N is given by (3.3.1) ϕ N = N γ i (t)ϕ i, i=1 were γ i C ((, T )), 1 i d. In view of (3.3.7a), for n N tere olds (u n, dϕ N dt ) H + a(t, u N, ϕ N ) = f(t), ϕ N V,V. Since C ((, T ); V N ) is dense in C ((, T ); V ), for all ϕ C ((, T ); V ) we tus ave (u n, ϕ t ) H + a(t, u N, ϕ) = f(t), ϕ V,V. Due to (3.3.9), integration over (, T ) and passing to te limit n implies tat for all ϕ C ((, T ); V ) (3.3.11) (u, ϕ t ) H dt + a(t, u, ϕ) dt = From (3.3.11) we deduce u/ t L 2 ((, T ); V ) and (3.3.12) u t, ϕ V,V dt = f(t), ϕ V,V dt. (u, ϕ t ) H dt, ϕ C ((, T ); V ). Consequently, u H 1 ((, T ); V ) wence u C([, T ]; H) by Lemma Using (3.3.12) in (3.3.11) it follows tat u satisfies te evolution equation (3.3.5a). It remains to be sown tat u fulfills te initial condition u() = u H. Let ϕ N as in (3.3.1) wit γ i C ([, T ]), γ i (T ) =, 1 i N, suc tat for n N (u n, dϕ N dt ) H + a(t, u N, ϕ N ) = (u n (), ϕ N ()) H + f(t), ϕ N V,V. Since te linear space of suc functions ϕ N is dense in {ϕ H 1 ((, T ); V ) ϕ(t ) = }, te previous equation olds true for all suc functions. Observing P n u u (n ) in H, integration over (, T ) and passing to te limit n yields (3.3.13) (u, ϕ t ) H dt + = (u, ϕ()) H + a(t, u, ϕ) dt = f(t), ϕ V,V dt.

13 3.4 Finite element metods 87 On te oter and, multiplication of (3.3.5a) by a function ϕ H 1 ((, T ); V ), integration over (, T ) and subsequent partial integration results in (3.3.14) (u, ϕ t ) H dt + = (u(), ϕ()) H + a(t, u, ϕ) dt = f(t), ϕ V,V dt. Inserting (3.3.14) into (3.3.13) gives (u(), ϕ()) H = (u, ϕ()) H wic proves u() = u. As far as te uniqueness of a solution is concerned, we obtain from (3.3.3) 1 ( ) t u(t) 2 H u() 2 H + a(s; u, u) ds = 2 t (f, u) H ds. Using te coercivity of te bilinear form results in te a priori estimate ) (3.3.15) u L 2 ((,T );V ) C ( f L 2 ((,T );V ) + u H. Now, if u 1, u 2 L 2 ((, T ); V ) H 1 ((, T ); V ) are two solutions of te evolution equation, it follows tat z := u 1 u 2 satisfies (3.3.5a),(3.3.5b) wit f = and z =. Te estimate (3.3.15) sows z =. 3.4 Finite element metods Let V H 1 (Ω) be a Hilbert space and let a(t,, ) : V V R be a bilinear form tat is uniformly coercive t [, T ], T R +,. Furter, assume u L 2 (Ω) and f L 2 ((, T ); V ). According to Teorem (??), te evolution equation (3.4.1a) (3.4.1b) (u t, v),ω + a(t, u, v) = < f(t), v > V,V, v V, (u(), v),ω = (u, v),ω, v V, admits a unique solution u L 2 ((, T ); V ) H 1 ((, T ); V ). Te standard finite element metod for te numerical solution of tis evolution equation consists in te discretization in space by restricting (3.4.1a),(3.4.1b) to a finite element space V V, dim V = n. Tis leads to te semidiscrete approximation (3.4.2a) (3.4.2b) ((u ) t, v ),Ω + a(t, u, v ) = < f(t), v > V,V, v V, (u (), v ),Ω = (u, v ),Ω, v V.

14 88 3 Parabolic Differential Equations As we already know from te proof of Teorem (3.3.6), te equations (3.4.2a), (3.4.2b) represent an initial value problem for a system of linear ordinary differential equations wic admits a unique solution: If (ϕ (i) )n i=1 is a basis of V, we can write te solution u (t) V, t [, T ], according to u (t) = n j=1 κ j(t)ϕ (i) and tus obtain (3.4.3a) (3.4.3b) M κ (t) + A (t)κ(t) = b (t), t [, T ], κ() = κ. Here, M R n n stands for te mass matrix M := (ϕ (1), ϕ(1) ),Ω (ϕ (n ), ϕ (1) ),Ω (ϕ (1), ϕ(n ) ),Ω (ϕ (n ), ϕ (n ) ),Ω and A (t) R n n, t [, T ], denotes te stiffness matrix A (t) := a(t, ϕ (1), ϕ(1) ) a(t, ϕ(n ), ϕ (1) ) a(t, ϕ (1), ϕ(n ) ) a(t, ϕ (n ), ϕ (n ) ), wereas te vectors b (t) R n, t [, T ], and κ R n are given by b (t) := (< f(t), ϕ (1) > V,V,, < f(t), ϕ (n ) > V,V ) T, κ = ((u, ϕ (1) ),Ω,, (u, ϕ (n ) ),Ω ) T. For te discretization in time wit respect to a partition Īk of te time interval [, T ] according to (3.2.4), we use an approximation of te time derivative by te forward or backward difference quotient or a convex combination of bot as in te Θ-metod described in capter??. In particular, te Θ-metod requires te computation of u,k V, k M + 1, as te solution of (3.4.4a) (3.4.4b) k 1 M (u,k+1 u,k ) + ΘA (t k+1 )u,k (1 Θ)A (t k )u,k = b Θ (t k, t k+1 ), k M, u, = u, were b Θ (t k, t k+1 ) := Θb (t k+1 ) + (1 Θ)b (t k ) and u = n j=1 κ j ϕ(j). We derive a priori estimates of te global discretization error u(t) u,k (t), t Īk, in te special case of te linear eat equation, i.e., A(t) = A = in (3.3.1a). For te general case we refer to Tomée (1997). We assume tat te finite element spaces V satisfy te following approximation property: For v H r+1 (Ω) tere olds

15 3.4 Finite element metods 89 ( ) inf v v,ω + v v 1,Ω C r+1, v V For te backward difference quotient in time, i.e., Θ = 1 in (3.4.4a),(3.4.4b), we first consider an estimate in te H 1 -seminorm. (3.4.5) Teorem: Let u H 1 ((, T ); H r+1 (Ω)) H 2 ((, T ); L 2 (Ω)) be te solution of te initial-boundary value problem (3.3.1a)-(3.3.1c) mit A(t) = A = and let u,k be te solution of (3.4.4a),(3.4.4b) in case Θ = 1. Ten, tere exists a constant C >, independent of and k, suc tat for all m M + 1 u(t m ) u,k (t m ) 1,Ω C( r + k). Proof. We set e(t m ) := u(t m ) u,k (t m ). Denoting by P a : V V te elliptic projection onto V, te error can be split according to e(t m ) = e 1 (t m ) + e 2 (t m ) mit (3.4.6) e 1 (t m ) := u(t m ) P a u(t m ), e 2 (t m ) := P a u(t m ) u,k (t m ). Te term e 1 (t m ) can be estimated from above as follows (3.4.7) e 1 (t m ) 1,Ω C r u(t m ) r+1,ω ( t m C r u r+1,ω + u t (t) r+1,ω dt. On te oter and, e 2 (t m ) satisfies te variational equation (3.4.8) (D k e 2(t m ), v ),Ω + ( e 2 (t m ), v ),Ω = (g(t m ), v ), v V, were g(t m ) := P a D k u(t m) u t (t m ). If we coose v = D k e 2(t m ) in (3.4.8), we obtain k D k e 2(t m ) 2,Ω + ( e 2 (t m ), e 2 (t m )),Ω = (e 2 (t m ), e 2 (t m 1 )) + k (g(t m ), D k e 2(t m )),Ω, wence ( ) e 2 (t m ) 2 1,Ω C e 2 (t m 1 ) 2 1,Ω + k g(t m ) 2,Ω. A repeated appliccation of tis reasoning results in ( (3.4.9) e 2 (t m ) 2 1,Ω C e 2 () 2 1,Ω + k m ) g(t l ) 2,Ω. In view of te assumption u u,k 1,Ω C r, for te first term on te rigt-and side in (3.4.9) it follows tat l=1

16 9 3 Parabolic Differential Equations (3.4.1) e 2 () 1,Ω u u,k 1,Ω + u P a u 1,Ω C r. In order to estimate te second term, we split g(t l ) according to g(t l ) = g 1 (t l ) + g 2 (t l ) were g 1 (t l ) := k 1 t l t l 1 (P a I)u t (t) dt, g 2 (t l ) := k 1 (u(t l ) u(t l 1 ) ku t (t l )). Anoter application of Teorem (??) yields k g 1 (t l ) 2,Ω C r t l t l 1 u t (t) 2 r+1,ω dt. Taking advantage of we get g 2 (t l ) = k 1 t l k g 2 (t l ) 2,Ω C k 2 t l 1 (t t l 1 )u tt (t) dt, t l t l 1 u tt (t) 2,Ω dt. If we sum over all l, te two previous estimates give (3.4.11) k C m g(t l ) 2,Ω l=1 ( 2r t m t m ) u t (t) 2 r+1,ω dt + k 2 u tt (t) 2,Ω dt. Te assertion now follows by using (3.4.1),(3.4.11) in (3.4.9) as well as in view of (3.4.7). Under stronger regularity assumptions on te classical solution of te initial-boundary value problem, for te Crank-Nicolsen-Galerkin metod, i.e., Θ = 1/2, te order of convergence 2 in k can be establised. (3.4.12) Teorem: Let u H 1 ((, T ); H r+1 (Ω)) H 2 ((, T ); H 2 (Ω)), u tt L 2 ((, T ); L 2 (Ω)) be te solution of te initial-boundary value problem (3.3.1a)-(3.3.1c) in case A(t) = A, = and let u,k be te solution of (3.4.4a),(3.4.4b) for Θ = 1/2.

17 3.4 Finite element metods 91 Ten, tere exists a constant C >=, independent of and k, suc tat for all m M + 1 u(t m ) u,k (t m ) 1,Ω C( r + k 2 ). Proof. We split te error e(t m ) = u(t m ) u,k (t m ) as in (3.4.6) and estimate e 1 (t m ) from above as in (3.4.7). Te second error term e 2 (t m ) satisfies te variational equation (3.4.13) (D k e 2(t m ), v ),Ω + ( (e 2 (t m ) + + e 2 (t m 1 ))/2, v ),Ω = (g(t m ), v ),Ω, v V, were g(t m ) := P a D k (u t(t m ) + u t (t m 1 ))/2. If we coose v := (e 2 (t m ) + e 2 (t m 1 ))/2 and split g(t m ) according to g(t m ) = (P a I)D k u(t m) + (D k u(t m) u t (t m k/2)) + (u(t m k/2) (u(t m ) + u(t m 1 ))/2) we can estimate e 2 (t m ) from above as in te proof of Teorem (3.4.5). We know tat for (r + 1)-regular elliptic boundary value problems we can acieve an optimal a priori estimate in te L 2 -norm, i.e., compared to te a priori estimate in te H 1 -norm we gain a power in. Tis result can be obtained for te linear eat equation as well. (3.4.14) Teorem: Assume tat te conditions of Teorem (3.4.5) and Teorem (3.4.12) old true and tat te boundary value problem for te Poisson equation is (r + 1)-regular. Ten, tere exists a constant C >=, independent of and k, suc tat for all m M + 1 u(t m ) u,k (t m ),Ω C( r+1 + k p ), were p = 1 for Θ = 1 and p = 2 for Θ = 1/2. Proof. Te proof can be accomplised as in Teorem (3.4.5) and as in Teorem (3.4.12), if we coose v = e 2 (t m ) in (3.4.8) and v = (e 2 (t m ) + e 2 (t m 1 ))/2 in (3.4.13). Remark ( ) As far as a priori estimates in te L -norm are concerned, we refer to Tomée (1997).