two-grid methods for mixed finite-element solution of coupled reaction-diffusion systems

Two-Grid Methods for Mixed Finite-ElementSolution of Coupled Reaction-Diffusion SystemsLi Wu, Myron B. Allen

Department of MathematicsUniversity of WyomingLaramie, Wyoming 82071

Received November 23, 1998; accepted April 25, 1999

We develop 2-grid schemes for solving nonlinear reaction-diffusion systems:

∂p∂t

− ∇ · (K∇p) = f(x,p),

wherep = (p, q) is an unknown vector-valued function. The schemes use discretizations based on a mixedfinite-element method. The 2-grid approach yields iterative procedures for solving the nonlinear discreteequations. The idea is to relegate all the Newton-like iterations to grids much coarser than the final one,with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlierfor single reaction-diffusion equations. An application to prepattern formation in mathematical biologyillustrates the method’s effectiveness.c© 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15:589–604, 1999

Keywords: reaction-diffusion systems; mixed finite element methods; two-grid methods; pattern formation

I. INTRODUCTION

Reaction-diffusion systems occur in numerous biogeochemical phenomena, especially those in-volving multispecies reactions and predator-prey systems. In the single-equation case, a reaction-diffusion system has the form

∂p

∂t−∇ · (K∇p) = f(x, p), (1.1)

in which p(x, t) is a real-valued density or concentration. This type of equation typically arisesfrom a mass balance law, in whichf is a nonlinear reaction term. The positive coefficientK(x)models spreading of the quantity of interest under the influence of a diffusive flux−K∇p.

Correspondence to:Li Wu, Civil & Env. Eng. Dept., UVM, College of EM, Votey 109, Burlington, VT 05405Contract grant sponsor: National Science FoundationContract grant number: DMS-9724437c© 1999 John Wiley & Sons, Inc. CCC 0749-159X/99/050589-16

590 WU AND ALLEN

Equation (1.1) generalizes to cases in which several species or chemicals interact. When twospecies interact, there is a vectorp = (p, q) of densities or concentrations, each associated with itsown diffusion coefficient. The reaction term in such a system is a vector-valued functionf(x,p).We have

∂p∂t−∇ · (K∇p) = f(x,p), (1.2)

whereK is a diagonal diffusion matrix that is diagonal in the absence of cross-diffusion. Reaction-diffusion systems of the form (1.2) model such mechanisms as the chemical basis of morphogene-sis, as originally proposed by Turing [1]. Owing to the interesting pattern-forming structures thatcan arise [2], systems of this form have attracted increasing interest over the past three decades.

In this article, we examine numerical techniques for solving systems of the form (1.2). Themain emphases of our methods are twofold: first, to obtain accurate discrete approximations forboth the concentrations and their fluxes; and, second, to implement efficient iterative schemes forsolving the nonlinear discretized equations.

To be more specific, we write Eqs. (1.2) as an equivalent first-order system:

∂p∂t

+∇ · v = f(x,p) mass balance,

K−1v +∇p = 0 flux law.

(1.3)

Herev is the diffusive flux. In this context, there are three numerical issues. First, in someapplications it is important to solve forp andv with comparable accuracy. Second, one mustaccommodate the nonlinear reaction termf(x,p) by linearizing and using an efficient iterativescheme. Third, one must solve the resulting linear algebraic systems effectively on fine spatialgrids, even whenK may be highly variable in space.

We examine a mixed finite-element approximation to the system (2.2) that addresses all threeissues. We use a mixed finite-element method with low-order trial spaces to discretize the partialdifferential equations and attain equal-order accuracy inp andv [3]. To linearize the discreteequations, we use a two-grid scheme, which allows us to iterate on a grid much coarser (and, hence,much cheaper) than that used for the final solution [4, 5]. To solve the resulting linear systems,we use a multigrid-based ‘‘inner’’ iteration to reduce sensitivity to fine grids and variations inK [6, 7]. All these methods extend similar techniques analyzed in detail for the single-equationcase in an earlier article [8]. The extensions raise additional issues, mainly associated with theorchestration of the various iterations.

In the remainder of the article, Section II reviews the two-grid method and associated numericalschemes, analyzed in detail in the earlier article [8]. Then Section III introduces a suite of iterativeorchestrations that extend the two-grid approach to coupled reaction-diffusion systems. We devotesome discussion to a comparison of the efficiencies of the orchestrations. Section IV illustratesthe method’s utility, examining a computational application to prepattern formation in animalcoats. Section V draws conclusions.

II. REVIEW OF THE TWO-GRID APPROACH

In this section, we review a two-grid method to solve the single reaction-diffusion equation

∂p

∂t−∇ · (K∇p) = f(x, p) (2.1)

...COUPLED REACTION-DIFFUSION SYSTEMS 591

on a rectangular domainΩ ∈ IR2. Here,p is the unknown potential,K is a positive functionof space,v = −K∇p is the flux, andf is the nonlinear reaction term. This review serves as afoundation for the extension to reaction-diffusion systems, introduced in the next section.

We owe the two-grid approach to Xu [5], who demonstrates the use of coarse and fine grids witha Galerkin finite-element method to solve nonlinear problems efficiently. Dawson and Wheeler [4]extend the two-grid idea to mixed finite-element discretizations in which, for the above problem,the coefficientK depends upon the unknown, butf does not. The method introduced here is alsobased on the mixed finite-element discretization, addressing problems having nonlinear reactiontermsf(p). For details, see our earlier article [8].

A. Mixed Finite-Element Discretization

We start with a brief review of the mixed finite-element discretization for Eq. (2.1). On therectangular domainΩ = (0, 1) × (0, 1), let4x = 0 = x0 < x1 < · · · < xm = 1 be a gridon x-axis, and let4y = 0 = y0 < y1 < · · · < yn = 1 be a grid ony-axis. Then4x ×4y

is a rectangular grid onΩ, having nodes(xi, yj). For convenience, we use grids having uniformmesh sizeh, but it is possible to relax this condition. Later, in the 2-grid linearization, we usehto denote the mesh size of the fine grid, which we denote as4h.

Based on this grid, we use standard, lowest-order Raviart–Thomas trial spacesWh andQh

to approximatep andv, respectively. The potential spaceWh consists of functions that arepiecewise constant on4h. The flux space is more complicated:Qh = Qx

h × Qyh, whereQx

h

contains functions that are piecewise linear and continuous on4x and piecewise constant on4y, while Qy

h contains functions that are piecewise constant on4x and piecewise linear andcontinuous on4y.

Equation (2.1) is equivalent to the first order system

∂p

∂t+∇ · v = f(x, p) mass balance;

K−1v +∇p = 0 flux law.

(2.2)

The mixed finite-element approximation to this system is as follows: Find an approximate po-tentialph ∈Wh and an approximate fluxvh ∈ Qh such that∫

ΩK−1vh · qh −

∫Ω

ph∇ · qh = 0, for all qh ∈ Qh,

∫Ω

wh∇ · vh +∫

Ωwh

∂ph

∂t=

∫Ω

whf(ph), for all wh ∈Wh.

(2.3)

Discretizing implicitly in time via the approximation(∂ph

∂t

)n

' 1∆t

(pnh − pn−1

h )

converts the above system to a matrix equation for the vector(V nh , Pn

h )T of unknown degrees offreedom invn

h andpnh at each new time leveln:[

Ah Nh

NTh Sh

] [Vh

Ph

]n

=[

0Fn

h + ShPn−1h

]. (2.4)

Although the block matrix on the left is indefinite,Ah is a symmetric, positive definite blockcontaining integrals involvingK; Nh is a differencing matrix;Sh is a positive definite, diagonal

592 WU AND ALLEN

block matrix containing integrals involving1/∆t. For detailed descriptions of the matrix entries,see [8]. So far, this system is nonlinear, since the right-side vector depends upon the unknownsin Pn

n through the termFnh .

B. Linearization via the Two-Grid Approach

To solve the mixed-method system, we use a two-grid scheme, which relegates the computationaleffort of nonlinear iterations to a coarse subgrid4H ⊂ 4h, having mesh sizeH >> h. Asso-ciated with4H are trial spacesQH andWH analogous toQh andWh. We compute the trialfunctionspn

H ∈ WH andvnH ∈ QH using an iterative Newton-like approximation, described

shortly, to the mixed-method equations on4H . Once we have computedpnH andvn

H , we usethem to initialize a single iteration for the unknownspn

h andvnh associated with the fine grid. We

now review this scheme.To obtain the coarse-grid iterative scheme, we adopt the following expansions: At time level

n, themth iterates are

pn,mH = pn,m−1

H + δpn,mH ; (2.5)

f(pn,mH ) ' f(pn,m−1

H ) + f ′(pn,m−1H )δpn,m

H , (2.6)

whereδpn,mH denotes an unknown correction. Substituting these two expressions into the weak

form (2.3) and rearranging, we have

rl

∫Ω

K−1vn,mH · qH −

∫Ω

δpn,mH ∇ · qH =

∫Ω

pn,m−1H ∇ · qH ,

for all qH ∈ QH ,

∫Ω

wH∇ · vn,mH +

∫Ω

wH

[1

∆t− f ′(pn,m−1

H )]

δpn,mH

=∫

ΩwH

[f(pn,m−1

H )− pn,m−1H − pn−1

H

∆t

], for all wH ∈WH .

The corresponding block matrix equation is[AH NH

NTH SH −Ψn,m−1

H

] [V n,m

H

δPn,mH

]=

[ −NHPn,m−1H

Fn,m−1H − SH(Pn,m−1

H − Pn−1H )

]. (2.7)

Here,Ψn,m−1H is a matrix arising from integrals of the form∫

ΩwHf ′(pn,m−1

H )δpn,mH .

We solve this system iteratively, untilδPn,mH is small enough in norm to indicate numerical

convergence. SettingPnH = Pn,m−1

H + δPn,mH now gives the updated nodal values ofpn

H .Next, to obtain the fine-grid approximationspn

h andvnh , we interpolatepn

H to4h and adoptthe expansions

pnh ' pn

h = pnH + δpn

h;

f(pnh) ' f(pn

H) + f ′(pnH)δpn

h.(2.8)


Substituting these expansions into the weak form (2.3), we get a block matrix equation analogousto that in Eq. (2.7):[

Ah Nh

NTh Sh −Ψn

h

] [V n

h

δPnh

]=

[ −NhPnH

Fnh − Sh(Pn

H − Pn−1h )

], (2.9)

where now the vectorPnH contains values of the coarse-grid solutionpn

H interpolated to the finegrid. Thus, one solution of a fine-grid matrix equation at each time-level suffices to compute thenodal values(Pn

h , V nh ) to the approximate fine-grid solution(pn

h,vnh).

This linearization is efficient only when the coarse grid4H can be much coarser than the finegrid4h without sacrificing the accuracy associated with the fine grid solution (pn

h, vnh). Under

reasonable hypotheses, the fine-grid solution obeys error estimates of the form

‖pNh − pN‖+

[ N∑n=1

∆t‖K1/2(∇pnh −∇pn)‖2

]1/2

≤ C(hk+1 + H2k+2 + ∆t

).

Here,k is related to the degree of polynomials used in the trial spacesWH , Wh, QH , andQh. Inparticular, for the lowest-order Raviart–Thomas spaces used here,k = 0. Hence, theL2 errorsin the fine grid potential and flux areO(h + H2 + ∆t). For proof and numerical corroboration,we refer to our previous article [8].

C. Numerical Linear Algebra

On both the coarse and fine grids, one must solve linear systems of the form[A N

NT M

] [VP

]=

[F1F2

]. (2.10)

This system is sparse and symmetric, but it is indefinite—a problem that is typical of mixedfinite-element discretizations. Our approach to this problem exploits the following heuristic [6]:If D is a diagonal approximation ofA, then the first block equation yields

V = D−1[F1 + (D −A)V ]−D−1NP.

Substituting this expression forV into the second block equation yields(M −NT D−1N

)P = F2 −NT D−1F1 −NT D−1(D −A)V. (2.11)

The matrixM−NT D−1N is sparse, symmetric, and, ifD is chosen reasonably, positive definite.For example, in computing the integrals inA, one can use the trapezoid rule to get an approximationto A that is automatically diagonal [7]. For any such choice ofD, the matrixM − NT D−1Nhas a block tridiagonal structure with at most five nonzero entries on any row.

Equation (2.11) furnishes the core of an inner iterative scheme for solving the system (2.10).To get from inner iterative level− 1 to level`, we use the following algorithm:

(i) Initialize the iterations:`← 1.(ii ) Load the right-side vector:G(`−1) ← F2 −NT D−1F1 −NT D−1(D −A)V (`−1).

(iii ) Solve(M −NT D−1N)P (`) = G(`−1) approximately forP (`).(iv) Compute fluxes:V (`) ← D−1F1 + D−1(D −A)V (`−1) −D−1NP (`).(v) If convergence test fails,`← ` + 1.

Steps (ii ) and (iv) require only sparse matrix multiplication. The inner step (iii ) involves a penta-diagonal matrix equation for which a variety of efficient solvers are available. We solve thisequation approximately at each iteration using several V-cycles of a multigrid algorithm [7].

594 WU AND ALLEN

III. TWO-GRID SCHEMES FOR COUPLED SYSTEMS

We now consider extensions of the ideas reviewed above to coupled systems involving two speciesor reactants. These systems have the form

∂p

∂t−∇ · (K1∇p) = f(x, p, q),

∂q

∂t−∇ · (K2∇q) = g(x, p, q).

(3.1)

Herep andq are concentrations of the reactants;f andg describe the nonlinear reaction kinetics,andK1 andK2 are the diffusion coefficients, assumed to be positive functions ofx that arebounded away from zero.

The discussion focuses on the effectiveness of various orchestrations of the iterative time-stepping scheme. Computational experiments suggest that error estimates comparable to thosefor the single-equation case hold for two coupled equations, although formal analysis for the2-equation case would be significantly more tedious than that presented for the single-equationcase [8].

A. Naive One-Grid Scheme

For brevity, let us denote

H1 = ∂/∂t−∇ · (K1∇),H2 = ∂/∂t−∇ · (K2∇).

In solving coupled problems like (3.1), one commonly orchestrates a sequence of iterations thatalternately fixp orq. For example, without using 2-grid techniques, we might calculate as followsto step from time leveln− 1 to leveln:

Scheme 0: Naive one-grid scheme.

(i) SolveH1(pnh) = f(x, pn

h, qn−1h ) for pn

h,(ii ) SolveH2(qn

h) = g(x, pnh, qn

h) for qnh ,

(iii ) SolveH1(pnh) = f(x, pn

h, qnh) for pn

h.

(‘‘Hatted’’ variables, such aspnh, denote temporary values.) Thisbootstrappingscheme involves

solving three individual nonlinear reaction-diffusion equations at each time level on the fine grid.Because all the iterations involve the fine-grid discrete equations, the scheme fails to exploit theefficiency associated with the 2-grid concept, but it serves as a template for the more efficientschemes developed below.

B. Incorporating the Two-Grid Approach

We now introduce three plausible iterative schemes that use the bootstrapping approach in conjunc-tion with 2-grid ideas, in different orchestrations. The following simple initial-boundary-valueproblem serves as a test bed for comparisons among the schemes:

∂p

∂t−∇ · (∇p) = p2q − 0.5p2 + g1(x, y), for t > 0, x = (x, y) ∈ Ω = (0, 1)2,

∂q

∂t−∇ · (∇q) = −pq + 2q + g2(x, y), for t > 0, x = (x, y) ∈ Ω = (0, 1)2,

(3.2)


TABLE I. Decay ofL2 errors with grid refinement, keepingH = h1/2, for Scheme 1.

h H‖ph − p‖

‖p‖‖uh − u‖

‖u‖‖qh − q‖

‖q‖‖vh − v‖

‖v‖2−2 2−1 4.986 × 10−2 2.626 × 10−2 2.362 × 10−1 1.307 × 10−1

2−4 2−2 3.873 × 10−3 1.704 × 10−3 2.580 × 10−2 9.295 × 10−3

2−6 2−3 2.538 × 10−4 1.212 × 10−4 1.685 × 10−3 5.872 × 10−4

2−8 2−4 1.774 × 10−5 1.127 × 10−5 1.120 × 10−4 4.366 × 10−5

subject to the initial conditionsp(x, 0) = q(x, 0) = 0 and the Dirichlet boundary conditionsp = q = 0 on ∂Ω. We denote the flux associated withp asu and that associated withq asv.We chooseg1(x) andg2(x) so that the exact solution isp(x, t) = −tex+yx(x − 1)y(y − 1),q(x, t) = tx sin(2πx)y sin(2πy). We use the uniform time-step∆t = 0.1 throughout. Fornumerical comparisons, we examine solutions computed att = 5∆t.

In an effort to assess computational efficiency, the discussion below reports CPU times recordedfor several of the runs. We recorded these times using an SGI Indigo–- a platform that is neitherespecially fast nor capable of parallel processing. Because we attempted neither to optimize thecode nor to filter out the effects of computational overhead, these CPU times are merely roughmeasures of the schemes’ efficiencies.

The first scheme mimics Scheme 0 on the coarse grid before transferring to the fine grid.

Scheme 1: Single bootstrapping on the coarse grid. First execute Newton-likeiterations on the coarse grid, in the following order, to getpn

H andqnH :

(i) SolveH1(pnH) = f(x, pn

H , qn−1H ) for pn

H ,(ii ) SolveH2(qn

H) = g(x, pnH , qn

H) for qnH ,

(iii ) SolveH1(pnH) = f(x, pn

H , qnH) for pn

H .

Then interpolatepnH andqn

H onto the fine grid to obtain starting values forpnh andqn

h . Next, onthe fine grid:


h, qnH) for pn

h,(ii ) SolveH2(qn

h) = g(x, pnh, qn

h) for qnh .

Applying this scheme to the test problem (3.2) provides a gauge of its effectiveness. Considerfirst the convergence of the scheme ash→ 0, keepingH = h1/2 in accordance with the single-equation error estimates [8]. Table I shows numerical results for this computational test. Ash decreases, the errors shrink sharply, confirming computationally that Scheme 1 preserves theconvergence of the mixed finite-element discretization. Although the convergence theorem inour previous article [8] guarantees that errors converge at least linearly withh whenk = 0, ournumerical results are much better than linear. Dawson, Wheeler, and Woodward in their article[9] show superconvergence properties for a 2-grid finite difference scheme for reaction-diffusionequations having nonlinear diffusion term but linear reaction term. We believe that, for ourequations and numerical schemes, there are some superconvergence properties behind.

Of equal interest is the efficiency of the scheme, that is, its ability to produce accurate resultswhile iterating on grids that are quite coarse. We examine the sensitivity of the computed resultsto the choice ofH by fixing the mesh size of the fine grid ash = 2−6, then varying the mesh sizeH of the coarse grid from2−1 down to2−6.

Table II shows theL2 errors and CPU times. The fine-grid errors appear to be reasonablyinsensitive to the coarseness of the grid on which we iterate, and the coarse-grid mesh sizeH can

596 WU AND ALLEN

be much coarser thanh without sacrificing accuracy. In the last row of the table,H = h, meaningthat we did not use the 2-grid idea to solve the problem in this instance. Instead, we used Scheme0. This case requires much more CPU than in other cases. When we use the 2-grid scheme, theCPU time drops sharply asH increases, effectively stabilizing forH ≥ 2−3.

We turn now to Scheme 2. In contrast with Scheme 1, this scheme executes one more boot-strapping step in an effort to improve the approximate solution on the coarse grid. That is, on thecoarse grid, we iterate to solve the following four equations in the given order.

Scheme 2: Double bootstrapping on the coarse grid. On the coarse grid,

(i) SolveH1(pnH) = f(x, pn

H , qn−1H ) for pn

H ,(ii ) SolveH2(qn

H) = g(x, pnH , qn

H) for qnH ,

(iii ) SolveH1(pnH) = f(x, pn

H , qnH) for pn

H ,(iv) SolveH2(qn

H) = g(x, pnH , qn

H) for qnH .

On the fine grid, the computations remain the same as in Scheme 1.

We discuss the performance of this scheme below.Scheme 3 uses the same procedures as Scheme 1 on the coarse grid to computepn

H andqnH .

But Scheme 3 sequentially solves the partial differential equations on the fine grid:

Scheme 3: Bootstrapping on the fine grid. Using two-grid methods in each step,


h, qnH) for pn

h,(ii ) SolveH2(qn

h) = g(x, pnh, qn

h) for qnh ,

(iii ) SolveH1(pnh) = f(x, pn

h, qnh) for pn

h.

Numerical results, similar to those reported for Scheme 1, confirm that both Scheme 2 andScheme 3 are convergent and preserve the expected error estimates ash→ 0. They also indicatethat, as in Scheme 1, the errors are fairly insensitive to the choice of the mesh sizeH of the coarsegrid.

C. Comparison Among Schemes

Since all these orchestrations produce numerical solutions that converge at the appropriate rateash → 0, the key issue is which scheme offers the greatest efficiency. Recall that Scheme 0,described in Section III.A, directly solves three nonlinear problems on the fine grid at each timelevel without using two-grid concepts. Schemes 1, 2, and 3 all exploit the 2-grid idea in various

TABLE II. L2 errors forh = 2−6 versus coarse mesh sizeH, for Scheme 1.

H‖ph − p‖

‖p‖‖uh − u‖

‖u‖‖qh − q‖

‖q‖‖vh − v‖

‖v‖ time (s)

2−1 3.843 × 10−4 1.646 × 10−4 1.685 × 10−3 5.843 × 10−4 88.7762−2 2.557 × 10−4 1.289 × 10−4 1.686 × 10−3 5.835 × 10−4 89.3572−3 2.538 × 10−4 1.212 × 10−4 1.685 × 10−3 5.872 × 10−4 91.9452−4 2.532 × 10−4 1.272 × 10−4 1.684 × 10−3 5.885 × 10−4 110.8352−5 2.532 × 10−4 1.203 × 10−4 1.684 × 10−3 5.905 × 10−4 200.3562−6 2.528 × 10−4 1.064 × 10−4 1.684 × 10−3 5.864 × 10−4 521.696


TABLE III. Error comparison ath = 2−7, H = 2−3, andt = 0.5.

Scheme‖ph − p‖

‖p‖‖uh − u‖

‖u‖‖qh − q‖

‖q‖‖vh − v‖

‖v‖ time (s)

Scheme0 6.400 × 10−5 2.672 × 10−5 4.202 × 10−4 1.456 × 10−4 2141.2Scheme1 6.423 × 10−5 3.090 × 10−5 4.152 × 10−4 1.436 × 10−4 379.08Scheme2 6.423 × 10−5 3.106 × 10−5 4.148 × 10−4 1.392 × 10−4 398.19Scheme3 6.384 × 10−5 3.044 × 10−5 4.145 × 10−4 1.424 × 10−4 523.50

orchestrations. In Table III, we compare all four schemes on the basis of CPU time, under theconditions thath = 2−7, H = 2−3. In all these results, we use∆t = 0.1 and examine thenumerical results att = 5∆t.

The table shows that the errors are fairly insensitive to the choice of schemes, and that all theschemes are capable of producing accurate solutions. Scheme 1 does the least work on both thecoarse grid and the fine grid, so it is the most efficient scheme. Since Scheme 2 takes slightlymore CPU time, it appears that the extra bootstrapping that it uses to get a better solution on thecoarse grid does not help much to improve the accuracy of the fine-grid solution, so the additionalcomputational effort is not justified. On the other hand, Scheme 3 devotes extra work to improvingthe fine-grid results, which is not worthwhile. Finally, even the least efficient orchestration usingthe 2-grid method is significantly more efficient than the single-grid approach used in Scheme 0.

IV. APPLICATION TO MATHEMATICAL BIOLOGY

We now illustrate a more intriguing application of the 2-grid scheme to coupled equations inmathematical biology. One interesting property of some reaction-diffusion systems is their abilityto produce pattern-forming solutions [2]. An example is a proposed model for the formation ofcoat markings in animals. The biochemical mechanism for this process is poorly understood.J. D. Murray [10] proposes the following mechanism for the formation ofprepatterns, which arethe biochemical conditions that trigger nonuniform pigmentation:

The pattern formation process could be controlled by two reaction-diffusion systems.One operates as a chemical switch in the form of an inhibitor, which, for example,continuously decreases and acts on a second system.

This chemical switch involves reactions between a substrates(x, t) and a cosubstratea(x, t).Because the effects of this switch decrease in time, the reaction has a finite time in which todevelop prepatterns.

A. Model of Prepattern Formation

Murray proposes a model for the substrate-cosubstrate reaction in the form of a coupled reaction-diffusion system in the following dimensionless form:

∂s

∂t−∇ · (∇s) = g(s, a),

∂a

∂t−∇ · (β∇a) = f(s, a).

(4.1)

598 WU AND ALLEN

Here,β is the ratio of the two diffusion coefficients on the reactant surface, andg andf are thedimensionless overall reaction rates, having the following forms:

g(s, a) = γ[s0 − s− ρF (s, a)],

f(s, a) = γ[α(a0 − a)− ρF (s, a)],

F (s, a) = sa/(1 + s + κs2).

(4.2)

Hereκ is a constant inhibition parameter, with stronger inhibition corresponding to larger valuesof κ. The parametersα, ρ, a0, ands0 are constants related to the experimental arrangement [10].

Several issues are peculiar to this pattern-formation model. First, as Murray points out,

When the inhibition is high, spatial patterns are obtained, but over the range wherethe inhibition gives a steady state in the appropriate range, diffusive instability ispossible and, hence, spatial structuring. As the inhibition decreases further, thepattern formation mechanism enters the domain where diffusive instability is notpossible. Thus, there is afinite time available for pattern formation.

Second, in nature, coat markings are typically not visible in an animal’s early fetal stage. Forexample, patterns of pandas are not visible until several weeks after they are born. Markingson zebras cannot be seen until their hair is developed during the gestation period. However,invisibility of patterns does not indicate that they are not present in some form. Zebras begin tolay down prepatterns around 21–35 days into their 360-day gestation period [11]. This type ofprepattern formation is the phenomenon modeled here.

Third, patterns on regions like necks, tails, and legs are associated with periodic boundaryconditions, and patterns on an animal’s back or underbelly are better modeled by zero-flux Neu-mann boundary conditions. All the results presented in this article are for the above system (4.1)and (4.2), subject to zero-flux boundary conditions on the concentrationss anda on a rectangulardomain(0, xmax)× (0, ymax). That is,

∂s

∂x=

∂a

∂x= 0 at x = 0, x = xmax, for all y ∈ [0, ymax];

∂s

∂y=

∂a

∂y= 0 at y = 0, y = ymax, for all x ∈ [0, xmax].

(4.3)

Although we normally assign zero initial values to boths anda, the behavior of the system isalmost insensitive to initial conditions [10], in the sense discussed below.

Fourth, the existence and structure of pattern-like solutions depends on the geometry and sizeof the spatial domain, according to a linearized stability analysis that Murray reviews [10]. Theparameterγ, which accounts for this geometry, controls whether pattern formation occurs as wellas the frequency of the patterns.

Fifth, as Murray points out, the patterns that emerge from the system (4.1)-(4.2) are not steadystates. Instead, they appear in finite time. Also, they are not strictly independent of initialconditions. Nearby initial conditions yield different patterns that are nevertheless qualitativelyvery similar. Murray argues that this property is appropriate for a model of animal coats, whichdiffer slightly from one individual to another.


B. Computational Results

We now present numerical results obtained by applying the 2-grid scheme to equations (4.1) and(4.2) with the boundary conditions (4.3). We fix the values of several parameters as follows:α = 1.5, κ = 0.125, ρ = 13, s0 = 103, a0 = 77, andβ = 7. We consider a rectangular domainΩ = (0, xmax)× (0, ymax) = (0, 10)× (0, 1). For all of the computations, we apply Scheme 1,

FIG. 1. Graph ofs(x, t) at different time levels, usingγ = 250, ∆t = 0.0005. The initial uniform solutiongradually evolves into a solution that exhibits longitudinal trends (stripes) in they-direction.

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described in Section III. We choose the coarse grid so thatH = 2−4, and the fine grid so thath = 2−6.

First we show how prepatterns emerge in the model. For the first two sets of results, shownin Figs. 1 and 2,γ = 250 and∆t = 0.0005. Figure 1 shows the graph ofs(x, t) at varioustimes, while Fig. 2 shows gray-scale plots fors at the same times. The concentrations remainsnearly constant at early times untilt = 20∆t. At this time level, tiny spots begin to appear, as

FIG. 2. Gray-scale plots of the solutions shown in Fig. 1, showing more clearly the emergence of stripes.


shown in (a). Att = 30∆t, more spots appear, and some of them merge with their neighbors.This appearance of longitudinal trends is the beginning of prepattern formation, shown in (b). Att = 40∆t, the basic shapes of stripes start to appear.

At t = 60∆t, t = 80∆t, and t = 95∆t, the structure of the stripes becomes clear andstable, except that the amplitude of the stripes keeps increasing. According to Murray’s proposedmechanism, this increase would be offset in nature by the decrease and eventual disappearance ofthe pre-pattern inhibition. In the model, no new stripes form betweent = 60∆t andt = 95∆t.This fact corresponds to the observation in nature that, after some time, individual animals stopproducing new patterns. Thus, having established a pre-pattern within a finite time, we simplystop the model.

Next we consider how the geometry of the domain affects the prepatterns. The parameterγcontrols the scale of the domain for a given geometry. Equivalently, if we keep the computationalregion fixed, changing the value ofγ changes the frequency of prepatterns. Murray gives a detailedanalysis of this effect [10]. One can see the effect graphically by comparing the prepatterns shownin Figs. 1 and 2 with those shown in Figs. 3 and 4. All parameters used in these last two figureskeep the values used to generate Figs. 1 and 2, except that we increase the value ofγ to 2000 and

FIG. 3. Graphs ofs(x, t) at different time levels, usingγ = 2000 and∆t = 0.0001. As in Fig. 6.1,the solution surface develops longitudinal trends (stripes), but they occur with a higher frequency in thex-direction.

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FIG. 4. Gray-scale plots of the solutions shown in Fig. 3, showing the emergence of high-frequency stripes.

use∆t = 0.0001. The frequency of stripes is clearly higher in Figs. 3 and 4 than in Figs. 1 and2, in agreement with Murray’s analysis.

As mentioned earlier, prepattern formation should be qualitatively insensitive to the initialconditions, even though different initial conditions yield slightly different patterns. Figure 5shows that, numerically, with different initial conditions, we get qualitatively similar spatialpatterns. This result is consistent with Murray’s [10] analysis of the reaction-diffusion system(4.1)-(4.2) and with his proposed explanation for the differences in individual animals’ coats.

Whether or not these results actually describe the prepattern formation observed in real animalsis a question that biologists must answer. The main conclusion of this section is that the 2-gridmixed finite-element scheme extends to reaction-diffusion systems whose properties are of interestin applications.

V. CONCLUSIONS

The results reported here show that 2-grid methods developed and analyzed for single reaction-diffusion equations admit efficient extensions to coupled reaction-diffusion systems.

The methods have the following properties:


FIG. 5. Gray-scale plots of solutions att = 80∆t, showing the effects of different initial conditions. Theinitial values are as follows: (a)s(x, 0) = 0; (b) s(x, 0) = 0.1; (c) s(x, 0) = 0.3; (d) s(x, 0) = 1. In thesecomputations,a(x, 0) = 0, γ = 250, and∆t = 0.0005.

• They preserve theO(h) error estimates associated with the mixed finite-element discretiza-tion, provided that the coarse-grid mesh sizeH satisfiesH = O(h1/2).

• They yield significant speedups over single-grid methods, owing to their ability to con-centrate most of the nonlinear iterations on coarse grids.

• Among several reasonable orchestrations, those that produce coarse-grid approximationsfor all unknowns before transferring to the fine grid appear to be most efficient.

Finally, the schemes furnish a computationally attractive way to generate numerical solutionsto reaction-diffusion systems of interest in applications.

The authors thank Professor Eun–Jae Park of Yonsei University, Seoul, Korea, for many valuablediscussions that initiated this work.

References

1. A. M. Turing, The chemical basis of morphogenesis, Phil Trans Roy Soc Lond B 237 (1952), 37–72.

2. J. D. Murray, Mathematical biology, 2nd Ed., Springer–Verlag, Berlin, 1993.

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3. J. Douglas, R. Ewing, and M. Wheeler, The approximation of the pressure by a mixed method in thesimulation of miscible displacement, RAIRO Anal Numer 17 (1983), 17–33.

4. C. N. Dawson and M. F. Wheeler, Two-grid methods for mixed finite element approximations ofnonlinear parabolic equations, Contemp Math 180 (1994), 191–203.

5. J. Xu, Two-grid finite element discretization techniques for linear and nonlinear PDE, SIAM J NumerAnal 33 (1996), 1759–1777.

6. M. B. Allen, R. E. Ewing, and P. Lu, Well conditioned iterative schemes for mixed finite-element modelsof porous-media flows, SIAM J Sci Stat Comp 13 (1992), 794–814.

7. M. B. Allen and Z. Wang, A multigrid-based solver for transient groundwater flows using cell-centereddifferences, A. Peters et al. (Editors), Proc Tenth Int Conf Comp Meth Water Resource, Vol. 2, Heidel-berg, Kluwer, Dordrecht, The Netherlands, 1994, pp. 1375–1392.

8. L. Wu and M. B. Allen, A two-grid method for mixed finite-element solution of reaction-diffusionequations, Numer Meth PDE, to appear.

9. C. N. Dawson, M. F. Wheeler, and C. S. Woodward, A two-grid finite difference scheme for nonlinearparabolic equations, SIAM J Numer Anal 35 (1998), 435–452.

10. J. D. Murray, A Prepattern formation mechanism for animal coat markings, J Theo Bio 88 (1981),161–199.

11. D. Thomas, Artificial enzyme membranes, transport, memory, and oscillatory phenomena, D. Thomasand J. P. Kernevez (Editors), Analysis and control of immobilized enzyme systems, Springer–Verlag,New York, 1975, pp. 115–150.

12. H. Meinhardt, Models of biological pattern formation, Academic, New York, 1982.

13. P. A. Raviart and J. M. Thomas, A mixed finite element method for second-order elliptic problems, I.Galligani and E. Magenes (Editors), Mathematical aspects of the finite element method, Lecture NotesMath 606, Springer–Verlag, New York, 1977.

two-grid methods for mixed finite-element solution of coupled reaction-diffusion systems

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