Key words. finite element approximation, error bounds, reaction-diffusion system, 2-SBDF, Turing patterns, mammal coat markings, Thomas system

Transcription

1 A SECOND-ORDER, THREE LEVEL FINITE ELEMENT APPROXIMATION OF AN EXPERIMENTAL SUBSTRATE-INHIBITION MODEL MARCUS R. GARVIE AND CATALIN TRENCHEA Abstract. This paper concerns a second-order, three level piecewise linear finite element scheme 2-SBDF [J. RUUTH, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., , pp ] for approximating the stationary Turing patterns of a wellknown experimental substrate-inhibition reaction-diffusion Thomas system [D. THOMAS, Artificial enzyme membranes, transport, memory and oscillatory phenomena, in Analysis and control of immobilized enzyme systems, D. Thomas and J.P. Kernevez, eds., Springer, 975, pp. 5-50]. A numerical analysis of the semi-discrete in time approximations leads to semi-discrete a priori bounds and an optimal error estimate. The analysis highlights the technical challenges in undertaking the numerical analysis of multi-level 3 schemes. We illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology, namely, to approximate the Turing patterns of the Thomas system over a schematic mammal skin domain with fixed geometry at various scales. We also make some comments on the correct procedure for simulating Turing patterns in general reaction-diffusion systems. Key words. finite element approximation, error bounds, reaction-diffusion system, 2-SBDF, Turing patterns, mammal coat markings, Thomas system AMS subject classifications. 65M60, 92C5, 35K57, 65M5. Introduction and motivation. In this paper we study the numerical solutions of a 2-component reaction-diffusion system of Thomas [4], which after nondimensionalization [33] has the following form: u t = u + a u hu, v, v = δ v + αb v hu, v,. t hu, v = ρuv +u+ku 2. Here ux, t and vx, t are morphogen concentrations at time t and vector position x. is the Laplacian operator in two space dimensions, and the parameters δ, α, a, b, ρ, and K are strictly positive. We assume homogeneous Neumann boundary conditions and appropriate initial data additional details given below. System. is a popular reaction-diffusion system for studying pattern formation and has been used to investigate the mechanisms governing the differentiation and growth of the structure of an organism morphogenesis and mammalian coat formation [32, 34]. Complex systems with interacting components frequently give rise to emergent properties, or pattern formation phenomena. The study of models in nature for pattern formation is an intensive area of research. Examples include the distribution of plankton in the ocean [3], morphogenesis of organisms [43], chemotaxis [25], and cardiac arrhythmias [22], to cite just a few. A large number of pattern formation phenomena, including those cited above, can be modeled by the Turing [43] mechanism in reaction-diffusion systems. Turing demonstrated that for appropriate conditions on Department of Mathematics and Statistics, MacNaughton Building, University of Guelph, Guelph, ON Canada NG 2W, mgarvie@uoguelph.ca Corresponding author Department of Mathematics, 30 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA 5260, trenchea@pitt.edu

2 2 M.R. GARVIE & C. TRENCHEA the reaction kinetics and distinct diffusion coefficients, systems of two or more chemical species can react and diffuse to produce spatially heterogeneous solutions, namely, a spatial pattern. This phenomena is also known as diffusion-driven instability, as in the spatially homogeneous situation there is a linearly stable equilibrium solution that can become linearly unstable in the presence of diffusion. The applicability of the Turing mechanism to interacting ecological species was first shown in [27]. For additional background material on pattern formation in biology see [30, 34]. Many reaction-diffusion systems have been used to investigate morphogenesis in the Turing sense, however the precise mechanisms responsible for pattern formation in embryonic development are still unknown. Murray [33] showed that the major difference between them are in the parameter ranges for diffusion driven instability Turing Space, and that the Thomas [4] system possesses the largest Turing Space among the practical systems. The effective numerical approximation of reaction-diffusion equations for pattern formation requires special treatment. Consider a generic system of reaction diffusion equations of the form w t = D w + fw,.2 where w is the vector of chemical concentrations, f is the vector of nonlinear reaction kinetics and D is a diagonal matrix of diffusion coefficients. Assume that.2 has been discretised in space, e.g. by the finite element or finite difference method, to give a system of ordinary differential equations of the form Ẇ = D h W + FW,.3 where FW and h W arise from the approximation of the reaction kinetics and the diffusion terms respectively, and h is the discrete Laplacian operator. Many time-stepping schemes have been proposed for the approximation of.3 see e.g. [42] for an introduction. For example, the Backward Euler scheme e.g., [3], the Crank-Nicolson method e.g., [9], Runge-Kutta methods e.g., [29] and various semiimplicit linear solvers e.g., [7] have been effectively employed in the 2-component case. Ruuth [39] cf. [3] analyzed the performance of several linear multistep implicitexplicit IMEX schemes for reaction-diffusion equations in pattern formation, and found that some popular first-order schemes, as well as schemes that lead to weak decay of high frequency spatial errors, may yield plausible results that are qualitatively incorrect. Ruuth recommends a second-order IMEX scheme with strong decay of high frequency errors, namely, the semi-implicit backward differentiation formula 2-SBDF given by 3W n+ 4W n + W n = D h W n+ + 2FW n FW n..4 This scheme has also been refered to in the literature as Extrapolated Gear see [44] and others. As this scheme has three time levels, in addition to an initial approximation W 0, the time stepping procedure requires an approximation at the first step, namely W. One way to kick-start this procedure is to use a first-order IMEX scheme, for example -SBDF [39]. For additional works on multistep IMEX schemes see [5,, 24]. The main aim of this paper is to undertake the numerical analysis of the semidiscrete in time weak formulation of.. For the temporal discretization we employ

3 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 3 the second order 2-SBDF scheme.4, with starting values computed using a firstorder scheme -SBDF, which leads to a sparse system of linear algebraic equations for each time step. Compared to standard first-order schemes the numerical analysis of the second order scheme.4 is more challenging. For ease of exposition we present a priori estimates and an error analysis for the semi-discrete in time weak formulation. The numerical analysis of the corresponding fully-discrete problem requires additional standard techniques from the finite element method for elliptic problems, e.g., interpolation error estimates, inverse estimates, and error estimates for numerical integration [8]. Furthermore, the main challenge in the numerical analysis of the fully-discrete problem is due to the temporal discretization, namely, the higher order approximation of the time derivative and the asymmetry of the approximate reaction kinetics. The methodology in this paper generalizes some techniques applied previously to first-order, two level finite element methods for reaction-diffusion systems [7, 8, 6] to the three level setting. In addition to the theoretical aims given above, we present numerical results for a fully-discrete finite element approximation of. that demonstrate the characteristic Turing patterns of this system. For the spatial discretization we use the standard Galerkin finite element method with piecewise linear continuous basis functions and Lumped Masses, [35]. We also comment on the correct procedure for simulating pattern formation in general reaction-diffusion systems. In Section 2 we initially consider the spatially homogeneous system, followed by the construction of an invariant rectangle in phase space that leads to the global wellposedness of the classical solutions for the full reaction-diffusion system. The semidiscrete in time estimates and optimal error bound are derived in Section 3, while the fully-discrete approximations and results of numerical experiments are presented in Section 4. Some concluding comments are made in Section Mathematical preliminaries. 2.. Function spaces. Standard notation is used for the Sobolev spaces. The usual L 2 Ω inner product over Ω with norm is denoted,, and L p Ω, p <, is the space of pth-integrable functions on Ω, with norm 0,p. Furthermore, we denote L p loc Ω by the set {f : f Lp K for every K c Ω}, where c denotes compact injection. Continuous injection is denoted. We let, denote the duality pairing between H Ω and H Ω. The H Ω semi-norm will be denoted by and the norm of the dual space H Ω is denoted. A standard Banach space we use is L Ω, with associated essential supremum norm u 0, u L Ω := inf{m : ux M a.e. on Ω}. We also define function spaces depending on space and time e.g., [40], p.45. Let X be a Banach space and p [, ]. Denote L p 0, T ; X to be the Banach space of all measurable functions u : 0, T X such that t ut X is in L p 0, T, with norm u Lp 0,T ;X := T 0 ut p X dt /p for p <, 2. u L 0,T ;X := ess sup ut X if p =. 2.2 t 0,T In addition, we write L p Ω T L p 0, T ; L p Ω.

4 4 M.R. GARVIE & C. TRENCHEA We assume the standard Hilbert space setup e.g., [40], p.55 V c H H V, 2.3 where each space is dense in the previous one, and indicates the explicit identification of elements in H and H. We recall the well-posedness of the Coercive Homogeneous Neuman Problem [4, p.225]: let f L 2 Ω and k be a finite constant, then there exists a unique solution u H Ω of the problem Ω u v + kuv dx = Ω fv dx, v H Ω Local analysis. In this section we briefly look at the spatially homogeneous system, namely. with no diffusion present. This will enable the construction of an invariant rectangle in phase space that provides an elementary proof of the wellposedness of the full reaction-diffusion system. The local dynamics can be analyzed by considering the nullclines zero-isoclines of this system, which are the solution curves for v = F u := a u + u + Ku2, ρu v = Gu := αb + u + Ku2 ρu + α + u + Ku 2, 2.5 corresponding to the first and second equations of.. nullclines leads to the following cubic equation The intersection of the αku 3 + aαk ρ αu 2 + aρ + aα αbρ αu + aα = 0, 2.6 which has either three, or one, positive real solutions, corresponding to the number of stationary points stable, or unstable. The typical nullclines are illustrated in Figure 2.. A consideration of the signs of the kinetic functions f and g on either side of their respective nullclines F and G, readily leads to the following arbitrarily large invariant region in the positive quadrant of phase space: A,B := { u, v [0, 2 : 0 u A, 0 v B, A a, B b }. 2.7 The case Σ a,b is illustrated in Figure Well-posedness of the reaction-diffusion system. Before proving wellposedness of the equations we need to establish the formal setting and re-state the substrate-inhibition system. with appropriate initial and boundary data. Let Ω be a bounded and open subset of R 2, with a boundary Ω of class C 2+s, s > 0, i.e., Ω is a dimensional C 2+ν manifold on which Ω lies locally on one side. The model problem is formulated as follows:

5 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 5 v = b nullcline for f = 0 nullcline for g = 0 f < 0 f > 0 g < 0 g > 0 0 u = a Figure 2.: Typical nullclines for the local kinetics f and g of. with a = 50, b = 5, ρ = 2, k = 0.07, and α =. Find the functions ux, t and vx, t such that u = u + a u hu, v in Q := Ω 0, T, 2.8a t v = δ v + αb v hu, v in Q, 2.8b t ρuv hu, v := + u + Ku 2, 2.8c ux, 0 = u 0 x, vx, 0 = v 0 x, x Ω 2.8d u ν = v ν = 0 on Ω 0, T, 2.8e where the parameters a, b, δ, α, ρ, and K are real and strictly positive, and ν denotes the outward normal to Ω. We assume the initial data u 0 x, v 0 x are in L Ω H Ω. It will be convenient to denote the reaction kinetics corresponding to 2.8a and 2.8b by fu, v and gu, v respectively. Theorem 2.. Let u 0 x, v 0 x L Ω. Then there exists a unique nonnegative classical solution of the substrate-inhibition system 2.8a-2.8e for all x, t Ω [0,. Furthermore, if the initial data is chosen in the invariant region A,B [0, 2 given by 2.7, then u, v A,B for all x, t Ω [0,. Proof. Local existence of solutions is based on well-known semigroup theory see for example Pazy [36], or Henry [20]. From Proposition in [23] it follows immediately that 2.8a-2.8e has a unique noncontinuable classical solution u, v for x, t Ω [0, T max. To prove global existence of solutions from local existence is straightforward. From Theorem 4.3 of [7] it follows that the invariant region A,B 2.7 is also invariant for the full PDE system. The invariant region yields an L - a priori bound that contradicts non-global existence as solutions either exist for all time, or blow-up in the sup-norm in finite time [5]. The substrate-inhibition system 2.8a-2.8e leads to the following weak formulations:

6 6 M.R. GARVIE & C. TRENCHEA P Find u, t, v, t H Ω L Ω such that {u, 0, v, 0} = {u 0, v 0 } and for almost every t 0, T u t, η + u, η = a u hu, v, η η H Ω, 2.9a v t, η + δ v, η = αb v hu, v, η η H Ω, 2.9b where hu, v is defined by 2.8c. The regularity of the classical solutions implies the following strong solution result, which facilitates estimates in later sections. Corollary 2.2. With the assumptions on the initial data in Theorem 2. the substrate-inhibition system 2.8a-2.8e possesses a unique strong solution {u, v} s.t. u, v L 2 0, T ; H 2 Ω C[0, T ]; H Ω, 2.0a u t, v t L2 Ω T. 2.0b 3. The semi-discrete in time approximations. We provide the analysis of a semi-discrete in time approximation of the problem discussed in the previous section. Discretization in time leads to a sequence of elliptic problems, that can be solved by marching in time from 0 to T. As the invariant region given by 2.7 is not necessarily invariant for the semidiscrete problem, it is advantageous to modify the reaction kinetics via ρuv + u + Ku 2 ρuv + u + Ku 2, which avoids the potential singularity of discrete approximations to hu, v and also facilitates the derivation of semi-discrete in time a priori estimates in Section 3.3. Of course for a correctly converged numerical solution such a modification makes no difference to the solutions. To help simplify the notation in the sequel we define modified reaction kinetics corresponding to 2.8a and 2.8b by fη, χ := a η ĥη, χ, ĝη, χ := αb χ ĥη, χ, 3. where ĥη, χ := ρηχ/ + η + Kη Thus the modified weak formulation can be written as: P: Find u, t, v, t H Ω L Ω such that {u, 0, v, 0} = {u 0, v 0 } and for almost every t 0, T u t, χ + u, χ = 2 fu, v fu, v, χ χ H Ω, 3.3a v t, χ + δ v, χ = 2ĝu, v ĝu, v, χ χ H Ω. 3.3b In later error analysis we shall need the following local Lipschitz condition on ĥ for an elementary proof see Appendix 6.: Lemma 3.. Let B be a convex compact subset of R 2. Then ĥu, v ĥu 2, v 2 ρ u u K u + u 2 v + ρ v v 2 u 2 for all u, v, u 2, v 2 B.

7 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 7 Lemma 3.2. For all a, b, c R 3a 4b + ca = 2 [a2 + 2a b 2 ] 2 [b2 + 2b c 2 ] + 2 a 2b + c2, 3.4a 3a 4b + ca c = 2[a b 2 b c 2 ] + a c b This lemma will be used with a, b and c set equal to terms at time levels t n+, t n and t n respectively. 3.. Semi-discrete in time weak formulation. Let N be a positive integer and := T/N be the fixed time step. Denote the partition of 0, T by σ N := {t n } N n=0 with t n := n. We consider the following semi-discrete in time approximations of Problem P: P For n =,..., N find U n, V n H Ω such that {U 0, V 0 } = {u, 0, v, 0} and χ H Ω U n U n, χ + U n, χ = fu n, V n, χ, 3.5a V n V n, χ + δ V n, χ = ĝu n, V n, χ, 3.5b P 2 For n = 2,..., N find U n, V n H Ω such that {U 0, V 0 } = {u, 0, v, 0}, {U, V } prescribed by P and χ H Ω 3U n+ 4U n +U n, χ + U n+, χ = 2 fu n, V n fu n, V n, χ, 3.6a 3V n+ 4V n +V n, χ + δ V n+, χ = 2ĝU n, V n ĝu n, V n, χ. 3.6b Note that in practice the time step in P is chosen much smaller than the time steps in P 2. For ease of notation we use the same symbol for both schemes Existence and uniqueness. The existence and uniqueness of the semidiscrete problem is given by the following result: Theorem 3.3. Given U 0, V 0 L Ω and fixed time steps, there exists a unique solution U, V H Ω of the st order semi-discrete problem P, and a unique solution {U n, V n } N n=2 of the 2nd order semi-discrete problem P 2, with U n, V n H Ω, n = 2,..., N. To prove existence and uniqueness of the semi-discrete in time weak formulations we recall a result for the Coercive Homogeneous Neumann Problem see 2.4 and apply induction. Proof. First observe that ĥu, v = ρuv + u + Ku 2 ρ u v = ρ v, 3.7 u thus fu, v, ĝu, v L 2 Ω if u, v L 2 Ω. Consider fixed time steps for both the first and second order schemes. Now by assumption U 0, V 0 in L Ω L 2 Ω. Existence and uniqueness of solutions to the Coercive Homogeneous Neumann Problem see 2.4 implies that there exists a unique solution U, V of P in H Ω L 2 Ω. As U 0, V 0, U, V L 2 Ω,

8 8 M.R. GARVIE & C. TRENCHEA existence and uniqueness of solutions to 2.4 yields that there exists a unique solution U, V of P2 in H Ω L 2 Ω. Proceeding inductively, we have a unique solution U n, V n H Ω, for n = 2,..., N. We note that this result does not bound the semi-discrete solutions uniformly with respect to, which is the purpose of the next section A priori estimates. To derive a semi-discrete error estimate in Section 3.4 we prove three a priori estimates bounding U n and V n independent of. We first require a lemma uniformly bounding solutions after the first time step. Lemma 3.4. Assume U 0, V 0 L Ω H Ω. Then the solution {U, V } of P satisfies the following uniform bounds: U U V V 0 0 C /2, 3.8 U U 0 + V V 0 C, 3.9 U + V C. 3.0 Proof. The proof of 3.8 and 3.9 follows from standard discrete energy techniques applied to scheme P see e.g. [7, 8]. Result 3.0 follows from the assumptions on the initial data and We also require a bound on the reaction-kinetics in L 2, namely: Lemma 3.5. fu, v 2 0 C + u v 2 0, ĝu, v 2 0 C + v 2 0, for all u, v L 2 Ω, for all u, v L 2 Ω, Proof. This follows in a straightforward manner after noting 3.7. We shall use the following discrete Grönwall lemma [6, Lemma 5..]. Lemma 3.6. Assume w n, α n, p n 0, 0 β <, satisfy n w n + p n α n + β w k+, n 0, where {α n } is non-decreasing with the convention that = 0. Then w n + p n β k=0 αn βw 0 exp β k=0 nβ β Theorem 3.7. Assume the results and assumptions of Lemma 3.4 hold, then for sufficiently small the solutions of scheme P 2 satisfy max { U n 2 + V n 2 } C, 3.2 n N max { U n+ U n V n+ V n 2 0} C, 3.3 n N N { U n+ U n V n+ V n 2 0} C, 3.4 n= N { U n+ U n 2 + V n+ V n 2 } C. 3.5 n=.

9 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 9 Proof. Estimate I. In scheme P 2 choose χ = U n+ in 3.6a, and χ = V n+ in 3.6b. Applying 3.4a, Cauchy-Schwarz inequality and Lemma 3.5 yields and 4 [ U n U n+ U n 2 0] 4 [ U n U n U n 2 0] + 4 U n+ 2U n + U n U n U n C + U n V n U n V n 2 0, 4 [ V n V n+ V n 2 0] 4 [ V n V n V n 2 0] + 4 V n+ 2V n + V n δ V n V n C + V n V n 2 0. Now add the inequalities 3.6 and 3.7, multiplying through by 4, discard terms of the form a 2b + c 2 and 2a b 2, change the notation from n to i, and sum over all i =,..., n to obtain n n U n V n U i δ V i+ 2 i= n C + C U i V i i= Applying the discrete Grönwall lemma 3.6 to 3.8 and noting that C < for sufficiently small yields 4 U n V n C C C exp Ctn C n i= U i+ 2 + δ V i+ 2 i= t n := n, 3.9 after recalling the assumptions on the initial data. This leads to the following uniform bound for the solutions of P 2 : max { U n 0 + V n 0 } C n N Estimate II. In scheme P 2 choose χ = U n+ U n / in 3.6a, and χ = V n+ V n / in 3.6b. Adding the resulting equations and applying the

10 0 M.R. GARVIE & C. TRENCHEA identity 3.4b yields [ U n+ 2 U n V n+ V n 2 0 U n U n V n V n 2 0 ] + [ U n+ 2 2 U n V n+ V n 2 ] [ 0 + U n+ U n 2 ] [ + U n+ 2 U n 2 δ [ ] + V n+ 2 V n 2 ] + δ V n+ V n 2 = 2 fu n, V n, U n+ U n fu n, V n, U n+ U n + 2 ĝu n, V n, V n+ V n ĝu n, V n, V n+ V n. 3.2 Application of Young s inequality and using Lemma 3.5 bounds the right hand side of 3.2 via 8 fu n, V n ĝu n, V n fu n, V n ĝu n, V n U n+ U n V n+ V n C + C [ U n V n U n V n 2 ] 0 + U n+ U n V n+ V n It follows from 3.2 and 3.22, kick-back of the last two terms in 3.22, and the uniform bound 3.20 that [ U n+ 2 U n V n+ V n 2 0 U n U n V n V n 2 0 ] + U n+ U n V n+ V n [ U n+ U n δ V n+ V n 2 ] [ + U n+ 2 U n 2 δ [ ] + V n+ 2 V n 2 ] C + C [ U n V n U n V n 2 0] C. After a change of notation from n to i, summing over all i =,..., n, noting the telescopic sum property n i= a i+ a i = a n + a n a + a 0, taking terms at t 0 and t to the right hand, multiplying through by 2, and recalling Lemma 3.4 yields U n U n V n V n n i= n { U i+ U i V i+ V i 2 0} i= { U i+ U i 2 + δ V i+ V i 2 } + 2 U n 2 + U n 2 + δ V n 2 + V n 2 C. Thus after noting Lemma 3.4 and 3.20 the results of Theorem 3.7 follow. Remark 3.8. The estimates in theorem 3.7 hold without the restriction on the time step after a small modification of P 2 see Proposition 6.3 in Appendix 6..

11 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 3.4. Error bound. Let ε n+ u, ε n+ v H Ω denote the local truncation errors see, e.g. [38, 37] at time-step t n+ such that ε n+ u, χ := ε n+ v, χ := 3ut n+ 4ut n + ut n, χ + ut n+, χ 2 fut n, vt n fut n, vt n, χ, 3vt n+ 4vt n + vt n, χ + δ vt n+, χ 2ĝut n, vt n ĝut n, vt n, χ, 3.23 where χ H Ω, and the pointwise errors e n u, e n v H Ω are defined by e n u = ut n U n, e n v = vt n V n, for 0 n N Lemma 3.9. Assume the classical solution of. has the following regularity d 2 u dt 2, d2 v dt 2 L2 0, T, H Ω, d 3 u dt 3, d3 v dt 3 L2 0, T, H Ω. Then the truncation error satisfies the following bound: N ε n u 2 + ε n v 2 2 n= C 2[ T 0 d2 u dt 2 t 2 + d2 v dt 2 t 2 + d3 u dt 3 t 2 + d3 v dt 3 t 2 dt ] Proof. First we expand ut n, vt n and ut n, vt n about t n+ by Taylor s formula to second order with integral remainder, to obtain 3ut n+ 4ut n + ut n 3vt n+ 4vt n + vt n = du tn+ dt t n+ + = dv dt t n+ + t n tn+ t n K t d3 u dt 3 tdt, K t d3 v dt 3 tdt, where K t is bounded by a constant independent of and u, v. Using the Taylor expansion to first order, relation 3., and recalling that f f, ĝ g on R 2 + we have 2 fut n, vt n fut n, vt n = a ut n+ + 2 tn+ t n 2ĝut n, vt n ĝut n, vt n = αb αvt n+ + α 2 tn+ K 2 t d2 u dt 2 tdt 2ĥut n, vt n + ĥut n, vt n, t n K 2 t d2 v dt 2 tdt 2αĥut n, vt n + ĥut n, vt n,

12 2 M.R. GARVIE & C. TRENCHEA where K 2 t is also bounded by a constant independent of u, v and. Substituting these expansions into 3.23 and using the fact that u, v > 0 we find that ε n+ u, χ = du tn+ dt t n+, χ + ut n+, χ + K t d3 u t n dt 3 tdt a + ut n+ tn+ 2 K 2 t d2 u t n dt 2 tdt + 2hut n, vt n hut n, vt n, χ, ε n+ v, χ = dv dt t n+, χ + δ vt n+, χ + + αvt n+ α 2 tn+ Using 2.8a,2.8b at t n+, we obtain ε n+ t n tn+ t n K t d3 v tdt αb dt3 t n K 2 t d2 v dt 2 tdt + 2hut n, vt n hut n, vt n, χ u,χ = hut n+, vt n+ +2hut n, vt n hut n, vt n, χ tn+ + K t d3 u tn+ tdt 2 K dt3 2 t d2 u dt 2 tdt, χ, ε n+ t n t n v,χ = hut n+, vt n+ +2hut n, vt n hut n, vt n, χ tn+ + K t d3 v tn+ tdt 2 K dt3 2 t d2 v dt 2 tdt, χ. t n By Taylor s theorem in two variables see e.g., [2], we have hut n+, vt n+ + 2hut n, vt n hut n, vt n = h u ut n+, vt n+ 2 tn+ + h v ut n+, vt n+ 2 t n tn+ K 2 t d2 u dt 2 dt t n with R n R n C 2. Then by 2.8c this gives t n K 2 t d2 v dt 2 dt + Rn R n, hut n+, vt n+ + 2hut n, vt n hut n, vt n, χ tn+ C 3 2 d2 u dt 2 t + d2 v dt 2 t dt χ, t n where the constant C 3 = C 3 v C[0,T ],H Ω is independent of n and. Therefore 3.26 and 3.27 imply tn+ ε n+ u C c d3 u tn+ dt 3 t dt + c 2 2 d2 u dt 2 t dt, tn+ ε n+ v C c d3 v t n dt 3 t dt + c 2 2 and finally ε n+ u 2 C 3 t n+ t n ε n+ v 2 C 3 t n+ t n t n tn+ t n d 3 u dt 3 t 2 + d2 u dt 2 t 2 dt, d 3 v dt 3 t 2 + d2 v dt 2 t 2 dt, d2 v dt 2 t dt,.

13 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 3 which yields To establish the error estimate we first prove a stability property. Lemma 3.0. For sufficiently small, the P 2 scheme satisfies the following stability property e N u 2 + e N v 2 + n=0 C N e N u 2 + δ e N v 2 n=0 e C u e v e u e 0 u e v e 0 v 2 0 N + 4 ε n+ u 2 + δ εn+ v 2 2NC exp C. Proof. We subtract 3.6a-3.6b from 3.23 to obtain 3en+ u 4e n u + e n u, χ + e n+ u, χ = ε n+ u, χ + 2 fut n, vt n fut n, vt n 2 fu n, V n + fu n, V n, χ, 3en+ v 4e n v + e n v, χ + δ e n+ v, χ = ε n+ v, χ + 2ĝut n, vt n ĝut n, vt n 2ĝU n, V n + ĝu n, V n, χ Then we take χ = e n+ u in the first equation, χ = e n+ v in the second, and use 3. to obtain 3en+ u 4e n u + e n u, e n+ u + e n+ u 2 = ε n+ u, e n+ u + 2 fut n, vt n 2 fu n, V n fut n, vt n + fu n, V n, e n+ u = ε n+ u, e n+ u 2 e n u + ĥut n, ut n ĥu n, V n, e n+ u + e n u + ĥut n, ut n ĥu n, V n, e n+ u, 3en+ v 4e n v + e n v, e n+ v + δ e n+ v 2 = ε n+ v, e n+ v + 2 ĝut n, vt n ĝu n, V n, e n+ v ĝut n, vt n ĝu n, V n, e n+ v = ε n+ v, e n+ v + 2 αe n v ĥut n, vt n + ĥu n, V n, e n+ v + αe n v + ĥut n, vt n ĥu n, V n, e n+ v. By use of Lemma 3. we have fut n, vt n fu n, V n = e n u ĥutn, vt n ĥu n, V n [ ] e n u + ρ2 + K ut n + U n vt n + ρ e n v U n, ĝut n, vt n ĝu n, V n = αe n v ĥutn, vt n ĥu n, V n [ ] e n v α + U n + e n u + ρ2 + K ut n + U n vt n.

14 4 M.R. GARVIE & C. TRENCHEA yielding 3en+ u Ω Ω 4e n u + e n u, e n+ u + e n+ u 2 ε n+ u, e n+ u e n+ u { e n u [ + ρ2 + K ut n + U n vt n ] + ρ e n v U n } dx e n+ u 3en+ v { e n [ + ρ2 + K ut n + U n vt n ] + ρ e n U n } dx, u 4e n v + e n v, e n+ v + δ e n+ v 2 { e n+ v 2 e n v α + ρ U n + 2 e n u [ + ρ2 + K ut n + U n vt n ] Ω } + e n v α + ρ U n + e n u [ + ρ2 + K ut n + U n vt n ] dx. ε n+ v, e n+ v + We add the two relations, use Young s inequality, the regularity of u, v, the uniform bound 3.2, the definition of the H Ω norm and Sobolev embedding yields 3en+ u 4e n u + e n u, e n+ u + 3en+ v 4e n v + e n v, e n+ v + e n+ u 2 + δ e n+ v 2 ε n+ u 2 + δ εn+ v en+ u δ 2 en+ v en+ u δ 2 en+ + C e n u e n v e n u e n v 2 0. After kick-back and multiplying by 2 we obtain 3en+ u 4e n u + e n u, e n+ u + 3en+ v 2 ε n+ u δ εn+ v 2 + e n+ u 2 + δ e n+ v 2 + C v v 2 0 4e n v + e n v, e n+ v + e n+ u 2 + δ e n+ v 2 e n u e n v e n u e n Finally, summing from to n, using Lemma 3.2, multiplying by and changing the summation index yields e N u 2 + e N v 2 + N e N u 2 + δ e N v 2 n=0 e u e v e u e 0 u e v e 0 v 2 N C N n=0 e n+ u e n+ v 2 0. n=0 ε n+ u 2 + δ εn+ v Application of the the discrete Grönwall lemma 3.6 concludes the proof. Combining Lemmata 3.9 and 3.0, under the assumption that e u, e v, 2e u e 0 u, 2e v e 0 v are of order 2, we derive the convergence and error estimate of the solution U n, V n of P 2. Theorem 3.. Under the assumptions of Lemma 3.9, there exists a constant 2 v 2 0.

15 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 5 Cu, v > 0 such that utn U n 0 + vt n V n 0 max 0 n N + N utn U n δ vt n V n n=0 Cu, v ut U 0 + vt V 0 + 2ut U ut 0 U vt V vt 0 V A fully discrete approximation. Before stating the fully discrete finite element approximations of the Thomas system we recall some standard definitions. Let T h be a quasi-uniform partitioning [8, page 32] of Ω into disjoint open simplices {τ} with h τ := diam τ and h := max τ T h h τ, so that Ω = τ T h τ. We use the standard Galerkin finite element space of piecewise linear continuous functions defined by: S h := {v CΩ : v τ is linear τ Ω h } H Ω. We shall also need the Lagrange interpolation operator π h : CΩ S h s.t. π h vx j = vx j for all j = 0,... J for nodes {x j } J j=0 of the triangulation. Let {ϕ j} J j=0 be the standard basis for S h, satisfying ϕ j x i = δ ij, where {x i } J i=0 is the set of nodes of T h. A discrete L 2 inner product on CΩ is then defined by u, v h := Ω π h uxvx dx J M jj ux j vx j, 4. where M jj :=, ϕ j ϕ j, ϕ j h > 0, corresponding to the diagonal lumped mass matrix M. We study the following fully discrete approximation of scheme P for the first time step, and scheme P 2 : P h, Find Uh, V h Sh with initial densities {Uh 0, V h 0} = {πh u 0 x, π h v 0 x} so that for all χ h S h we have P h, j=0 U h Uh, 0 h h χ h + U h, χ h = fu 0 h, Vh 0, χ h, 4.2a V h Vh 0 h ĝu, χ h + δ V h, χ h = 0 h, Vh 0 h, χ h. 4.2b 2 For n = 2,..., N find Uh n, V h n Sh such that {Uh 0, V h 0} = {πh u 0 x, π h v 0 x}, {Uh, V h } prescribed by Ph, so that for all χ h S h we have 3U n+ h 3V n+ h 4U n h + U n h, χ h h + U n+ h, χ h = 2 fu n h, V n h 4V n h + V n h, χ h h + δ V n+ h, χ h = h n fu h, V n h, χ h, 4.3a 2ĝU n h, Vh n ĝu n h, V n h h, χ h. 4.3b

16 6 M.R. GARVIE & C. TRENCHEA Choosing Uh n = h j=0 U j nϕ j, Vh n = h j=0 V h nϕ j, χ h = ϕ i, i = 0,..., T in P h, 2, where Uj n ux j, n, Vj n vx j, n, leads to a sparse linear system with the following block matrix form of 2J + 2 linear equations: M O U n+ ΛU n, V n, U n, V n = O M2 ΞU n, V n, U n, V n, V n+ where {U n } i = Ui n, {Vn } i = Vi n. M and M2 are strictly diagonally dominant and do not change from one time level to the next. Furthermore, because of the block structure of the linear system, we solve the approximate solutions for u and v independently at each time step using the Generalized Minimal Residual Method GMRES in MATLAB R2009b, preconditioned by incomplete LU factorization. 4.. Numerical experiments Simulation of prepattern formation. We repeat a numerical experiment of Murray [32] who used the Thomas system for illustrative purposes to simulate prepattern formation over a schematic mammal skin with fixed geometry and various domain sizes. We provide sufficient details of the numerical procedure so that results can be verified in future work, using the numerical schemes presented in this paper, or using different convergent numerical methods. Schemes P h, with P h, 2 were solved over Ω 0, T ], where Ω is a schematic mammal skin domain fitting in the rectangles [0, Lx] [0, Ly], with Lx := 52/2 i and Ly := 768/2 i, i = 3, 5, 7, 9, Figure 4.0. Homogeneous Neumann boundary conditions were employed with initial data u 0, v 0 prescribed as perturbations of the stationary states u c, v c of the corresponding spatially homogeneous system on every point of the computational domain. The stationary states u c, v c were perturbed using a truncated double Fourier series 20 px, y := s= { 20 z rs sin r= sπx cos Lx } rπy, 4.4 Ly scaled to be on [, ], denoted px, y, with coefficients z rs drawn from a simple pseudo random number generator D_UNIFORM_0 [28] see Appendix 6.2 for more details. In particular, we prescribed u 0 x, y = u c + px, y/0, v 0 x, y = v c + px, y/0, i.e., the stationary states are perturbed up to a maximum of ± 0%. We solved the schemes until the transient solutions died out T = 500, and checked that the solutions were unchanged at t = 2T, which confirmed that the patterns represented stationary solutions. In order to verify that the patterns represented correctly converged solutions we also checked that solutions were unchanged with refinement of the spatial and temporal discretization parameters. In all calculations we used the unstructured mesh generator MESH2D v24. See for further details.

17 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 7 // a b c d Figure 4.0: Approximate solutions U n for scheme P 2 at time T = 500. We used an unstructured mesh with triangles and nodes. Time steps were / SBDF and 0 8 -SBDF. Initial data was prescribed as perturbations of the stationary solutions see text for further details. The schematic mammal skin domains are bounded by rectangles with the following dimensions: a , b 8 2, c 32 48, d

18 8 M.R. GARVIE & C. TRENCHEA Rates of convergence. We present numerical evidence in 2 space dimensions to test the optimal rate of convergence given in Theorem 3.. As no exact solution of the reaction-diffusion system is known, we compared approximate solutions u n computed with a small time step fine, with the corresponding solutions U n generated with a sequence of larger time steps {}. In all simulations we used a fine triangulation T h with a maximum diameter of all triangles equal to h additional details below. For notational convenience we extend the approximate solutions in time via u + t : = u n, t t n, t n ], t n := n fine, n =, 2,..., N fine, U + t : = U n, t t n, t n ], t n := n, n =, 2,..., N, N fine : = T/ fine, and define the errors N := T/, η 0 h, : = u + U + 2 L 2 0,T ;H Ω = N η h, : = u + U + 2 L 0,T ;L 2 Ω = n= u n U n u n U n 2, max n N un U n 2 0. Using Lumped Mass Qaudrature [4, p. 340] we computed the ratios Ri := η ih, η i h, /2, i = 0,, 4.5 η i h, /2 η i h, /4 for a typical example computed in Section 4.. see Table 4. for additional details. With the assumption that η 0 h, and η h, can be expressed in the form C h p + C 2 q, p, q, C, C 2 R, the ratios 4.5 simplify to R0 = R = 2 q. Thus the results in Table 4. indicate that the rate of convergence is O 2, which is consistent with the error bound in Theorem 3.. η 0 3/000, η 3/000, R0 R / e e / e e / e e /200.80e e / e e /800.6e e / e e-08 / e e-08 Table 4.: Verification of Theorem 3.: fine = /2800; h = 0.003; Lx = ; Ly =.5; = 2 j /25, j = 0,,..., 7. For details concerning the initial data and other parameter values see Section 4...

19 FINITE ELEMENT APPROXIMATION OF A SUBSTRATE-INHIBITION MODEL 9 5. Conclusions. We studied a second-order, three level finite element approximation of a well-known reaction-diffusion Thomas system for patterning in nature. For the spatial discretization we used the standard Galerkin finite element method with piecewise linear continuous basis functions. For the temporal discretization we employed the second-order 2-SBDF finite difference scheme, with starting values computed from the first-order scheme -SBDF scheme [39]. The main contribution of this paper is the numerical analysis of the semi-discrete in time weak formulation of the Thomas system. The three time levels of the scheme and the asymmetry of the approximate reaction-kinetics presents significant technical challenges to the derivation of a priori estimates and the optimal semi-discrete in time error estimate. To the best of our knowledge, our study provides the first comprehensive numerical analysis of a multi-level 3 semi-discrete in time weak formulation for a system of nonlinear reaction-diffusion equations. The application of the results in this paper show promise in the numerical analysis of the Extrapolated Gear scheme.4 applied to other reaction-diffusion systems. The analysis in our paper should cover reaction-diffusion systems where the reactionkinetics satisfy Lemma 3.5 and the local Lipschitz condition in Lemma 3.. Further work is needed to generalize the numerical analysis to cover systems that do not satisfy these Lemmata. However, much of our work deals with discrete energy estimates of the three level time derivative, and thus is applicable to general systems of semi-linear PDEs. In order to illustrate the numerical performance of the finite element method studied in this paper, we repeated an experiment of Murray [32], who simulated Turing patterns over a schematic mammal skin with fixed geometry and various domain sizes. This is a classic experiment in mathematical biology, and the numerical results have been reproduced in numerous sources, e.g. [34, 2, 0]. However, in Murray s original work insufficient details were given for the numerical simulations to be accurately duplicated. Furthermore, the traditional approach used to prescribe initial data for diffusion-driven instability is also not repeatable. Typically with this approach, the spatially homogeneous equilibrium solutions are perturbed at every point on the computational grid, using an unspecified random number generator with an unspecified seed value. As the numerical solution is sensitive to small changes in the initial data, even results obtained with the same random number generator, but with different seeds, vary significantly. Another problem with this procedure is that mesh refinement cannot be used to demonstrate convergence as successive grids use essentially different initially conditions. To overcome these problems we provided a consistent procedure for generating initial data that can be used to generate Turing patterns, which is useful for future comparative work using the Thomas system or other Turing systems. We hope the work in this paper stimulates further numerical analysis of multi-level finite element schemes for nonlinear reaction-diffusion systems. Acknowledgments. We thank James Blowey University of Durham, UK for some helpful comments during the preparation of this manuscript. 6. Appendix. 6.. Proof of Lemma 3.. Let u := u, v T, u 2 := u 2, v 2 T, z := z, z 2 T with z = u 2 s + su for 0 s, and denote e v := v v 2 and e u := u u 2. The physics review paper [0] has been cited 3527 times ISI Web of Knowledge.

20 20 M.R. GARVIE & C. TRENCHEA Then elementary calculation yields ĥu ĥu e u v u 2 e v 2 = ρ + u + Ku 2 + ρ + u + Ku 2 + u 2 + Ku 2 2 u 2 v u 2 u + ρ + u + Ku 2 + u 2 + Ku ρk u 2 v u 2 2 u 2 + u + Ku 2 + u 2 + Ku 2 2 ρ e u v + u 2 e v + v e u + K v e u u + u 2 = ρ e u 2 + K u + u 2 v + ρ e v u Pseudo random number generation. In the interests of repeatability, we give details of the pseudo random number generator D_UNIFORM_0 [28] used to perturb the coefficients z rs of the double Fourier series 4.4. It is not the most efficient random number generator, but it is simple enough to be easily implemented in different languages. We take z rs equal to the nth random number r n drawn from D_UNIFORM_0, where n = r + 20s. The random numbers are calculated recursively via r n = s n /2 3, s n = 6807 s n mod 2 3, for n =, 2,... seeded with s 0. In all our simulations we used s 0 = Unconditional stability of a modified scheme. We consider the following modification of the semi-discrete in time, Extrapolated Gear scheme P 2 : Q 2 For n = 2,..., N find U n, V n H Ω such that {U 0, V 0 } = {u, 0, v, 0}, {U, V } prescribed by P and χ H Ω 3U n+ 4U n +U n, χ + U n+, χ + U n+, χ 6.a = a 2ĥU n, V n +ĥu n, V n, χ, 3V n+ 4V n +V n, χ + δ V n+, χ + αv n+, χ 6.b = αb 2ĥU n, V n +ĥu n, V n, χ. For the modified scheme we have the following unconditional stability result. Proposition 6.. Assume the results and assumptions of Lemma 3.4 hold, then the solutions of scheme Q 2 satisfy the estimates Proof. Choose χ = U n+ in 6.a, and χ = V n+ in 6.b. Applying the elementary identity 3.4a, the Cauchy-Schwarz inequality and 3.7 yields 4 [ U n U n+ U n 2 0] 4 [ U n U n U n 2 0] + 4 U n+ 2U n + U n U n+ 2 + U n U n a2 Ω 4 + ρ2 2 2 V n 0 + V n 0 2, 4 [ V n V n+ V n 2 0] 4 [ V n V n V n 2 0] + 4 V n+ 2V n + V n δ V n+ 2 + α V n α V n αb2 Ω 4 + ρ2 2 V n 0 + V n 2, 0 2

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