Age-time continuous Galerkin methods for a model of population dynamics

Transcription

1 Journal of Computational and Applied Mathematics 223 (29) Age-time continuous Galerkin methods for a model of population dynamics Mi-Young Kim, Tsendauysh Selenge Department of Mathematics, Inha University, Incheon , South Korea Received 2 March 26; received in revised form 6 November 27 Abstract Continuous Galerkin finite element methods in the age-time domain are proposed to approximate the solution to the model of population dynamics with unbounded mortality (coefficient) function. Stability of the method is established and a priori L 2 - error estimates are obtained. Treatment of the nonlocal boundary condition is straightforward in this framework. The approximate solution is computed strip by strip marching in time. Some numerical examples are presented. c 28 Elsevier B.V. All rights reserved. MSC: 65M12; 92D25; 65C2 Keywords: Age-dependent population dynamics; Integro-differential equation; Nonstandard finite element method; Error estimate; Stability of the scheme 1. Introduction We consider the following linear initial-boundary value problem due to Lotka McKendrick [7]: u t + u a + µ(a)u =, < a < a Ď, < t < T, u(, t) = aď β(a)u(a, t)da, < t < T, u(a, ) = u (a), a < a Ď. Here, the function u(a, t) denotes the age-specific density of individuals of age a at time t. The demographic parameters µ = µ(a) and β = β(a) are the age-specific mortality and natality rates, respectively. System (1.1) describes the evolution of the age density u(a, t) of a population within a maximum age a Ď <, whose growth is regulated by the vital rates β(a) and µ(a), [6,7,21]. While there has been considerable interest in the numerical solution of (1.1) by the finite difference methods, [1 3,5,9,1,13 17,19], little attention has been devoted to applying (1.1) Corresponding author. Tel.: ; fax: address: mikim@inha.ac.kr (M.-Y. Kim) /$ - see front matter c 28 Elsevier B.V. All rights reserved. doi:1.116/j.cam

2 66 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) finite element methods. The reason for this is probably that the standard Galerkin method does not yield optimal order of error estimates when it is applied to first-order hyperbolic equations [4,18] and that it requires special care to treat the boundary condition. Meanwhile, in [18] Milner has applied a standard Galerkin finite element method in age combining with backward Euler in time to approximate the solution to the model of population dynamics. Milner obtained a O(h r )-convergent rate. Here, r is the degree of the polynomials in the approximate space. In [11,12], one of the authors has proposed discontinuous Galerkin (DG) methods. The author obtained a O(h r+1/2 )- convergent rate when the vital rate was bounded, which is known to be optimal in DG methods. In this paper, we propose continuous Galerkin (CG) methods in the age-time domain to approximate the solution to (1.1) and we obtain a O(h r+1 )-convergent rate. The CG methods proposed here allow adaptive meshes in age and time stepping as in the DG methods [11,12]. Treatment of the nonlocal boundary condition (1.1) 2 is straightforward. The approximate solution is computed by marching through successive time levels as in the DG method [12]. In the analysis of the method we will follow the idea of [22]. The plan of the paper is as follows. In Section 2, we describe the finite element spaces and define the finite element method. In the approximation, we use continuous trial functions and discontinuous test functions both in age and time. In Section 3, we provide some local properties of the approximate solution. In Section 4, we prove stability of the method using local properties shown in Section 3. In Section 5, we derive error estimates for the method using the stability result. Finally, in Section 6, we present some numerical results. 2. Numerical method Throughout the paper we assume the following hypotheses [7]. (H1) The nonnegative function u belongs to L 2 [, a Ď ]. (H2) β L [, a Ď ] and β(a) β for some positive constant β. (H3) µ(a) and µ L 1 loc [, a Ď]. Under the assumptions (H1) (H3), it is known [7] that problem (1.1) has a unique nonnegative solution, global in time. Due to the finite maximum age a Ď, the mortality rate µ is unbounded near the maximum age [7]. Following [8], we assume the following general growth rates near a Ď. (H4) There exist positive constants a (< a Ď ) and µ such that µ(a) µ for a [, a ]. (H5) µ(a) = λ/(a Ď a) α on (a, a Ď ], for some α 1, λ > 1. Under the assumption (H4) (H5), the solution to (1.1) is regular enough provided that the compatibility condition at the origin is satisfied with smooth data [13] and the CG method can be applied to approximate the solution [13]. Here we note that, if < α < 1 and λ >, then the solution is not regular. In that case, DG methods would be a proper choice for the effective computation [11,12]. We now describe the finite element spaces used in this paper. Let be a family of partitions of [, a Ď ]; i.e., if δ, then δ = {a i } M+1 i=, where = a < a 1 < < a M+1 = a Ď. We shall use the notation I i = (a i 1, a i ), h i = a i a i 1, and h = max 1 i M+1 h i. For the approximations U of u, we first note that u(a Ď, t) [13]. We thus let U(a, t), for (a, t) I M+1 J. We then assume that the computational domain is I J = [, A] [, T ] with A = a M. For integers q 1 and 1 r q 1, define S δ (q, r) = {ϕ C (r) (I ) ϕ Ii P q, 1 i M} where P q denotes the set of polynomials of degree < q, q 1, and C ( 1) (I ) denotes the set of piecewise continuous functions on I. Similarly, let Γ be a family of partitions of J. If γ = {t n } N n= Γ, we let J n = (, t n ), k n = t n and k = max 1 n N k n, and, if p 1 and 1 s < p 1, then the spaces S γ (p, s) are defined in the same way as above.

3 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) We are interested in partitions (δ, γ ) belonging to a subset Ω of Γ. We assume that there is a constant τ > 1 such that, for all (δ, γ ) Ω, τ 1 k n h i τ, 1 i M + 1, 1 n N. (2.1) In the remainder of this paper, we assume that p and q are fixed integers 1 such that p q + 1. For each (δ, γ ) Ω, we now let W h = S δ (q + 1, ), W k = S γ (p + 1, ), V h = S δ (q, 1), and V k = S γ (p, 1). We then define W h,k and V h,k to be the tensor product spaces, respectively, W h,k = W h W k, V h,k = V h V k. Note that if U W h,k then U at V h,k. For any integer r, let H r (I ) denote the standard Sobolev space of L 2 -functions on I with r derivatives in L 2 (I ), and, if r <, then H r (I ) be the dual of H r (I ) with respect to the inner product on L 2 (I ). We shall frequently use the notation r = H r (I ) and =. Also, if r, s are integers and ϕ C (I J), define ϕ 2 r,s = r i= j= s ( a ) i ( t ) j ϕ 2 L 2 (I J). We let H r,s (I J) be the completion of C (I J) in this norm. Throughout this paper, C will denote a generic constant, not necessarily the same at different occurrences. We then observe that the spaces S δ (q, r) have the approximation property that, for any integer j, r + 1 j q, there is a constant C > such that inf ϕ S δ (q,r) i= r+1 h i ω ϕ i Ch j ω j, for all ω H j (I ). (2.2) We also observe from (2.1) and (2.2) that there is a constant C > such that, for all ϕ W h and j q, i=1 ϕ H j (I i ) { C h j ϕ, C k j ϕ, (2.3) and similar inverse properties hold for the other spaces above. The continuous Galerkin method we shall analyze is then given as follows. Find U W h,k such that T U(, t) = (U t + U a + µu)ϕda dt =, for ϕ V h,k, U(a, ) = U (a). β(a)u(a, t)da, (2.4) Here U W h is chosen as an approximation of u and we assume that the compatibility condition at the origin, U () = β(a)u (a)da, is satisfied. We note that dim(w h,k ) = (Mp + 1)(Nq + 1), dim(v h,k ) = MpNq, and that the initial and boundary conditions in (2.4) represent Mp + Nq + 1 linear equations. We also observe that the method (2.4) is a time-stepping method in the sense that U Jn can be computed from U(, ) and U(, t) Jn.

4 662 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) Local properties In this section we derive some preliminary results to prove stability of the method. Since ϕ V h,k varies independently on each I i J n, 1 i M, 1 n N, we see that (2.4) 1 is equivalent to 1 1 ( Û t + k ) n Û a + k n µû ϕda dt =, ϕ P q ([, 1]) P p ([, 1]), (3.1) h i where Û(a, t) = U(a i 1 + h i a, + k n t) and µ(a) = µ(a i 1 + h i a). We now define the bilinear form L : M N R by 1 1 ( L(w, v) = w t + k ) n w a vdadt, h i where M = P q+1 ([, 1]) P p+1 ([, 1]) and N = P q ([, 1]) P p ([, 1]). Concerning the property of L, we have the following Lemmas 3.1 and 3.2. Lemma 3.1. Let p and q be integers such that p, q 1 and p q + 1. Assume that w M = {ϕ M ϕ(a, ) ϕ(a, 1) } is the solution of L (w, v) =, v N. Then w. For the proof of Lemma 3.1, see [22]. Lemma 3.2. The conditions inf sup L(w, v) v N w M w L 2 ([,1] [,1]) v ς 1 > L 2 ([,1] [,1]) and sup v N inf w M L(w, v) w L 2 ([,1] [,1]) v ς 2 > L 2 ([,1] [,1]) are necessary and sufficient to guarantee that there there exists a unique solution in M such that L(w, v) = F(v) for all v N, where F is a linear functional on N. See [2] for the proof. Lemma 3.3. Let p, q 1, p q + 1 and let w M be a solution of L(w, v) = for all v N. Then there is a constant C such that Proof. Let w 2 L 2 ([,1] [,1]) C{ w(, ) 2 L 2 ([,1]) + w(, 1) 2 L 2 ([,1]) }. Z(a, t) = w(a, t) w(a, 1)t w(a, )(1 t). (3.2) We then see that L(Z, ϕ) = 1 1 { w(a, 1) + w(a, ) w a (a, 1)t w a (a, )(1 t)}ϕdadt, and Z M. Therefore, we have, by (2.1), Lemmas 3.1 and 3.2, and the inverse estimate, that Z L 2 ([,1] [,1]) sup ϕ N sup ϕ N L(Z, ϕ) ϕ L 2 ([,1] [,1]) 1 for all ϕ N 1 {w(a, 1) + w(a, ) + w a(a, 1)t + w a (a, )(1 t)}ϕ dadt ϕ L 2 ([,1] [,1]) C{ w(, ) L 2 ([,1]) + w(, 1) L 2 ([,1]) },

5 or, by (3.2), that M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) w L 2 ([,1] [,1]) C{ w(, ) L 2 ([,1]) + w(, 1) L 2 ([,1]) }, where C is independent of the mesh. This completes the proof. We are now ready to prove the following lemma. Lemma 3.4. Let p, q 1, p q + 1. Assume that U W h,k is a solution of (2.4). Then there is a constant C such that for sufficiently small h and for 1 n N, U 2 L 2 (I J n ) Ck n{ U(, ) 2 + U(, t n ) 2 }. Proof. Let w M be the solution of L(w, v) = k n 1 1 We see, by Lemma 3.2, that w L 2 ([,1] [,1]) sup v N µwvdadt. 1 1 k n µwv dadt v L 2 ([,1] [,1]) Ck n w L 2 ([,1] [,1]). We then see, by (2.1), that w, for sufficiently small h > and thus that the solution of (3.3) is unique in M. We now note that Û is the solution of (3.3) with (given) Û(, ) and Û(, 1) at t = and t = 1, respectively. We also note that the solution is unique, since L(, v) is linear. Let z(a, t) = Û(a, t) Û(a, 1)t Û(a, )(1 t). We then see that z M and thus by following the proof of Lemma 3.3, that or Û L 2 ([,1] [,1]) C{ Û(, ) L 2 ([,1]) + Û(, 1) L 2 ([,1]) }, U L 2 (I i J n ) Ck n{ U(, ) L 2 (I i ) + U(, t n) L 2 (I i )}. (3.4) Summing on i we obtain that U L 2 (I J n ) Ck n{ U(, ) + U(, t n ) }. It completes the proof. 4. L 2 -stability We let P h and P k be the L 2 -projections onto the spaces V h and V k, respectively. Then, P h and P k can be extended to functions of a and t in an L 2 -sense. We define the extended projection P h : L 2 (I J) V h L 2 (J) by (P h ν, ϕ) = (ν, ϕ), for ϕ V h L 2 (J), and P k : L 2 (I J) L 2 (I ) V k by (P k ν, ϕ) = (ν, ϕ), for ϕ L 2 (I ) V k, where (, ) is the usual L 2 -inner product. Define P h,k : L 2 (I J) V h V k by P h,k = P h P k. We note that the operators in age and time commute. Let ε (, 1) be a fixed number. We now assume that if δ, then δ = {a i } M+1 i= satisfies = a < a 1 < < a M = a < < a M+1 = a Ď. We define a piecewise constant function g on I by g(a) = 1 A (1 ε)a i 1 + ε, for a I i. (3.3)

6 664 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) We observe that g(a + i ) g(a i ) = 1 A (1 ε)h i, (4.1) and ε g 1. We then define a norm h on W h by (4.2) ω 2 h = g(p h ω) 2 + h 2 gω a 2 + h α ω 2 L 2 ([a,a Ď ]). (4.3) We note from (2.3) and (4.2) that h is uniformly equivalent to + h α/2 L 2 ([a,a Ď ]) on W h for h sufficiently small. We are now ready to prove the following stability result. Theorem 4.1. Assume that U W h,k is a solution of (2.4). Then, there is a constant C independent of U such that, for sufficiently small h, max{ U(, t) + h α/2 U(, t) L 2 ([a,a Ď ]) } C U. (4.4) We note that, since (2.4) is a system of linear equations with as many equations as unknowns, the following corollary follows directly from Theorem 4.1. Corollary 4.2. The system (2.4) has a unique solution U W h,k for h sufficiently small. Proof of Theorem 4.1. We prove the estimate (4.4) by energy arguments. Let ϕ V h,k be given by { g(a)ph,k U(a, t), t J ϕ(a, t) = n,, otherwise. By using ϕ as a test function in (2.4), we obtain that (U t + U a + µu)g(a)p h,k Uda dt =. (4.5) First observe that, since for each a I, U t (a, ) V k, we have that U t g(a)p h,k Uda dt = Also, note that, by (4.2), we obtain = = 1 2 g(a)u t P h Uda dt d dt g P h U 2 dt g(a)p h U t P h Uda dt = 1 2 { g(p h U)(, t n ) 2 g(p h U)(, ) 2}. (4.6) µug(a)p h,k Uda dt C µu 2 L 2 (I J n ), (4.7) where the constant C is independent of U. Finally, we obtain from (4.1) that, for any n, 1 n N, = 1 2 U a g(a)p h,k Uda dt = 1 2 i=1 i=1 ai a i 1 d da {g(a)(p ku) 2 }da dt {g(ai )(P k U) 2 (a i, t) g(a + i 1 )(P ku) 2 (a i 1, t)}dt

7 Since M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) = 1 g(a )(P k U) 2 (a, t)dt 1 g( + )(P k U) 2 (, t)dt M 1 {g(a i + ) g(ai )}(P k U) 2 (a i, t)dt 2 i=1 = 1 { } 1 tn 2 A (1 ε)a M 1 + ε (P k U) 2 (a, t)dt ε (P k U) 2 (, t)dt 1 ε M 1 h i (P k U) 2 (a i, t)dt 2 2A i=1 ε (P k U) 2 (, t)dt 1 ε M 1 h i (P k U) 2 (a i, t)dt. (4.8) 2 2A i=1 M 1 i=1 h i {P k U(a i, t)} 2 dt C U 2 L 2 (I J n ), for some C >, it follows from (4.5) (4.8) that g(p h U)(, t n ) 2 g(p h U)(, ) 2 ε (P k U) 2 (, t)dt C{ U 2 L 2 (I J n ) + µu 2 L 2 }. (4.9) (I J n ) In order to obtain the control over the L 2 -norm of U(, t n ), we now use the function { g(a)uat (a, t), t J ϕ(a, t) = n,, otherwise, as a test function in (2.4) and we obtain that We observe that (U t + U a + µu)g(a)u at da dt =. (4.1) U a g(a)u at da dt = 1 2 = 1 2 We also have, by (2.3) and (4.2), that µug(a)u at da dt C d dt gu a 2 dt { gu a (, t n ) 2 gu a (, ) 2}. (4.11) µu U at da dt Ch 2 µu L 2 (I J n ), (4.12) where the constant C is independent of U. In the same way as above, we also obtain from (4.1) that ai U t g(a)u at dadt = 1 d 2 i=1 a i 1 da { gu t (a, ) L 2 (J n ) }2 da 1 [{ } ] 1 2 A (1 ε)a M 1 + ε U t (A, ) 2 L 2 (J n ) ε U t(, ) 2 L 2 (J n ) ε 2 U t(, ) 2 L 2 (J n ). (4.13)

8 666 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) We see from (4.1) (4.13) that there is a constant C > satisfying gu a (, t n ) 2 gu a (, ) 2 ε U t (, ) 2 L 2 (J n ) Ch 2 µu 2 L 2 (I J n ). (4.14) Adding (4.9) and h 2 times (4.14), we then obtain, by (2.4) 2 and (4.3), that U(, t n ) 2 h U(, ) 2 h ε P k U(, ) 2 L 2 (J n ) + εh2 U t (, ) 2 L 2 (J n ) + C{ U 2 L 2 (I J n ) + µu 2 L 2 (I J n ) } C{ε U(, ) 2 L 2 (J n ) + U 2 L 2 (I J n ) + µu 2 L 2 (I J n ) } C{ U 2 L 2 (I J n ) + h α U 2 L 2 }. (4.15) ([a,a Ď ] J n ) On the other hand, by summing (3.4) on i = M + 1,..., M, we have that U 2 L 2 ([a,a Ď ] J n ) Ck n{ U(, ) 2 L 2 ([a,a Ď ]) + U(, t n) 2 L 2 }. (4.16) ([a,a Ď ]) We thus see from (4.15) and (4.16) and Lemma 3.4 that U(, t n ) 2 h U(, ) 2 h Ck n { U(, t n ) 2 + U(, ) 2 } + Ck n h α {h α U(, t n ) 2 L 2 ([a,a Ď ]) + h α U(, ) 2 L 2 ([a,a Ď ]) }. We then have, by the equivalence of the norms on W h, that or (1 Ck n ) U(, t n ) 2 + (1 Ck n h α ) U(, t n ) 2 L 2 ([a,a Ď ]) (1 + Ck n ) U(, ) 2 + (1 + Ck n h α ) U(, ) 2 L 2 ([a,a Ď ]) (1 Ck n ) U(, t n ) 2 h (1 + Ck n) U(, ) 2 h. (4.17) Since (4.17) holds for n = 1,..., N, the result follows from the discrete analog of Gronwall s lemma. The following Corollary 4.3 is a direct consequence of Theorem 4.1. Corollary 4.3. Assume that V is a solution of the nonhomogeneous problem with data f (a, t) associated with (2.4). Then, under the same assumptions in Theorem 4.1, there exists a constant C > such that max{ V (, t) + h α/2 V (, t) } C{ U + f L 2 (I J n )}. (4.18) 5. L 2 -error analysis In this section, we shall use Theorem 4.1 to prove L 2 -error estimates for the approximation to (2.4). These estimates will be derived under certain smoothness assumptions on the solution u of (1.1) and hence the data in (1.1) has to satisfy certain compatibility conditions at the origin (, ). Lemma 5.1. Assume that w W h such that w() = and w a vda =, for all v V h. Then, for h sufficiently small, w. See [22] for the proof. Define π h : H 1 (I ) W h by (π h ω)() = ω() and (π h ω) a vda = ω a vda, for v V h.

9 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) We then note from Lemma 5.1, that the projection π h is well-defined for h sufficiently small. For any integer ν, let H ν (δ) denote the piecewise Sobolev space given by H ν (δ) = {w L 2 (I ) w Ii and let ν,δ be the associated norm; that is, w 2 ν,δ = w 2 H ν (I i ). i=1 sup ϕ H ν (δ) H ν (I i ), 1 i M}, If ν < is an integer, then we define H ν (δ) by duality with respect to the inner product on L 2 (I ); that is, H ν (δ) is the completion of L 2 (I ) in the norm w ν,δ = wϕda. ϕ ν,δ In the same way as above we define the spaces H ν (γ ), with norms ν,γ consisting of functions defined on J. We observe that if w H ν (δ) for some ν <, then w H ν (I ) and w ν w ν,δ. The following approximation property follows by a standard duality argument [22]. Lemma 5.2. There is a constant C such that if h is sufficiently small and ω H q+1 (I ), ω π h ω ν,δ Ch q+ν+1 ω q+1, for 1 ν q 1. The following super-convergence result for the operator π h now follows directly from Lemma 5.2. Lemma 5.3. There is a constant C such that, if h is sufficiently small and ω H q+1 (I ), max a δ ω(a) (π hω)(a) Ch 2q ω q+1. Proof. For a fixed a δ, define G(a, ) L 2 (I ) by { 1 if a a, G(a, a) = otherwise. Observe that for any ψ H 1 (I ) such that ψ() =, we have ψ(a) = ψ a G(a, a)da. Therefore, since G(a, ) H q 1 (δ), it follows from Lemma 5.2 that ω(a) (π h ω)(a) = {ω a (π h ω) a }G(a, a)da Ch2q ω q+1. This completes the proof. and Similar to the operator π h, define π k : H 1 (J) W k by (π k v)() = v() and T (π k v) t ϕdt = T v t ϕdt, for ϕ V k. (5.1) Hence, similar to the operator π h, we obtain that there is a constant C such that, for k sufficiently small, v π k v ν,γ Ck p+ν+1 v p+1 for 1 ν p 1 (5.2) max v(t) (π k v)(t) Ck 2p v H p+1 (J). (5.3)

10 668 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) We now extend π h to apply it to functions of a and t in an L 2 -sense. Abusing the notation, we define π h : H 1, (I J) W h L 2 (J) by T (π h ω) a ϕda dt = T ω a ϕda dt, for ϕ V h L 2 (J) (5.4) with (π h ω(, ), ϕ) = (ω(, ), ϕ) for ϕ L 2 (J) (the trace of ω H 1, (I J) is defined at a = ). Note, in particular, that if ω is smooth enough that ω(, t) H 1 (I ) for each t, we can define π h ω pointwise in time, and error estimates for the extended projection follow from Lemma 5.2. We also extend π k to apply it to functions of a and t in an L 2 -sense. Thus, define π k : H,1 (I J) L 2 (I ) W k by T (π k v) t ϕda dt = T v t ϕda dt, for ϕ L 2 (I ) V k (5.5) with (π k v(, ), ϕ) = (v(, ), ϕ) for ϕ L 2 (I ). We now define π h,k : H 1, (I J) H,1 (I J) W h,k by π h,k = π h π k. We observe that it follows from the identity I π h,k = I π h + I π k (I π h ) (I π k ), (5.6) and from Lemma 5.2 and (5.2) that there is a constant C such that, for any ω H q+1, (I J) H 1,p+1 (I J), Since (I π h,k )ω, C{h q+1 ω q+1, + k p+1 ω 1,p+1 }. (5.7) max (I π h )ω(, t) ω,1 by the Sobolev imbedding theorem, we have by (5.3) that max (I π h,k )ω(, t) C{h q+1 ω q+1, + k 2p ω 1,p+1 }. (5.8) We note that the operators π h and π k commute. We are now ready to prove the following convergence result for method (2.4). Theorem 5.4. Assume that u H q+1,1 (I J) H 1,p+1 (I J). There is a constant C, independent of u, such that max{ (u U)(t) + h α/2 (u U)(t) } C{h q+1 α u q+1,1 + h α k p+1 u 1,p+1 + π h u U }. Proof. Choose W = π h,k u such that W (, t) = β(a)π h,kuda. It is then enough by (5.8) to estimate θ = W U. We now note that where T (W t + W a + µw + ρ)ϕda dt =, for ϕ V h,k, (5.9) ρ = (I π h )u t + (I π k )u a + µ(i π h,k )u. We see, by Lemma 5.2, (5.2) and (5.6), that there exists a constant C satisfying ρ, (I π h )u t, + (I π k )u a, + µ(i π h,k )u, (I π h )u t, + (I π k )u a, { M + C (I π h,k )u L 2 (I i J) + h α i= i=m +1 (I π h,k )u L 2 (I i J) }

11 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) { M C h q+1 u H q+1,1 (I i J) + k M p+1 u H 1,p+1 (I i J) i= i= } + h q+1 α u H q+1,1 (I i J) + h α k p+1 u H 1,p+1 (I i J), (5.1) i=m +1 i=m +1 where M is an integer such that a M < a < a M +1. We see, by (2.4) and (5.9), that θ satisfies the following. T θ(, t) = (θ t + θ a + µθ + ρ)ϕda dt =, for ϕ V h,k, β(a)θ(a, t)da, θ(a, ) = π h,k u (a) U (a). We here note that θ(, ) W k. Hence, it follows from Corollary 4.3, Lemma 5.3, and (5.1) that max{ θ(, t) + h α/2 θ(, t) } C{h q+1 α u q+1,1 + h α k p+1 u 1,p+1 + π h u U }. This completes the proof. Remark 5.5. When µ and β depend on the total population, that is, µ = µ(a, z) and β = β(a, z) with z(t) = aď u(a, t)da, one can modify the method (2.4) by using an extrapolation of the same order p that is the degree of polynomial in time as follows: T P,n+1 = U(, t) = P(t) = (U t + U a + µ(a, P,n+1 ))Uϕda dt =, for ϕ V h,k, p ( ( 1) i 1 p ) P n i+1, i i=1 aď U(a, ) = U (a). β(a, P n+1 )U(a, t)da, U(a, t)da, (5.11) For the initialization, one may use the schemes based on method of characteristics ([14], for example) together with Richardson extrapolations. We here note that (5.11) 3 is nonlinear because β(a, P n+1 ) is not known in advance; in order to compute P n+1 we also need the value U n+1. To solve (5.11) 3, one can do a simple iteration as given in [13]. 6. Numerical examples In this section we present some numerical examples. In all the tests, we have used the bilinear polynomial space as an approximate space. In the tests, we computed the rate of convergence using the usual formula r i (h, k) = log ( E i (h, k)/e i ( h 2, k 2 )) log 2 Here E i (h, k) and E i (h) are the errors defined by, r i (h) = log ( E i (h)/e i ( h 2 )), i = 2,. log 2 E i (h, k) = U(, ) u(, ) l i (l i ) and E i (h) = U(, t ) u(, t ) l i, i = 2,, for some t and l i (l i ) and l i are the discrete li ([, T ]; l i ([, a Ď ])) and l i ([, a Ď ]) norms, respectively.

12 67 M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) Table 6.1 Convergence estimates for Example 6.1 k h E (h, k) r (h, k) E 2 (h, k) r 2 (h, k) 1/8 1/ E E 2 1/16 1/ E E /32 1/ E E /64 1/ E E /128 1/ E E /256 1/ E E Table 6.2 Convergence estimates with fixed k = 1/256 and t = 1/2 for Example 6.1 h E (h) r (h) E 2 (h) r 2 (h) 1/ E E 2 1/ E E / E E / E E Table 6.3 Convergence estimates for Example 6.2 k h E (h, k) r (h, k) E 2 (h, k) r 2 (h, k) 1/8 1/ E E 3 1/16 1/ E E /32 1/ E E /64 1/ E E /128 1/ E E /256 1/ E E Table 6.4 Convergence estimates with fixed k = 1/256 and t = 1/2 for Example 6.2 h E (h) r (h) E 2 (h) r 2 (h) 1/ E E 3 1/ E E / E E / E E / E E / E E Example 6.1. We solve problem (1.1) with the following data: a Ď = 1, β(a) = 2a(1 a), µ(a) = 1 exp( 1(1 a)). u (a) is chosen so that u(a, t) = ω exp( a µ(ξ)dξ α a) exp(α t) is the exact solution, where α and ω We took u (a) = ω exp(α t) as the initial data. We note that the compatibility condition at (, ) is satisfied, which guarantees the continuity of the solution u(a, t). The results are presented in Tables 6.1 and 6.2. In the next example we consider the case that the mortality function µ is unbounded. Example 6.2. We solve problem (1.1) with the following data: a Ď = 1, β(a, p) = e, µ(a) = 1 1 a, and u (a) = (1 a)e a. The results of the convergence rates are shown to Tables 6.3 and 6.4. Results of Examples 6.1 and 6.2 show the expected result from the theoretical estimate.

13 Acknowledgement M.-Y. Kim, T. Selenge / Journal of Computational and Applied Mathematics 223 (29) The research of the first author was supported by KRF(C14). References [1] S. Anita, M. Iannelli, M.-Y. Kim, E.-J. Park, Optimal harvesting for periodic age-dependent population dynamics, SIAM J. Appl. Math. 8 (5) (1998) [2] T. Arbogast, F.A. Milner, A finite difference method for a two-sex model of population dynamics, SIAM J. Numer. Anal. 26 (1989) [3] J. Douglas Jr., F.A. Milner, Numerical methods for a model of population dynamics, Calcolo 24 (1987) [4] T. Dupont, Galerkin methods for first order hyperbolics: An example, SIAM J. Numer. Anal. 1 (1973) [5] G. Fairweather, J.C. López-Marcos, A box method for a nonlinear equation of population dynamics, IMA J. Numer. Anal. 11 (1991) [6] M. Gurtin, R.C. MacCamy, Non-linear age-dependent population dynamics, Arch. Ration. Mech. Anal. 54 (1974) [7] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, in: Applied Mathematics Monographs, Comitato Nazionale per le Scienze Matematiche, Consiglio Nazionale delle Ricerche (C.N.R.), vol. 7, Giardini-Pisa, [8] M. Iannelli, F. Milner, On the approximation of the Lotka Mckendrick equation with finite life-span, J. Comput. Appl. Math. 136 (21) [9] M. Iannelli, M.-Y. Kim, E.-J. Park, A. Pugliese, Global boundedness of the solutions to a Gurtin MacCamy system, NODEA Nonlinear Differential Equations Appl. 9 (22) [1] M. Iannelli, F. Milner, A. Pugliese, Analytical and numerical results for the age structured SI S epidemic model with mixed inter-intracohort transmission, SIAM J. Math. Anal. 23 (1992) [11] M.-Y. Kim, Discontinuous Galerkin method for the Lotka McKendrick equation with finite life-span, Math. Models Methods Appl. Sci. 16 (2) (26) [12] M.-Y. Kim, Discontinuous Galerkin methods for a model of population dynamics with unbounded mortality, SIAM J. Sci. Comput. 27 (4) (26) [13] M.-Y. Kim, Y. Kwon, A collocation method for the Gurtin MacCamy equation with finite life-span, SIAM J. Numer. Anal. 39 (6) (22) [14] M.-Y. Kim, E.-J. Park, An upwind scheme for a nonlinear model in age structured population dynamics, Comput. Math. Appl. 3 (8) (1995) [15] Y. Kwon, C.-K. Cho, Second-order accurate difference methods for a one-sex model of population dynamics, SIAM J. Numer. Anal. 3 (1993) [16] T. Lafaye, M. Langlais, Threshold methods for threshold models in age dependent population dynamics and epidemiology, Calcolo 29 (1992) [17] L. Lopez, D. Trigiante, A hybrid scheme for solving a model of population dynamics, Calcolo 19 (1982) [18] F.A. Milner, A finite element method for a two-sex model of population dynamics, Numer. Methods Partial Differential Equations 4 (1988) [19] F.A. Milner, G. Rabbiolo, Rapidly converging numerical algorithms for models of population dynamics, J. Math. Biol. 3 (1992) [2] Ch. Schwab, p- and hp- Finite Element Methods, Oxford University Press Inc, [21] G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, Inc., New York, [22] R. Winther, A stable finite element method for initial-boundary value problems for first-order hyperbolic systems, Math. Comp. 36 (153) (1981)