NONNEGATIVITY OF EXACT AND NUMERICAL SOLUTIONS OF SOME CHEMOTACTIC MODELS

Transcription

1 NONNEGATIVITY OF EXACT AND NUMERICAL SOLUTIONS OF SOME CHEMOTACTIC MODELS PATRICK DE LEENHEER, JAY GOPALAKRISHNAN, AND ERICA ZUHR Abstract. We investigate nonnegativity of exact and numerical solutions to a generalized Keller-Segel model. Tis model includes te so-called minimal Keller-Segel model, but can cover more general cemistry. We use maximum principles and invariant sets to prove tat all components of te solution of te generalized model are nonnegative. We ten derive numerical metods, using finite element tecniques, for te generalized Keller-Segel model. Adapting te ideas in our proof of nonnegativity of exact solutions to te discrete setting, we are able to sow nonnegativity of discrete solutions from te numerical metods under certain standard assumptions. One of te numerical metods is ten applied to te minimal Keller-Segel model. Recalling known results on te qualitative beavior of tis model, we are able to coose parameters tat yield convergence to a non-omogeneous stationary solution. Wile proceeding to exibit tese stationary patterns, we also demonstrate ow naive coices of numerical metods can give pysically unrealistic solutions, tereby justifying te need to study positivity preserving metods. 1. Introduction Negative approximations of intrinsically nonnegative quantities, suc as density, are erroneous. Tey reduce confidence in simulation tecniques, even wen te negative values are close to zero. Tey often generate instabilities in nonlinear iterations twarting convergence to a solution. Accordingly, nonnegativity of simulated solutions as become te first sanity ceck for any computer simulation of biological cell densities or cemical concentrations. However, not all nonlinear systems of partial differential equations come wit a guarantee tat teir exact solutions are nonnegative. Teir numerical discretization introduces a furter layer of difficulty before one can certify tat te simulated solution will be nonnegative. In tis work, we examine tese difficulties in te context of a specific class of nonlinear systems, wic extend te influential cemotactic model proposed by Patlak [28] and later studied by Keller and Segel [21]. We provide a numerical approximation tecnique and prove tat bot te exact and te numerical solutions of te model are nonnegative. We begin by describing a generalization of te Keller-Segel model tat we sall focus on in later sections. It involves a species of density u occupying a domain Ω R n and N reacting cemicals of concentrations v i, i = 1, 2,..., N, represented as components of a vector function v. We are interested in te situation were one of tese cemicals is a cemoattractant for te species. Tis situation is modeled by te following system Key words and prases. cemotaxis, Keller-Segel, nonnegative solutions, pattern formation, monotone, numerical metods, spurious, blow up. Tis work was partially supported by te NSF under grant DMS and te Flemis Academic Centre for Science and te Arts (VLAC). 1

2 2 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR of equations, taken from [10], wic we refer to as a generalized Keller-Segel system trougout tis paper: (1a) (1b) (1c) (1d) t u = (D u χu v N ) x Ω, t > 0, t v = D v + αu + g( v) x Ω, t > 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x) x Ω, u(x, t) n = v i(x, t) n = 0 x Ω, t 0, 1 i N. Te rate of cange of v is determined by a cemical reaction network, represented by te (nonlinear) function g( v). Te parameter χ describes te cemotactic sensitivity. Te term αu, wit a constant vector α R N satisfying α 0, indicates tat some or all of te cemicals can be produced by te species. (Here and trougout, te notation w 0 signifies tat eac component of w is nonnegative.) Above, we ave augmented te differential equations wit no-flux boundary conditions and initial conditions. We assume tat te initial data u 0 0 and v 0 0 are nontrivial functions on Ω. Additionally, n generically denotes te unit outward normal on te boundary of any domain under consideration and / n = n. (E.g., in (1), te domain under consideration is Ω and n is te outward unit normal on te boundary Ω.) For now, we assume tat Ω is Lipscitz so tat n is defined a.e. on Ω, but we will place furter assumptions on Ω in later sections for teoretical reasons. Trougout, we also use te abbreviated notations t and i for / t and / x i, respectively. We note tat oter generalizations of te Keller-Segel system as recently been proposed in [15] and [20], but most of teir analysis is limited to a particular number of cemicals, and moreover teir focus is not on te nonnegativity questions we intend to study ere. Let us now consider a few examples tat fit te generalized Keller-Segel model (1). Example 1.1 (Minimal Keller-Segel model). Tis very well-known nonlinear system of two equations is obtained by coosing, in (1), N = 1, α = [ α 1 ], g( v) = [ γv1 ], wit α 1 > 0 and γ > 0. In tis setting, u represents te density of te amoeba Dictysotelium discoideum and v te density of te cemoattractant cyclic adenosine monopospate (camp). Keller and Segel proposed tis model to understand aggregation by cemotaxis in [21]. Te extensive reviews in [19] and [29] puts tis model in perspective and points to many known results on te beavior of its solutions. Example 1.2 (Full Keller-Segel model wit decay). Te minimal Keller-Segel model from te previous example is a simplified version of a four-equation model, also originally derived in [21]. As in te case of te minimal model we let u denote te density of te amoeba, but te density of te camp is now denoted by v 3. Additionally, an enzyme wic degrades te camp, and is also emitted by te amoeba, enters te model. We denote te density of tis enzyme by v 1. As explained in [21] or [19], te camp and te enzyme undergo a reversible reaction to form a complex, wit density v 2. (Tis complex may ten degrade into te enzyme plus a degraded product wic is not typically

3 POSITIVE EXACT AND NUMERICAL SOLUTIONS 3 considered in tis model.) Te reactions just described can be briefly written as (2) v 1 + v 3 v 2 v 2 + degraded product. Moreover, we assume te enzyme decays at some positive rate γ 1, as justified biologically in [22]. Assuming a linear (in u) cemotactic sensitivity function, as in te minimal model, as well as constant production rates, we can now express te four equation model in te setting of (1). If te forwards, backwards and decay reaction rate constants from (2) are given by k i for 1 i 3 respectively, using te laws of mass action, te reaction network is described by k 1 v 1 v 3 + (k 2 + k 3 )v 2 γ 1 v 1 g( v) = (k 2 + k 3 )v 2 + k 1 v 1 v 3 k 1 v 1 v 3 + k 2 v 2. Using tis g in (1), and setting cemical production rates by α 1 α = 0, α 3 wit α 1 > 0 and α 3 > 0, we obtain te full Keller-Segel model. Except for te decay term, tis is te same as te four-equation model of [21]. Example 1.3 (Dimerization). Te cemistry of dimerization involves two molecules of a cemical v 1 combining reversibly to form anoter cemical v 2. Tis reaction can be written as 2v 1 v 2 wit te forwards and backwards reaction rate constants given by k 1 > 0 and k 2 0 respectively. If we also assume tat te cemicals decay wit rate constants γ 1 0 and γ 2 0, ten te laws of mass action yield [ ] 2k1 v1 (3) g( v) = 2 + 2k 2 v 2 γ 1 v 1 k 1 v1 2 k 2 v 2 γ 2 v 2 to describe te reaction kinetics of te network. Letting [ ] α1 α =, 0 te system (1) describes a scenario in wic an organism produces a cemical, wic undergoes dimerization to form anoter cemical, wic is in turn a cemoattractant for te organism. In te next section, we sow tat under a reasonable assumption on g (satisfied for all te above examples), solutions of te generalized Keller-Segel system (1) ave nonnegative components. Nonnegativity of solutions of te minimal model of Example 1.1 seems to be widely recognized among researcers to follow from te maximum principle. However, we ave found it difficult to locate a reference wic details te application of te appropriate maximum principle to be used. Wile te result seems to be known for te minimal model, nonnegativity of solutions of te full Keller-Segel model and te generalized model (1) were not known previously. Accordingly, in Teorem 2.2 below, we prove a nonnegativity

4 4 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR result for te general model. We include all te essential details (and include an appendix collecting te more standard results we use in te proof). Te arguments we present are elementary and self-contained. Tese arguments ten serve as our motivation in te design and nonnegativity analyses of a numerical metod we present in Section 3. Tere are a few oter factors driving te design of our numerical metod. We aim for te simplest positivity preserving metod tat can be obtained using te lowest order Lagrange finite element approximation space, for two reasons: First, finite element spaces allow us to compute on unstructured meses and visualize solutions on complicated domains, unlike te finite difference models wic are limited to simple domains like te square or to one space dimension [7]. Second, te lowest order finite element space (wic as been around since [8]) is now available in almost any computational package for partial differential equations. We ope to lower te overead of implementing a new numerical metod for cemotaxis by designing a metod using standard tools. Indeed, te metod we propose is a combination of various tools, all of wic by itself, are standard, but teir combination, as we sall sow, is particularly suited for te cemotaxis application. Apart from te Lagrange finite element space, te oter standard ingredients we use include a discretization of [42], a mass lumping tecnique [39], and simple backward differencing in time. Te numerical treatment of nonlinear reaction term is motivated by [41]. Te need for good numerical metods in understanding cemotaxis ave not been underestimated by oter researcers. A comparison of te performance and efficiency of some of te various (not necessarily positivity preserving) numerical scemes for cemotaxis is presented in [38]. Among oter numerous previous works are upwind finite element and central upwind finite volume metods [6, 31, 32], mixed finite elements [25] (tat borrows tecniques from similar semiconductor models [4, 5, 17]), discontinuous Galerkin (DG) metods [11, 12], and a fractional step metod [40]. Te mixed and DG metods are particularly interesting since tey are extendable to iger order (unlike our metod, wic is essentially first order). Tese metods also possess mass conservation properties. However, most of tese metods use tecniques tat are not as standard as te simple Lagrange finite element, and ave an additional overead in solution cost, primarily due to increased system size. Furtermore, wile conservation is of critical importance in many applications were total output is important (e.g., oil flow), it is currently unclear if it is important in cemotaxis, especially if it requires extra expense. Te finite element models of [31, 32] seem to be te closest to tis paper. Considering te minimal model and assuming no time dependence in cemical concentration, [31] solves for te cemical concentration using a discrete Greens function, and uses it to obtain a nonlocal approximation of (1a). Te approac is extended in [32] to te case were bot quantities vary in time. Mass lumping and mes angle conditions are used in [32] (very similar to te our treatment later) to provide a convergence teory. Tere, stability is gained (see [32, eq.(4)]) by an upwinding approximation of [2] applied only to te last term of (1a). In contrast, we use te approac of [42] to approximate te entire flux term in (1a). Anoter finite element approac is considered in [37]. Tere, te focus is on maintaining positivity of solutions and conservation properties via a flux correction, resulting in a different metod from wat we propose in tis paper. To indicate oter points of departure from te existing literature, many previous numerical studies ave

5 POSITIVE EXACT AND NUMERICAL SOLUTIONS 5 v 2 Figure 1. Te directions of te vector field g( v) of Example 1.3 see (3) point inward at te points marked on te axes. (Te parameters in (3) were set to k 1 = 0.1, k 2 = 0.2, γ 1 = γ 2 = 0 for tis illustration.) v 1 focused on solutions of te minimal model tat blow up [6, 12, 37]. We instead focus on obtaining a bounded stationary pattern. Also, several studies like [25, 31] consider models even furter simplified from te minimal model. None of te above mentioned numerical studies consider a system of more tan two partial differential equations. In contrast, our aim is to develop tecniques tat apply to te more general system (1). In te next section, we prove a non-negativity result for te solutions of te nonlinear boundary value problem (1). In Section 3, we consider a finite element discretization of (1), motivating te derivation of matrices involved and our treatment of te nonlinear term. In Section 4, we prove tat, under certain mes angle conditions, te numerical solutions are nonnegative. We ten apply te numerical metod to visualize a stationary solution of te Keller-Segel model. 2. Nonnegativity of exact solutions In tis section, we prove tat solutions of (1) are nonnegative. Trougout tis section we assume tat Ω is a bounded connected set wose boundary Ω is smoot and in particular, satisfies te interior ball condition. We also assume tat te u 0 and v 0 are smoot functions. Assuming tat a smoot solution to (1) exists, we now proceed to determine te sign of te solution. Our analysis is under te following assumption on te reaction term g. Assumption 2.1. Assume tat g is uniformly Lipscitz on compact subsets of R N and tat for any v 0, we ave g j ( v ) 0 wenever v j = 0, for any j = 1, 2,..., N. Loosely speaking, tis assumption requires te vector field g to be inward pointing at te boundaries of te positive ortant in te v-coordinate system (see Figure 1). Te following teorem is te main result of tis section.

6 6 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR Teorem 2.2. Suppose Assumption 2.1 olds. Assume tat (u, v) is a smoot solution to (1) wit nontrivial u 0 0 and v 0 0. Ten u > 0 for all x Ω and t > 0 and moreover, v 0 for all x Ω and t 0. Te proof of tis teorem appears at te end of tis section. We will need a few standard, and some not so standard, maximum principles for parabolic problems. Tese results are collected in Appendix A in a form appropriate for use in te arguments of tis section. In order to prove te teorem, we will need to develop a few intermediate lemmas below. Before proceeding to tese lemmas, let us first sow tat for a large class of practically important reaction networks, Assumption 2.1 olds. A general reaction network, wit N cemicals v i, is composed of many reactions of te type (4) v i1 + v i2 + + v ik κ v j1 + v j2 + v jl were i s and j s (are possibly repeated) indices contained in {1, 2,..., N}. We consider any g( v) derived from te laws of mass action kinetics. Fix any index i, and consider a component reaction of te form (4), were i appears M 1 times on te left and side of (4) and M 2 times on te rigt and side of (4). Wen te kinetics of a reaction is determined by mass action, te rate at wic te reaction occurs is directly proportional to te products of te concentrations of te reactants, and te constant of proportionality is called te rate constant, denoted by κ in (4). Tus, te i t component of g( v) satisfies (5) g i ( v) = M 1 κv i1 v i2 v ik + M 2 κv i1 v i2 v ik = (M 2 M 1 )κv i1 v i2 v ik. Tis gives te rate of production of v i in te reaction network. Next, we consider two cases: First, if M 2 M 1, it is obvious from (5) tat (6) g i ( v) 0 for all v 0. In te remaining case, if M 2 < M 1, ten since bot M 1 and M 2 are non-negative integers, obviously M 1 1. In oter words, v i appears on te left and side of (4) at least once, and ence appears as a factor in te product v i1 v i2 v ik. Terefore, (7) g i ( v) 0 wenever v i = 0, for all v 0. Since i was an arbitrary index, we conclude tat eiter (6) or (7) olds for every component of g. Terefore, we ave just establised tat Assumption 2.1 olds for any g obtained from mass action kinetics. (Note tat any function derived by mass action, being a polynomial, obviously satisfies te local Lipscitz continuity required in Assumption 2.1.) Finally, observe tat te functions g in Examples 1.2 and 1.3 were derived using te law of mass action, so te positivity result of Teorem 2.2 applies to tose examples. Tat Assumption 2.1 is satisfied in te case of Example 1.1 is trivial to sow. Let us now develop te intermediate results required to prove Teorem 2.2. We begin wit a result on a scalar equation and eventually proceed to generalize it to te system of equations defining v. Altoug results like te next lemma seem to be known [9] in te community, we aim to give an elementary and self-contained presentation.

7 POSITIVE EXACT AND NUMERICAL SOLUTIONS 7 Lemma 2.3. Suppose tere is an ε 0 > 0 suc tat for all 0 < ε < ε 0, te continuous function f : R Ω [0, T ] R satisfies (8) f( ε, x, t) > 0, (x, t) Ω [0, T ] Let w : Ω [0, T ] R be continuously differentiable wit respect to t, twice continuously differentiable wit respect to x, and satisfy (9a) (9b) t w = d w + f(w, x, t), x Ω, t (0, T ] w(x,t) 0 x Ω, t = 0, for some number d > 0, togeter wit te following boundary condition: (9c) Eiter w(x, t) 0, x Ω, t (0, T ], (9d) w or 0, x Ω, t (0, T ]. n Ten w(x, t) 0 for all x Ω and t [0, T ]. Proof. If not, tere is an ( x 0, t 0 ) Ω [0, T ] suc tat w( x 0, t 0 ) < 0. Ten w( x 0, τ) is negative at τ = t 0 but nonnegative at τ = 0 due to (9b). Hence, tere are (small enoug) values of ε (0, ε 0 ) suc tat w attains te value of ε at one or more τ (0, t 0 ). Fix suc an ε and let t 0 be te first time wen w attains te value of ε. Let x 0 be any point in Ω suc tat w(x 0, t 0 ) = ε. Ten, as τ increases to t 0 and is sufficiently close to t 0, te function w(x 0, τ) cannot increase, so (10) t w(x 0, t 0 ) 0. Note also tat (11) min w(x, t) = w(x 0, t 0 ) = ε (x,t) Ω [0,t 0 ] (because if w took a value lesser tan ε in Ω [0, t 0 ], ten t 0 would not be te first time wen w attains ε). Te remainder of te proof is split in two parts. First, suppose (9c) olds. Ten, x 0 Ω. In view of (11), we terefore ave (12) w(x 0, t 0 ) 0. Combining (10) and (12), we find tat 0 ( t w d w) (x0,t 0 ) = f( ε, x 0, t 0 ). Tis contradicts (8) and finises te proof in te case of te boundary condition (9c). To complete te proof for te case of te boundary condition (9d), first note tat if x 0 is an interior point of Ω, ten we obtain a contradiction using (10) and (12) as above, so we need only consider x 0 Ω. By (9a) and (8), te inequality t w d w = f > 0 olds at (x 0, t 0 ), and so by continuity, it olds in Ω 0 [t 1, t 0 ] Ω (0, t 0 ], were Ω 0 is a ball wose boundary contains x 0 (possible due to our assumption tat te interior ball condition olds). By (11), we know tat te minimum of w in Ω 0 [t 1, t 0 ] is attained at (x 0, t 0 ). If tis minimum is also attained at anoter point (x 0, t 0) in te same neigborood, ten x 0 is an interior point of Ω, so (10) and (12) finis te proof as before. Hence it only remains to consider te situation wen x 0 Ω 0 is te sole point in Ω 0 [t 1, t 0 ] were te

8 8 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR minimum is attained. But in tis situation, all conditions of Lemma A.1 are satisfied in a sufficiently small parabolic frustum contained in Ω 0 (t 1, t 0 ]. Hence, (56) implies tat w/ n < 0 at x 0 wic contradicts (9d). In order to strengten tis result to a more useful form for systems, we need a result on existence and uniqueness of a PDE of te form (9). Te needed results can be found in [16, Teorems 6 and 10 in 7.4]. To fit our application, we sligtly modify and restate tem in te teorem below. Recall tat f(w, x, t) is said to be locally Hölder continuous in (x, t) if tere is a C and 0 < α < 1 suc tat f(w, x 1, t 1 ) f(w, x 2, t 2 ) C (x 1, t 1 ) (x 2, t 2 ) α for all (x 1, t 1 ) and (x 2, t 2 ) in every closed bounded subset B of Ω T. Teorem 2.4. Consider (9a) togeter wit te initial and boundary condition (13) w(x, t) = w 0 (x, t), (x, t) ( Ω {t = 0}) ( Ω [0, T ]) for some smoot w 0. Assume tat f(w, x, t) is Lipscitz continuous in w uniformly wit respect to bounded subsets of R Ω [0, T ], and tat f(w, x, t) is locally Hölder continuous in (x, t). Suppose te equation t w 0 = d w 0 + f(w 0, x, 0) olds on Ω for some number d > 0. Ten tere is a 0 < T 0 T suc tat a unique solution to (9a), satisfying te initial and boundary condition (13), exists in Ω T0. Let us now proceed to systems of partial differential equations. We say tat f : R N Ω [0, T ] is locally Lipscitz continuous in y, uniformly in (x, t) if for every bounded subset D R N, tere is a constant M f > 0 suc tat (14) max f i ( y, x, t) f i ( z, x, t) M f max y i z i i i for all y, z D and all (x, t) Ω [0, T ]. Note tat M f does not depend on (x, t). Extensions of results like Lemma 2.3 for systems can be found in [1]. However, tere is a long list of assumptions in [1], to be verified before one can conclude positivity. Some of tose assumptions do not apply to (1), yet is tecniques, rooted on te positive invariance of te positive cone, appear to be powerful enoug to extend to (1). Noneteless, instead of adapting is tecnique, we ave cosen to present anoter elementary proof, inspired by Weinberger [41], one of te original arcitects of suc tecniques. Furtermore, te details of te ensuing argument will serve as motivation to construct a discrete version of te same argument to prove tat numerical solutions are also nonnegative (in Section 4). Lemma 2.5. Suppose tat f : R N Ω [0, T ] is locally Lipscitz continuous in y, uniformly in (x, t), and locally Hölder continuous in (x, t). Assume tat for all (x, t) Ω T, (15) f i (y 1, y 2,..., y i 1, 0, y i+1,..., y N, x, t) 0, i = 1, 2,..., N wenever y j 0 for all j i. Let w : R N Ω [0, T ] R N be a smoot solution of (16a) (16b) t w i = D i w i + f i ( w, x, t), x Ω, t (0, T ], i = 1, 2,..., N, w(x,t) 0 x Ω, t = 0,

9 POSITIVE EXACT AND NUMERICAL SOLUTIONS 9 togeter wit te following boundary condition: (16c) Eiter w(x, t) 0, x Ω, t (0, T ], (16d) w or 0, x Ω, t (0, T ]. n Ten w(x, t) 0 for all x Ω and t [0, T ]. Proof. Te idea is to construct a new function F i from f i suc tat Lemma 2.3 can be applied. To tis end, first let a + = max(a, 0). It can be easily verified, case by case, tat for any two numbers a and b, (17) a + b + a b. Next, define F i : R Ω [0, T ] R by (18) F i (v, x, t) = f i ( w 1 (x, t),..., w i 1 (x, t), v +, w i+1 (x, t),..., w N (x, t), x, t ) + v + v. By (17) and (14), F i is locally Lipscitz continuous in v, uniformly in (x, t). We now prove tat, for an arbitrary ε > 0, (19) F i ( ε, x, t) > 0, (x, t) K ε 1 i for any ε 1 < ε/m f. Here Ki α = {(x, t) Ω [0, T ] : w j (x, t) α for all j i}. Before proving tis, note tat altoug (15) immediately implies te inequality F i ( ε, x, t) ε, tis inequality olds in general only for (x, t) Ki 0. To obtain a similar inequality in a larger set, we use (14). Wence, for any ε 1 > 0 and any (x, t) K ε 1 i, f i ( 0, x, t) f i (w 1,..., w i 1, 0, w i+1,..., w N, x, t) M f ε 1, Since te first term above is nonnegative due to (15), tis implies tat F i ( ε, x, t) = f i (w 1,..., w i 1, 0, w i+1,..., w N, x, t) + ε M f ε 1 + ε. and (19) follows. Accordingly, we fix ε 1 < ε/m f and proceed. To prove te lemma by way of contradiction, suppose w 0. Ten tere exists some ε 2, sufficiently small, and cosen so tat 0 < ε 2 ε 1, suc tat at least one of te components of w attain te value ε 2. Let t 1 > 0 be te first time tat any of te components of w attain te value ε 2, and let i and x 1 Ω be suc tat w i (x 1, t 1 ) = ε 2. Ten (cf. (11)) (20) min i Clearly, tis implies tat min (x,t) Ω [0,t 1 ] w i (x, t) = w i (x 1, t 1 ) = ε 2. (21) Ω [0, t1 ] K ε 2 i K ε 1 i. Now, let v i (22a) (22b) be te solution to t v i = D i v i + F i (v i, x, t), x Ω, t > 0, v i = w i, (x, t) ( Ω {t = 0}) ( Ω {t > 0}). By Teorem 2.4 and te aforementioned continuity properties of eac F i, v i exists in an interval [0, t 2 ], were t 2 > 0 is te maximal time of existence of te solution. Te remainder of te proof is split into two cases: t 2 t 1 and t 2 < t 1.

10 10 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR Consider te first case t 2 t 1. Because of (21), we find tat te inequality in (19) olds for all x Ω and all 0 t t 1. So we can apply Lemma 2.3 to conclude tat v i 0 on [0, t 1 ), wic in turn implies tat F i (v i, x, t) f i (w 1,..., w i 1, v i, w i+1,..., w N, x, t). But ten, te (i t) equations in (16a) (16c) sow tat w i also solves (22) on [0, t 1 ). By uniqueness (Teorem 2.4) we conclude tat w i = v i 0 on [0, t 1 ), a contradiction to (20). Te oter case, t 2 < t 1, also leads to a contradiction, as we now sow. As above, we conclude tat w i = v i 0 on [0, t 2 ). Now, due to te smootness assumptions on w i on [0, T ] (and noting tat t 2 < t 1 T ) te solution v i is smoot at t = t 2. But ten we can extend te solution v i to some interval [0, t 3 ) wit t 3 > t 2 by again invoking Teorem 2.4. Tis is a contradiction to te maximality of t 2 and finises te proof. Finally, we are in a position to prove our main result. Proof of Teorem 2.2. First, let us prove tat u > 0. From (1a) and te product rule, t u D u + χ u v N + χu v N = 0. Terefore, by Lemma A.4 in Appendix A, applied wit Lu := D u + χ u v N + χu v N, we obtain tat u > 0 on Ω T. (Note tat te lowest order term c = χ v N, being a continuous function over a compact set, is bounded below.) Next, we prove tat v 0. Due to (1b), we may apply Lemma 2.5 wit w = v and f = αu + g( v). Condition (15) is satisfied because wenever v i = 0, we ave f i = α i u + g i ( v) 0 by virtue of te assumption on g and te already proved positivity of u. Te remaining assumptions of te lemma are easily verified, so its conclusion v 0 olds on Ω T. 3. Numerical metod In tis section, we describe two computational scemes for solving (1). Under certain conditions, te resulting numerical solutions are non-negative, as proved in te succeeding section. To focus on te numerical approximation, from now on, we assume tat Ω is a bounded connected polygon in R 2, wic is partitioned into triangles. Te collection of tese triangles (or elements ) is denoted by T and we assume tat tey satisfy te standard assumptions (see e.g., [39]) of a geometrically conforming finite element mes Description of te metod. Te solution functions u and v i are approximated in te lowest order Lagrange finite element space S = {w C(Ω) : w K is linear on K for every mes triangle K}. For any l = 1, 2,..., P, let φ l S denote te function tat equals one at te lt mes vertex and equals 0 at all oter mes vertices. We expand te approximating functions, at time t n = kn, in te basis {φ l } u(x, t n ) u n = P Ul n φ l, v i (x, t n ) v,i n = l=1 P l=1 V n i,lφ l.

11 POSITIVE EXACT AND NUMERICAL SOLUTIONS 11 Given te vectors U n and V i n, wose lt components are Ul n and Vi,l n, resp., we propose to compute te approximations at time k(n + 1) by solving te following system: (23a) (23b) 1 k M( U n+1 U n ) = DA(v n+1,n ) U n+1, 1 k M( V n+1 i V i n ) = D i LV n+1 i + α i MU n + MG + n+1 i ( v ). Te P P matrices A( ), L, and M are defined below. Te vector G + i ( v n ) is te vector wose lt component equals (24) g +,n i,l g i ( (V n 1,l ) +, (V n 2,l) +,..., (V n N,l) +) + (V n i,l) + V n i,l and for any number a, as before, a + = max(a, 0). Tis construction is motivated by te adaption of Weinberger s tecnique described in te previous section cf. (18). Note tat te metod (23) is an implicit metod: Given U n and V i n, we first solve te (second) nonlinear equation (23b) for V n+1 i,l. Ten we solve a P P linear system for U n+1 given by (23a). Note tat existence of numerical solutions for (23b) is not as straigtforward and is postponed for future studies. In te remainder, we take te approac of te previous section, wereby we study non-negativity properties of solutions, assuming tey exist. Let us now describe te matrices featured above in bot te metods. Te P P matrix M is a diagonal matrix obtained after lumping te masses [39], i.e., M jj = P k=1 M jk, were M jk = φ j φ k. Ω Clearly, M is te standard mass matrix wile M is te so-called lumped mass matrix. Te remaining matrices are stiffness matrices, given by L jk = L K l(j),l(k), L K l(j),l(k) = φ l(j) φ l(k), K T K (25) A jk (z) = K T A K,z l(j),l(k) were A K,z l(j),l(k) is defined below and l(j) {0, 1, 2} denotes te local vertex number of te l t global vertex. To describe te 3 3 matrix A K,z for any triangle K and any linear function z on K, we use te following notations associated to K. Let a l, l {0, 1, 2} denote te vertices of K, E lm denote te edge connecting a l and a m, and θ lm denote te interior angle opposite to te edge E lm. Ten define c lm (z) = 1 E lm E lm e (χ/d)z

12 12 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR Te local indices l, m {0, 1, 2} are always calculated mod 3 (so, e.g., A K l,l+3 = AK ll ), and tus all nine entries of A K,z are defined by (26a) (26b) (26c) A K,z ll = L K p l (z) l,l+1 c l,l+1 (z) p l (z) LK l,l+2 c l,l+2 (z) p l+1 (z) c l,l+1 (z) p l+2 (z) c l+2,l (z) A K,z l,l+1 = LK l,l+1 A K,z l,l+2 = LK l,l+2 were p l (z) = e (χ/d)z( a l). Tis matrix arises from a spatial discretization proposed in, as we will clarify in 3.2. Tese definitions complete te prescription of bot te metods (23) and (33). In te next section, we will prove tat bot te above proposed metods ave monotonicity properties under certain assumptions. Remark 3.1. In a computer implementation, we ave to be careful wile computing te ratios p l (z)/c lm (z), so as not to divide by small numbers. Te problem is evident wen considering p l (z) c lm (z) = 1 E lm e (χ/d)z( al) E lm e (χ/d)z were te denominator can be very small even wen z takes moderately large positive values on E lm. Noneteless, observe tat a l is an endpoint of te edge E lm and z is linear on E lm. Hence te quantity z z( a l ) is better suited for exponentiation. It is terefore better to compute te above ratio by implementing te following equivalent formula: p l (z) c lm (z) = ( 1 E lm E lm e (χ/d)(z z( a l)) ) Derivation of te metod. We begin te derivation by obtaining a variational form of (1). Multiply te equations of (1) by a test function φ H 1 (Ω) and integrate by parts. Ten, defining te flux J = D u χu v N, we find tat u and v i satisfy (27a) ( t u)φ = J φ, Ω Ω (27b) ( t v i )φ = D i v i φ + α i Ω Ω Ω uφ + Ω g i ( v)φ, for all i = 1, 2,..., N and all φ H 1 (Ω). Te discrete solution {u n, vn,i } is obtained by using a suitably approximated version of te above equations for all φ in te previously defined finite element subspace S H 1 (Ω). We now describe ow eac term in (27) is approximated, beginning wit te last term. Let G i ( v n) denote te vector wose lt component equals gn i,l g i(v1,l n,..., V N,l n ). At time

13 POSITIVE EXACT AND NUMERICAL SOLUTIONS 13 t = k(n + 1), setting φ equal to a basis function φ m, ( P ) g i ( v)φ m g i (V n+1 1,l,..., V n+1 N,l )φ l φ m = Ω Ω l=1 M G i ( v n+1 ). P l=1 M ml g n+1 i,l = M G i ( v n+1 ) In te last step above, we ave approximated te action of M by tat of M. Tis so-called process of lumping te mass is a well-known first order approximation [39]. Note also tat in te first step above, we ave approximated g i by its Lagrange interpolant on te finite element mes. However, comparing wit te last term in (23b), we find tat in place of te term G i ( v n+1 ) derived above, we ave G + n+1 i ( v ) instead. Noneteless, we sall prove in te next section, under a condition on te mes, tat any solution of (23) is nonnegative. Since G + i ( v n+1 ) = G i ( v n+1 ) wenever v n+1 0, we conclude, a posteriori, tat G + i ) can be replaced by G i ( v n+1 ), wenever te mes conditions old. Te oter two terms on te rigt and side of (27b) are approximated in a straigtforward manner using te standard finite element metod, so tose derivations are omitted. To describe ow te rigt and side of (27a) is approximated, we recall a succinct presentation of [42] and modify it to account for our different boundary conditions and nonlinear terms. Te main idea, going back to [24], is to approximate te term involving flux ( v n+1 (28) J = D u χu vn as if te flux were spatially constant on eac mes element. Consider a mes element K in T. Let a l, l {0, 1, 2} denote te vertices of K and let θ lm denote te interior angle opposite to te edge connecting vertices a l and a m. Let λ l denote te linear function on K wose value at te vertex a m is δ lm. It is easy to prove tat (29) L K lm λ m λ l = 1 2 cot θ lm K for all l m in {0, 1, 2}. It is also easy to sow, using te symmetry and zero-row-sum property of te 3 3 matrix L K, tat for any two linear functions z and w, (30) z w = L K ml d lm (z) d lm (w), m<l K were d lm (w) = w( a l ) w( a m ). Let c be any constant vector. Ten, combining (30) and (29), we obtain (31) c w = 1 K 2 c ( a m a l ) d ml (w) cot θ ml. m<l

14 14 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR Tis motivates te following approximation of te J-term in (27a) wen φ S. Te contribution to te integral on rigt and side of (27a) from an element K T is J φ 1 (32) J K 2 ( a m a l ) d ml (φ) cot θ ml. m<l In view of (31), we expect tis to be a reasonable approximation wenever ( J is smoot enoug and K is small enoug so tat) J is almost constant on eac mes element K. Next, we describe ow te term J ( a m a l ) in (32) is furter approximated. Letting t lm = ( a l a m )/ a l a m, it is easy to see from (28) tat te following identity olds on te edge E lm connecting a l to a m : De (χ/d)v N J t ml = (e (χ/d)v N u) t ml = d ml (e (χ/d)v N u) E lm E lm Tis motivates te approximation J ( a m a l ) = J t ml E lm E lm d ml (e (χ/d)v N u). E lm 1 D e (χ/d)v N Substituting tis into (32), and using te approximate solution components v,n n at te nt time-step in place of v N, and te approximate solution component u n+1 in place of u, J φ 1 E lm d ml (e (χ/d)vn,n u n+1 ) d ml (φ) cot θ ml K 2 m<l E lm 1 D e (χ/d)vn,n = DLK ml c lm (v n m<l,n ) d ml(e (χ/d)vn,n u n+1 ) d ml (φ) = l,m DA K,vn,N lm u n+1 ( a m ) φ( a l ) for any φ S. Tis completes te derivation of (23a), once we also approximate te time derivative in (27a) by a standard backward finite difference and lump te masses into te diagonal matrix M. Te derivation of (23b) from (27b) is similar and easier, so we omit te details A variant of te metod. Te metod (23) requires te solution of a nonlinear system at every time step and consequently can be expensive. A simple semi-implicit variant tat does not require nonlinear solvers at eac time step is te following: Given Ul n and Vi,l n n+1, let Ul and V n+1 i,l satisfy (33a) (33b) 1 k M( U n+1 U n ) = DA(v,N) n U n+1 1 k M( V n+1 i V i n ) = D i LV n+1 i + α i MU n + MG i ( v n ) In tis metod, in contrast to (23), we first solve te linear system (33a) to obtain U n+1, and ten solve anoter linear system given by (33b) to obtain V n+1 i. Hence existence of numerical solutions to (33) follows immediately from te invertibility of te two linear

15 POSITIVE EXACT AND NUMERICAL SOLUTIONS 15 systems. (Tis invertibility, under certain mes assumptions, is a simple consequence of te arguments in te next section.) 4. Non-negativity of numerical solutions In tis section, we prove tat te numerical solutions obtained by solving eiter (23) or (33) are non-negative. We will need to assume certain conditions on te angles of mes elements to proceed wit tis proof. Te edges of all triangles in T form te collection of mes edges E. Wen an edge e E is sared by two triangles in T, we denote by θ e ± te two angles subtended by e at te two vertices opposite to e. We assume tat (34a) θ + e + θ e π. Wen e E is on te boundary Ω, tere is only triangle adjacent to it and te angle subtended by e at its sole opposite vertex is denoted by θ e. We assume tat (34b) θ e π/2 for all suc edges e Ω. Several software packages exist tat generate meses satisfying (34). E.g., Delaunay triangulations generated by [34] satisfy (34a). Oter examples include software for generating acute triangulations [13] aving elements wit interior angles less tan π/2 obviously, suc meses satisfy (34a) and (34b). It is well known tat some suc angle condition is necessary for obtaining monotonicity properties in te finite element context (see, e.g. [39]) Non-negativity of te numerical cell density. We now sow tat te approximations u n given by metods (23) and (33) are non-negative provided te initial cell density is non-negative. Teorem 4.1. Suppose tat (34) olds. Fix some natural number m > 0 and assume tat for every 0 n m, solutions u n and vn,i to (33) exist at time t = kn > 0. Ten (for any time step size k > 0) a solution u m+1 to (33a) exists and we ave u m+1 0 on Ω, wenever u 0 0 on Ω. Similarly, if for every 0 n m, solutions un exist, ten a solution u m+1 to (23a) exists and we ave wenever u 0 0 on Ω. u m+1 0 on Ω, Proof. Fix any 0 n m. Consider (33) first. From (33a), (35) U n+1 = (M + kda) 1 M U n and vn+1,i to (23) were, for convenience, we ave abbreviated A = A(v n,n ). We will now prove tat te inverse of B = M + kda, appearing above, is a nonnegative matrix for any v n,n S. To tis end, consider an off-diagonal entry A jk, wic is nonzero only if tere is a mes edge e E connecting te jt and kt mes vertices. First, consider te case of boundary edges e Ω. Ten by (26b) or (26c), we conclude tat te sign of A jk is te same as te sign of an off-diagonal entry of L K. But, tis entry cannot be positive in

16 16 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR view of (29) and (34b). Second, consider te case wen e is an interior mes edge. Ten, again using (26), we find tat te sign of A jk is te same as te sign of (36) (cot θ + e + cot θ e ) = sin(θ+ e + θ e ) sin θ + e sin θ e 0 were we ave used (34a). Tus (37) A jk 0 for all j k. Next, observe tat (26) implies te column sum of te element matrices are zero indeed, for any l {0, 1, 2}, moving indices mod 3 in (26), A K,z ll = L K p l (z) l,l+1 c l,l+1 (z) p l (z) LK l,l+2 c l,l+2 (z), AK l+1,l = L K p l (z) l,l+1 c l,l+1 (z), AK l+2,k = L K p l (z) l,l+2 c l,l+2 (z) so teir sum vanises for any function z for wic te above quantities are well defined. Terefore, te matrix A obtained by summing A K,z over all K as in (25), as te property tat (38) A jj = j k A jk. In particular, tis implies tat te diagonal entries A jj 0. Returning to B, we now find tat all its diagonal entries B jj = M jj +kda jj are positive as M jj > 0. Moreover, B jj = M jj + kd k j ( A kj ) (by (38)) = M jj + kd A kj (by (37)) k j > kd A kj (as M jj > 0) k j = B kj (as M kj = 0). k j Tus, B is strictly diagonally dominant wit positive diagonal entries. Hence [3, M 35 N 38 ], we conclude tat B 1 0. Since M 0, by (35), we find tat U n+1 0 wenever U n 0. Since tis olds for any 0 n m, ten recalling u 0 0 and using a simple induction step proves te statement of te teorem for metod (33). Finally, observe tat te above argument olds verbatim if we instead set A = A(v n+1,n ). Hence, we conclude tat te teorem also olds for metod (23) Non-negativity of numerical signal concentrations. To prove tat v n,i 0 for te first metod (23), we develop a discrete version of te arguments used to prove Lemmas 2.3 and 2.5.

17 POSITIVE EXACT AND NUMERICAL SOLUTIONS 17 Teorem 4.2. Suppose Assumption 2.1 and (34) old and fix some natural number m > 0. Suppose tat for any 0 n m solutions u n and vn+1,i solve (23) at any time t = kn > 0. Ten for i = 1, 2,..., N, and for any time step size k > 0, we ave v n,i 0 on Ω, for all 0 < n m + 1 wenever v,i 0 0 on Ω. Proof. Supposing te result is not true, we proceed to find a contradiction. Let n 0 be an integer suc tat tere is an i for wic v n +1,i 0, but v,i n 0 for all n n and all i, i.e., k(n + 1) is te very first time wen a numerical solution component becomes negative. (Cf. proof of Lemma 2.5 to see te analogy between te discrete and exact arguments.) Now, set l suc tat (39) min V n +1 i l,l = V n +1 i,l < 0. and let ε V n +1 i,l. By (23b), (40) 1 k M ( l l V n +1 i,l V n i i,l ) = Di [L V n +1 ] l + α i M l l U n l + M l l g +,n +1 i,l We proceed to establis a contradiction by examining te sign of eac term above. Beginning ( wit te last term, we observe, in view of (24) and Assumption 2.1, tat g n +1 i,l = g i (V n +1 1,l ) +,..., (V n +1 i 1,l ) +, 0, (V n +1 i +1,l ) +,..., (V n +1 N,l ) +) + (0 ( ε)) ε > 0. Since M l l is also positive, te last term in (40) is positive. Next, by Teorem 4.1, we know tat U n 0, so te penultimate term in (40) is non-negative. To study te remaining term on te rigt and side of (40), we begin wit (41) [LV n +1 i ] l = Now, due to (39), V n +1 i,l P l=1 L l lv n +1 i,l = L l l V n +1 i,l l l L l lv n +1 i,l. V n +1 i,l. Furtermore, due to (34), te off-diagonal entries of L, being a cotangent or a sum of cotangents of interior angles, are nonnegative (see (29) and (36)). Hence and (41) yields L l lv n +1 i,l [LV n +1 i ] l V n +1 i,l L l lv n +1 i,l ( P were we ave also used te fact tat te row sums of L vanis. Combining tese observations, we find tat te rigt and side of (40) is (strictly) positive. But te left and side of (40) is (strictly) negative: Indeed, by te coice of n and l, we ave V n +1 i,l < 0 and V n i,l 0. Since M l l > 0, tis implies tat l=1 L l l ) 1 k M ( l l V n +1 ) i,l V n i,l < 0. = 0 In view of (40), tis is a contradiction and completes te proof.

18 18 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR For te variant (33), in contrast to te fully implicit metod (23), we are not able to prove an unconditional non-negativity result tat olds for any time step size k > 0. However, it is possible to obtain a result under a constraint on k. At te nt step, let k n = min i=1,...,n min l=1,...,p ( V n i,l min i,l (g i (V n 1,l,..., V n N,l ) + α iu n l, 0) Note tat tis number can be computationally evaluated at any given time step. If te denominator above is zero, ten k n is defined to be +. Te non-negativity of te next time iterate can be ensured if te next time step k is cosen so tat k k n, as we sow below. If k n = +, ten any k would satisfy te constraint k k n. Tis is te case if g i (V1,l n,..., V N,l n ) 0 and te conditions of Teorem 4.1 old (so U l n 0). Note also tat te constraint k k n is weaker tan te usual conditions for explicit metods (suc as k O( 2 )) for explicit metods, wic can be muc more stringent depending on te spatial mes size. Proposition 4.3. Suppose (34) olds. Given u n 0 and vn,i 0 (for all i = 1, 2,..., N), suppose we calculate v n+1,i using (33b) wit any 0 < k k n. Ten for all i = 1, 2,..., N. Proof. Clearly, (33b) implies tat (42) (M + k D i L) V n+1 i v n+1,i 0 on Ω, = M V n i + kα i M U n + km G i ( v n ) An argument similar to tat in te proof of Teorem 4.1 sows tat M + k D i L as a non-negative inverse. Hence, it suffices to prove tat te rigt and side of (42) is non-negative. By te definition of k n, we ave Vi,l n k n min(g i (V1,l n,..., V N,l n ) + α iul n, 0) for any i and l, so V n i,l + kα i U n l + kg i (V n 1,l,..., V n N,l) ( k n + k) min(g i (V n 1,l,..., V n N,l) + α i U n l, 0) 0. Tus te rigt and side of (42) is non-negative. 5. Application to te minimal Keller-Segel model In tis section, we apply te previously developed numerical metod to te minimal Keller-Segel system (of Example 1.1). Tis will lead us to exibit a nonomogeneous stationary state of te system, wic to te best of our knowledge, as not been seen before. Tis system of two coupled partial differential equations was considered in te original work of Keller and Segel [21] to model te spontaneous aggregation of slime mold Dictyostelium discoideum. Tey obtained tis system from a larger system by certain simplifying assumptions and tey stress tat tese simplifications, wile reasonable, are made principally to avoid obscuring essential features wit eavy calculations. Ever since, te qualitative beavior of tis system and its solutions ave been extensively studied [19, 35]. Considering te wealt of known results, tis model forms a good test bed ).

19 POSITIVE EXACT AND NUMERICAL SOLUTIONS 19 for numerical metods for cemotaxis. To simplify our discussion, we set all parameters to unity and consider te following minimal system: (43a) (43b) u t = ( u u v), x Ω, t > 0 v t = v v + u, x Ω, t > 0 (43c) u n = v = 0, x Ω, t > 0 n (43d) u(0, x) = u 0 (x), v(0, x) = v 0 (x), x Ω. Even wit tese simplifications, te system is sensitive to te size of te domain Ω and te initial conditions, as we sall now see. Let us fix Ω to be a disk of radius R centered at te origin, and review tree known results on ow te beavior of solutions of (43) depend on R and u 0. First, it is known tat if (44) u 0 (x)dx < 4π, Ω ten te solution to (43) exists and is uniformly bounded globally in time [19, 26]. Let m(w) denote te mean value of a function w on Ω. Inequality (44), in terms of te mean value of te initial iterate m(u 0 ), is te same as (45) m(u 0 ) < 4/R 2. Second, if (46) 4π meas(ω) < m(u 0) < 8π meas(ω), ten tere exist initial data {u 0, v 0 } for wic te solution of (43) blows up at te boundary of Ω in finite or infinite time (see [19, Table 5], [18] or [33]). Tis reinforces te need to satisfy (45) if we want to find te smallest value of m(u 0 ) tat yields (bounded) pattern formation. Tird, an analysis of [35] (or its extension in [10]) sows tat any constant solution u of (43) is unstable if u > 1 + µ 1 (Ω) were µ 1 (Ω) is te smallest nonzero eigenvalue of te Laplacian wit Neumann boundary conditions on Ω. Tese results suggest tat to avoid convergence to te biologically uninteresting constant stationary states, we sould coose te initial iterate to ave mean value m(u 0 ) > 1 + µ 0. Since te mean value of u remains uncanged (equals m(u 0 )) at all times, te instability of te constant solution wit te same mean value suggests tat u will not approac tis constant solution as t. Guided by tese tree results, we are led to perform numerical computations wit initial data u 0 satisfying 1+µ 1 (Ω) < m(u 0 ) < 4/R 2. Te eigenvalue µ 1 (Ω), in our case of te disk Ω, can be found by a simple calculation involving a Bessel root: Namely, µ 1 (Ω) = ˆµ 1 /R 2, were ˆµ is te eigenvalue µ 1 ( ˆD) on te unit disk ˆD. Tus, we coose u 0 satisfying (47) 1 + ˆµ 1 R 2 < m(u 0) < 4 R 2. Note tat te unit disk does not satisfy (47). We set R = 0.5 so as to satisfy (47).

20 20 DE LEENHEER, GOPALAKRISHNAN, AND ZUHR v n vn 2 0 u n un 1 1 v n vn 1 1 m(u n ) m(u 0) u n U 0 dim(s ) , , , , ,313 Table 1. Indicators of convergence Next, we come to te question of selection of initial condition for te time iteration. Since our goal in tis section is to compute a stationary state, we are now motivated by an analysis of [18, 19], tat establises te existence of nonomogeneous stationary states using a Mountain Pass argument. Te argument proves existence by locating te critical point of a functional using a Palais-Smale sequence of functions (48) z ε = w ε m(w ε ), as ε goes to 0, were ( ) ε 2 (49) w ε = log (ε 2 + π(x x 0 ) 2 ) 2 and x 0 is a point in Ω. Motivated by tis construction, we set our initial iterates by (50) u 0 = v 0 = z ε + c wit x 0 = ( R, 0), ε = 10 10, and set te constant c so tat m(u 0 ) = 15, tus satisfying (47). Wit te above described settings, we use te implicit metod (23). We use an almost uniform spatial mes consisting of triangles wit non-obtuse interior angles. (One of te meses used is visible in Figure 3.) All triangles of te mes ave an approximate diameter obtained by dividing te perimeter of Ω by number of mes vertices on Ω. We ten set te time step size by k = and compute approximations u n and vn at time kn, for large n, by solving (23). Note tat (23b) does not require us to solve a nonlinear system because g is linear for te minimal Keller-Segel model. We observed tat te u and v iterates at every time step were non-negative, as predicted by te teory in Section 4. We performed enoug time iterations to reac t = 20, on a sequence of spatial meses wit decreasing. Te values of for te meses we used, and te corresponding size of te discrete linear systems to be inverted at eac time step, are displayed in te first and te last columns of Table 1. To describe te results in te remainder of te table, first recall te definitions of te standard L 2 (Ω) and H 1 (Ω) norms, ( ( w 0 = ) 1/2 w 2 and w 1 = Ω ) 1/2 w 2 + w 2. Ω We report, in te second and tird column of Table 1, te H 1 (Ω) norms of te differences between te last two successive iterates, namely u n un 1 1 and v n vn 1 1, were n denotes te final time iteration number. Since tese differences are small, it appears tat we are close to a stationary state. We also examine ow tese approximately computed stationary states differ as te mes size decreases. Note tat is alved as we go down

21 POSITIVE EXACT AND NUMERICAL SOLUTIONS 21 Figure 2. Plot of final u and v and istory of teir norms a row in Table 1, wic corresponds to a uniform mes refinement wit te boundary points adjusted to lie on Ω. Te first column of te table suggests tat te difference, in L 2 (Ω), between te stationary v computed on two successive mes refinements decreases like O( 2 ), for small enoug values of. Next, we describe two furter tests we performed to ceck te simulation. First, it is easy to see, by integration by parts applied to (43a), tat t m(u) = 0. Tus, te mean value of u(, t) at any time t must equal te initial mean m(u 0 ), i.e., (51) m(u) m(u 0 ) = 0. We verify tat tis property is preserved to good accuracy by our discrete approximation of u in te fourt column of Table 1. To describe te second test, let us begin by noting tat an application of te maximum principle to te steady state version of (43a) sows tat te stationary u and v satisfy u = Ke v for some constant K. Moreover, integrating we find tat K = m(u 0 )/m(e v ). Tus, if u and v are te exact stationary states, ten defining (52) U = m(u 0) m(e v ) ev, we ave (53) u = U.