Key words. Finite element method; convection-diffusion-reaction; nonnegativity; boundedness

Transcription

1 PRESERVING NONNEGATIVITY OF AN AFFINE FINITE ELEMENT APPROXIMATION FOR A CONVECTION-DIFFUSION-REACTION PROBLEM JAVIER RUIZ-RAMÍREZ Abstract. An affine finite element sceme approximation of a time dependent linear convectiondiffusion-reaction problem in D and 3D is presented. Specific conditions are given in terms of te coefficient functions, te computational grid and te discretization parameters to ensure tat te nonnegativity property of te true solution is also satisfied by its approximation. Numerical examples are given wic confirm te necessity and sufficiency of te discretization conditions to ensure te nonnegativity of te approximation. Key words. Finite element metod; convection-diffusion-reaction; nonnegativity; boundedness AMS subject classifications. 65L6. Introduction. In tis paper we consider te finite element approximation of te linear convection-diffusion-reaction problem: Determine u(x, t) satisfying u(x, t) t a(x, t) u(x, t) + b(x, t) u(x, t) + g(x, t)u(x, t) = f(x, t), (x, t) (, T ], (.) u(x, t) =, (x, t) (, T ], (.) u(x, ) = u (x), x. (.3) Suc equations arise in modeling many pysical penomena. Often te unknown quantity u in (.)-(.3) represents a nonnegative pysical quantity. In suc cases, it is igly desirable tat te numerical approximation to u also be nonnegative. Herein we derive sufficient conditions on te discretization parameters wic guarantee te approximation of a nonnegative solution of (.) is also nonnegative. Our interest in tis problem is motivated by te finite element approximation of te generalized (nonlinear) Burgers-Huxley equation [4], a model tat arises in several fields. For example, in electrodynamics, te Burgers-Huxley equation describes te motion of te domain wall of a ferroelectric material in an electric field [9], in biology it is used to model te nerve pulse propagation in nerve fibers [7], and it is also used as a prototype model in te study of interactions between diffusion transport, convection and reaction []. For te system of equations (.)-(.3) we assume R d, d =, 3, is a bounded domain, < a min a(x, t) a max, b(x, t) L ((, T ], H ()), g(x, t),f(x, t) L ((, T ], L ()) and u (x). It is straigt forward to sow tat under te additional assumption tat εg + f > in [, T ] for ε (, ε ), te solution u(x, t) is nonnegative [5]. Hence, it is important for pysical relevance, tat under suc conditions te numerical approximation also sould be nonnegative on. Over te years tere as been considerable work done on numerical approximation scemes for elliptic and parabolic differential equations tat inerit a maximum principle satisfied by te continuous equation being approximated. Here we refer to Department of Matematical Sciences, Clemson University, Clemson, SC 9634, USA. (javier@clemson.edu)

2 an equation as satisfying a maximum principle if te maximum of te solution (approximation) can be bounded by a constant multiple of te maximum of te initial data, te boundary data, or te rigt and side function. In suc cases te nonnegativity of te solution (approximation) typically follows from te nonnegativity of te data. Te equivalence of aving te (discrete) nonnegative property and a (discrete) maximum principle as been studied in [7]. Under tese assumptions (.)-(.3) does not satisfy a maximum principle. Following we give a brief summary of recent work on discrete maximum principles for te approximation of elliptic and parabolic differential equations. A detailed description of te development in tis area is given in [3, 9]. Farago, Horvat, Korotov and collaborators ave made significant contributions to te development of finite element approximation scemes for parabolic problems tat inerit a maximum principle from te continuous problem. In [8] tey establised sufficient conditions for te θ-metod time discretization of te linear finite element approximation to te time dependent diffusion-reaction problem to satisfy a discrete maximum principle. Discrete maximum principles for a nonlinear parabolic equation and a nonlinear system of equations were analyzed in [9] and [], respectively. Scatz, Tomée, and Walbin in [5] used a semi-group approac to investigate te positivity and maximum-norm contractivity in time stepping metods for parabolic equations. A discontinuous Galerkin metod, satisfying a strict maximum principle, was introduced by Zang, Zang, and Su in [8] for a nonlinear convection-diffusion equation. For te case of an anisotropic diffusion-convection-reaction problem, Lu, Huang and Qiu in [3] derived a sufficient condition suc tat te linear finite element approximation satisfied a discrete maximum principle. In [5], for a coupled system of nonlinear parabolic equations in R, De Leeneer, Gopalakrisnan and Zur proposed and analyzed a linear finite element approximation using a backward Euler time discretization. Tey establised sufficient conditions suc tat te numerical approximation to teir coupled system inerited te nonnegativity property of te continuous solution. In te next section we introduce our numerical approximation sceme and derive te sufficient conditions tat guarantee te nonnegativity of te approximations. In section 3 we develop te error analysis wic sows tat our metod is first order bot, in time and space. Te fourt section illustrates te validity of our results troug some numerical simulations. In particular, we include an example were one of our conditions is not met and te corresponding numerical approximation exibits negative values. Te concluding remarks are presented in section 5.. Nonnegativity of te approximation. In tis section we derive sufficiency conditions to guarantee tat te approximation of a time dependent convectiondiffusion-reaction equation remains nonnegative... Notation. Let T denote a regular triangulation of wit mes parameter. For Sobolev spaces we use te usual L p () and H p () notation []. Let P (T ) denote te set of affine functions on T. To describe te weak formulation of te problem we use X = H () = { f H () : f = }, and X = { f C () : f T P (T ), T T, f = }.

3 For f, g scalar functions, and f, g vector functions, (f, g) := fg d, and (f, g) := f g d. Te L () and L () norms will be denoted by and, respectively. Te vector norms corresponding to l and l will be denoted likewise. We will also use te following norm for q L ((, T ], L ()) q, = sup q(t). t (,T ] For te temporal discretization, we partition te interval [, T ] into M equally spaced subintervals of widt, wit t k = k, k =,..., M, and q k := q(x, t k )... Numerical approximation. Multiplying (.) troug by v X and rearranging we obtain te weak formulation of (.)-(.) as: Find u(x, t) X, t (, T ] suc tat for all v X subject to ( u, v) + (a(x, t) u, v) + (b(x, t) u, v) + (g(x, t)u, v) = (f, v), (.) t u(x, ) = u (x), x. (.) Using te backward Euler temporal discretization we obtain a fully discrete approximation to (.)-(.3) as: Determine u k (x) X, k =,,..., M, suc tat for all v X (uk, v) + (a k u k, v) + (b k u k, v) + (g k u, v) = (uk, v) + (f k, v), (.3) subject to u (x i ) = u (x i ), i =,..., N, (.4) were x i, i =,..., N denotes te vertices in te triangulation T. Let φ j (x) X be defined by {, x = xj φ j (x) =., x x j Ten {φ j (x)} N j= forms a Lagrangian basis for X. Wit u k (x) = N j= c jφ j (x), and v = φ i, i =,..., N, equation (.3) becomes a N N linear system of equations Ac = r (.5) were A ij = φ j φ i d + a k φ j φ i d + b k φ j φ i d + g k φ j φ i d, and (.6) r i = u k φ i d + f k φ i d. (.7) 3

4 For u k (x) and f(x, t k ), for all x, a sufficient condition to guarantee te nonnegativity of c, and ence of u k (x), is tat te matrix A represents an M-matrix, i.e. A ij, j i (nonpositive off diagonal entries), (.8) N A ij > (diagonally dominant). (.9) j= Properties (.8)-(.9) guarantee tat A exist wit A ij, i, j =,..., N, i.e., all te entries in A are nonnegative []. In order to establis properties (.8)-(.9) we must consider te computation of eac of te entries A ij of A..3. Computation of A ij..3.. D setting. In Finite Element Metod (FEM) computations, integration over is performed by first writing as te sum of te triangles T T, and ten performing te integration over eac T. Te integration over T is done by introducing a cange of variables ξ = (ξ, η) wic transforms te integral over T to an integral over te reference triangle : [ x y ] = J(ξ) = [ (x x ) (x 3 x ) (y y ) (y 3 y ) ] [ ξ η ] [ x + y ] = Jξ + x. (.) For notational simplicity, we let J = det(j) = (x x )(y 3 y ) (x 3 x )(y y ), and J t denote te inverse of te transpose of J. We consider a triangulation T of suc tat for every triangle T not adjacent to, tere are positive constants c, C, c θ, c J and C J under wic te following conditions are satisfied: A. For e i te edge opposite to te i-t vertex, c l(e i ) C, were l denotes te lengt. A. Te angle ϕ subtended at any vertex satisfies < c θ cos ϕ <. A3. c J J C J. We note tat A can be relaxed by requiring only tat te sum of te angles oposite to te common edge sared by two triangles be less tan π [6]. On we ave φ (ξ, η) = ξ η, φ (ξ, η) = ξ, φ3 (ξ, η) = η. (.) Note also for (x, y) T (equivalent to (ξ, η) ): (x,y) φ j (x, y) = J t (ξ,η) φj (ξ, η), j =,, 3 (.) φ j (ξ, η) = φ j (x, y) =, and φ j = J t (ξ,η) φj =. (.3) j= j= To establis tat A in (.6) is an M-matrix, we carefully consider its construction. In view of (.) and Figure., wit i = n, te n -t row of te coefficient matrix is obtained by mapping eac triangle T, wic as n as a vertex, to te reference triangle, and tere evaluating te integrals. Let A T n,n j denote te same computations as in (.6) wit te region of integration being T instead of. In tis setting, te diagonal contribution to te n -t row is A T n,n and te contributions to te 4 j= j=

5 φ = ξ η, φ = [ ] t φ = ξ, φ = [ ] t φ 3 = η, φ 3 = [ ] t (x 3,y 3 ) n 3 F y T η (,) n (x,y ) n (x,y ) ˆT (,) ξ x Fig... Transformation of te reference triangle to te triangle T. off-diagonal terms are A T n,n and A T n,n 3. Tus, to establis tat A is an M-matrix, it is sufficient to sow tat (see (.8) and (.9)) A T n,j, j = n, n 3 and (.4) A T n,n + A T n,n + A T n,n 3 >. (.5) In order to facilitate te analysis, we introduce te following quantities wic arise in A T n,n j a,j = φ j φ J dξ, (.6) a,j = â k (ξ) J t (ξ,η) φj J t (ξ,η) φ J dξ, (.7) 3 a,j = b(ξ) k J t (ξ,η) φj φ J dξ, (.8) 4 a,j = ĝ k (ξ) φ j φ J dξ, (.9) were f(ξ) = f(j (ξ)). Lemma.. For all T T a,j a min C J c c θ for j =, 3 (.) 5

6 Proof. Let a T = We first consider a,. Using (.) and Figure. we ave Consequently, â k (ξ) dξ a min. (.) J t (ξ,η) φ = J [(y 3 y ) (x x 3 )] t, J t (ξ,η) φ = J [(y y 3 ) (x 3 x )] t. a, = at ((x x 3 ) (x x 3 ) + (y y 3 ) (y y 3 )) J In a similar fasion we obtain (.) a,3 = at ((x x ) (x 3 x ) + (y y ) (y 3 y )). (.3) J In order to see tat a, and a,3 are negative and strictly bounded away from zero, we consider te polynomials in its numerator. Let P i denote te point (x i, y i ) and P i P j te vector from P i to P j. For a, we ave [ ] [ ] x x (x x 3 ) (x x 3 ) + (y y 3 ) (y y 3 ) = 3 x x 3 y y 3 y y 3 = P 3 P P 3 P = P 3 P P 3 P cos θ c c θ >, (.4) were in te last inequality we ave used A and A. Applying A3, te result now follows from (.), (.), (.3) and (.4). Lemma.. For T T, and j =, 3, Proof. As, for j =, 3, a,j C J 4, (.5) 3 a,j b k 6 C, (.6) 4 a,j C J g k. (.7) 4 φ φj dξ = 4, using A3 we establis (.5). From (.8), using A and 3 a b k (ξ) J t (ξ,η) φ φ J dξ = b k P 3 P φ dξ b k 6 C. 6 φ dξ = /6, [ b k y3 y (ξ) x x 3 ] φ dξ

7 Te bound for 3 a T 3 is obtained analogously. In a similar manner as was used to establis (.5), we obtain (.7). Teorem.3. For < C J 4 ( ) a min c C c θ b k J 6 C g k 4 C J te coefficient matrix A is an M-matrix. Proof. First note tat (.4) is equivalent to sowing tat 4 r a T,j <, j =, 3. r= Using Lemmas. and. we obtain 4 r= r a T,j a min c C c θ + C J J 4 Tus, for sufficiently small and C J g k b k C + g k 4 C J, j =, 3. (.8) ( ) a min c C c θ b k J 6 C g k 4 C J, we can guarantee tat te off-diagonal entries of A are nonpositive. 3 To sow (.5), observe tat from φ j = (see (.3)), it follows tat 3 j= T a k φ j φ i dx =, and 3 j= A T n,n j = j= 3 j= T b k φ j φ i dx =. Hence, for < g k, φ J dξ dη + ĝ k (ξ) φ J dξ dη >. (.9) We close tis subsection wit te following corollary. Corollary.4. For and satisfying Teorem.3, and for f k, u k, we ave tat u k. Proof. From (.5), noting tat r is nonnegative and A an M-matrix, te stated result follows D setting. Following te same structure as in te D case, we consider a triangulation T of suc tat for every tetraedron T not adjacent to, tere are positive constants c, C, c θ, c J and C J satisfying: A4. For e i te edges of te tetraedron, c l(e i ) C. A5. Te angle subtended between any two distinct faces F i and F j, denoted F i F j, satisfies < c θ cos( F i F j ) <. A6. c J 3 J C J 3. 7

8 φ = ξ η ζ, φ = [ ] t φ = ξ, φ = [ ] t φ 3 = η, φ 3 = [ ] t φ 4 = ζ, φ 4 = [ ] t P 3 (x 3,y 3,z 3 ) ζ (x 4,y 4,z 4 ) P 4 P (x,y,z ) F T ˆT η P (x,y,z ) ξ Fig... Transformation of te reference tetraedron to te tetraedron T. Te corresponding cange of variables ξ = (ξ, η, ζ) to compute te integrals on te reference tetraedron is: x y = (x x ) (x 3 x ) (x 4 x ) (y y ) (y 3 y ) (y 4 y ) ξ η + x y = Jξ + x (.3) z (z z ) (z 3 z ) (z 4 z ) ζ z 3 wit sape functions ˆφ = ξ η ζ ˆφ = ξ ˆφ3 = η ˆφ4 = ζ (.3) Keeping te same notation, we state parallel results to tose from te D setting. Lemma.5. For T T, a,j a min 3C J c 4 c θ for j =, 3, 4. (.3) Proof. Let a T = â(ξ) dξ a min 6. (.33) Note tat, from (.7) a T,j = â k (ξ)j t (ξ,η) φj J t (ξ,η) φ J dξ, 8

9 and note tat te matrix vector products J t (ξ,η) φj can be written as cross products between te vectors representing te edges of te tetraedron P k P l. For example, J t (ξ,η) φ = y z 3 y 3 z y z 4 + y 4 z + y 3 z 4 y 4 z 3 x 3 z x z 3 + x z 4 x 4 z x 3 z 4 + x 4 z 3 x y 3 x 3 y x y 4 + x 4 y + x 3 y 4 x 4 y 3 J t (ξ,η) φ = = P P 3 P P 4 y 3 z y z 3 + y z 4 y 4 z y 3 z 4 + y 4 z 3 x z 3 x 3 z x z 4 + x 4 z + x 3 z 4 x 4 z 3 x 3 y x y 3 + x y 4 x 4 y x 3 y 4 + x 4 y 3 = P 4 P 3 P 4 P. Moreover, we relate tose products to te area of te faces, F i, and te angle between tem, F i F j, using te geometrical interpretations of te cross and dot product, respectively. Tus, applying A4 -A6 we get a = a T J P P 3 P P 4 P 4 P 3 P 4 P = 4aT Area(F )Area(F ) cos ( F F ) J a min 3C J c 4 c θ. Similarly, we obtain te same bound for a 3 and a 4. Lemma.6. For T T and j =, 3, 4, a,j C J 3, 3 a,j b k 6 4 C, 4 a,j C J 3 g k. Proof. Te proof follows in an analogous manner as tat for Lemma.. Teorem.7. For sufficiently small and C J ( ) a min 6 c 4 6C c θ b k J 4 C C J g k < < g k te coefficient matrix A in (.6) is an M-matrix. Proof. Te proof is analogous to tat of Teoem.3. Corollary.8. For and satisfying Teorem.7, and for f k, u k, we ave tat u k. Proof. Te proof is analogous to tat of Corollary Boundedness of te approximation. In tis section we sow tat te numerical approximation u k obtained using (.5) remains bounded as te discretization parameters, and go to zero. 9

10 Teorem.9. For and satisfying te conditions of Teorem.3 (or Teorem.7), te discrete approximations u k remain bounded as,. Proof. Let c g = g, and let c be a vector wit nonnegative entries suc tat c = c l for some l. Observe tat for A te coefficient matrix in (.5) N N N (Ac) l = A lj c j = A ll c l + A lj c j (A ll + A lj )c l = c l j= j= j l Now recall from te discussion in Teorem.3 tat te contribution of eac triangle T in te support of φ l to N j= A lj is, (see (.9)) φ J dξ dη + j= j l N j= A lj. ĝ k (ξ) φ J dξ dη = φ l dt + g k φ l dt T T φ l ( c g) dt. Summing over all triangles T T, we obtain te following bound c l φ l ( c g) d c l N j= T A lj (Ac) l. (.34) Noting tat te l-t row of te system of equations (.7) resulting from our numerical sceme at time t k is (Ac k ) l = u k φ l dx + f k φ l dx, (.35) and using (.34) wit c = u k = c l, we obtain ( ) c g u k φ i dx u k φ l dx + Simplifying and solving for u k yields ( ) c g u k uk + f k, u k = c u k + g c g u k + c g f k f k φ l dx. c g f k. (.36) Wit k replaced by k in (.36) we obtain an expression for u k. Tus, ( ) ( u k u k + c g ( ) + f k. c g c g ) f k

11 Continuing, leads to te following expression ( ) k u k u c + g k ( i= c g ) k+ i f i. (.37) Now as M = T, and using te fact tat for any two positive numbers p and w, wit p < we ave tat lim M Also, observe tat Tus, ( c g ( )w p ( ) pw exp, p ) M ( ) lim exp cg M = exp (c g T ). (.38) M c g ( ) M k ( ) ( ) cg (M k) cg (T k) exp = exp. c g c g c g lim M k= M ( ) M+ k T f k e (T s)c g f(s) ds. (.39) c g Owing to (.38) and (.39), we obtain te bound T u k exp(c g T ) u + e (T s)cg f(s) ds. (.4) 3. Error analysis. In tis section we establis tat te numerical approximation is of first order, bot in space and time. Te following discrete norm ( M / v s = v k H ()) s, (3.) k= and te discrete version of Gronwall s lemma [] will be used in te analysis. Lemma 3.. Let, H, and a n, b n, c n, γ n (for integers n ), be nonnegative numbers suc tat a l + b n n= γ n a n + n= c n + H for l n= Suppose tat γ n <, for all n, and set σ n = ( γ n ). Ten a l + ( b n exp n= ) σ n γ n ( n= ) c n + H n= for l.

12 We will also use te following inequality, wic for completeness, we supply its simple proof. Lemma 3.. For u, u t, u tt in L ((, T ], L ()) Proof. From we obtain Now observe tat ( tk u k t uk u k 3 tk tk u k = u k u k t + u tt (t t k ) dt, t k u k t uk u k t k u tt (t t k ) dt = ) tk t k u tt dt. (3.) t k u tt (t t k ) dt. tk t k u tt dt tk t k (t t k ) dt Te inequality (3.) now follows. = ()3 3 tk t k u tt. Teorem 3.3. Let u(x, t) L ((, T ], H ()), u t (x, t) L ((, T ], H ()), u tt (x, t) L ((, T ], L ()), satisfy (.)-(.3) on te interval (, T ]. Ten, te finite element approximation u k converges to u as,. In addition, tere exists a constant C = C(), suc tat te approximation u k satisfies te following error estimate: ( ( ) ) T 5 + c g + b, u k u k exp ( ) 5 + c g + b, { ( ) a C max + 9 b, u a min + () 3 T ( u tt dt + C 4 T u t H () dt + c g u ) } + C 4 u k H (), k =,,..., M. (3.3) Proof. At t = t k, te weak formulation (.) becomes ( uk t, v) + (ak u k, v) + (b k u k, v) + (u k g k, v) = (f k, v), (3.4) and its fully discrete approximation is given by ( uk uk, v) + (a k u k, v) + (b k u k, v) + (u k g k, v) = (f k, v). (3.5)

13 Subtracting (3.5) from (3.4) we obtain ( uk t uk uk, v) + (a k (u k u k ), v) + (b k (u k u k ), v) (3.6) + ((u k u k )g k, v) =. Defining e k = u k u k and substituting in (3.6), we get ( uk t uk u k + ek e k, v) + (a k e k, v) + (b k e k, v) + (e k g k, v) =. (3.7) For U k X, let e k = Λ k + E k were Λ k = u k U k and E k = U k u k. Ten (3.7) becomes ( uk t uk u k, v) + ( Λk Λ k + E k E k, v) + ( (Λ k + E k ), a k v) +(b k (Λ k + E k ), v) + ((Λ k + E k )g k, v) =. Coosing v = E k, and multiplying by, from (3.8) we obtain (3.8) ( uk t uk u k, E k ) + (Λ k Λ k + E k E k, E k ) +( (Λ k + E k ), a k E k ) + (b k (Λ k + E k ), E k ) + ((Λ k + E k )g k, E k ) =, or, equivalently (E k E k, E k ) + ( E k, a k E k ) = ( uk t uk u k, E k ) (Λ k Λ k, E k ) (a k Λ k, E k ) (b k Λ k, E k ) (b k E k, E k ) (Λ k g k, E k ) (E k g k, E k ). Using Young s inequality and Caucy-Scwarz yields ( Ek E k ) + a min E k ( uk t uk u k + E k ) + 4 Λk Λ k + E k + ( a max Λ k + a min a min Ek ) + ( bk Λ k + E k ) + bk E k + (c g Λ k + E k ) + c g E k. (3.9) Summing (3.9) from k = up to k = l, regrouping and using (3.), we get ( E k E k ) + a min k= + ( 5 + c g + b k ) l E k + k= 3 k= k= E k () 3 k= tk t k u tt dt ( Λ k Λk + a max Λ k a min + b k Λ k + c g Λ k ). (3.)

14 Note tat (Λ k Λ k ) = ( tk t k Λ t dt ) tk (Λ t ) dt, t k implies Simplifying (3.) we obtain Λk Λ k tk Λ t dt. t k E l + a min + + () 3 E k ( 5 + c g + b k ) l E k n= ( a max Λ k + b k Λ k + c a g Λ k min k= T u tt. In view of Lemma 3., (3.) implies E l + a min K { + k= T E k k= k= ) + T ( a max a min Λ k + b k Λ k + c g Λ k Λ t dt + () 3 T u tt dt }, Λ t ) (3.) (3.) were ( ) T (5 + cg + b, ) K = exp (5 + c g + b, ) and te product l tat appears in te exponential as been replaced by T. From te teory of finite element interpolation [3, 4], we ave tat for I te interpolant of te exact solution u in te space of piecewise linear continuous polynomials, and Λ k = u k I u k, tere exist a constant C suc tat Observing tat in R d for d =, 3, and using (3.3) in (3.) we obtain Λ k + Λ k C u k H (). (3.3) b k Λ k 3 b, Λ k, a max Λ k + b k Λ k + c a g Λ k min ( ) a C max + 3 b, u k H a () + C c g 4 u k H (), min (3.4) 4

15 and T Λ t dt C 4 T u t H () dt. (3.5) Using (3.), (3.4) and (3.5) to simplify (3.) yields { ( ) a E l + a min E K C max + 3 b, u a min ( T ) } + () u tt + C 4 T u t H 3 () + c g u. Finally, noting tat we obtain (3.3). 4. Numerical simulations. u l u l E l + Λ l, Example. As a first example, we let = (, ) (, ) and consider te linear partial differential equation (PDE) [ ] u sin(x) t u + u u = f(x, t), (x, t) (, ]. (4.) cos(y) For, u(x, y, t) = ( + sin( π xy))et, (4.) we approximate (4.) wit te sceme (.3)-(.4), using =. and =.5. Note tat te true solution is nonnegative and tat and satisfy te conditions of Corollary.4. Computations were performed using two meses, illustrated in te bottom frame of Figure 4.. Note tat te mes on te rigt and side violates condition A. Te top frame of Figure 4. sows te computed approximation (on te respective meses) at t =. Presented in te middle frame of Figure 4. is te time evolution of te minimum value of te approximation over as a function of time. Observe tat te nonnegativity of te approximation is lost using te mes wic violates A. Example. For a second example, we consider te nonlinear PDE tat motivated tis work, te generalized Burgers-Huxley equation, for = ( 4, 8) ( 4, 8) were u t u + αup (u x + u y ) ug(u) =, (x, t) (, ], (4.3) g(u) = ( u p )(u p γ) for γ (, ). We use for te boundary data a traveling wave solution [6] of (4.3) given by u(x, t) = ( γ + γ tan (a (z a t))) /p, 5

16 5 5 u(x, ) 5 u(x, ) x.5 y.5.5 x.5 y.5 x minu(x,t) minu(x,t) t t y y x.5.5 x Fig. 4.. Comparison between te numerical approximation to u(x, t) at time t =, and te evolution troug time of te minimum of te approximation using a mes satisfying (.3.) (left column) v.s. a not conforming mes (rigt column). Note tat te mes on te rigt violates te angle condition A. were α = α, z = (x + y)/, a = αp + p α + 4( + p) γ, 4( + p) a = ( + p γ) α + 4( + p) + α( + p + γ). ( + p) 6

17 We approximate (4.3) using a simple sceme in wic all nonlinearities ave been lagged, wit pysical parameters α =, γ =.5, p = and computational parameters =.9375, =., wic satisfy te conditions of Corollary.4. Tis corresponds [ to] te approximation sceme (.3) wit a k = a(x, t k ) =, b k = b(x, t k ) = u k, g k = g(x, t k ) = ( u k )(u k.5). Figure 4. sows te numerical approximation at time t = and te evolution of its minimum troug time. We observe tat te nonnegativity property olds as expected. x 6. u(x, ) y 5 5 x 5 minu(x,t) t Fig. 4.. Approximation to u(x, t) for t = (left). Evolution of te minimum of te approximation for t [, ]. 5. Conclusion. In tis work, we considered a finite element sceme for a linear convection-diffusion-reaction problem in D and 3D. It was establised tat for a carefully constructed mes, assuming some boundedness conditions on te functions involved in te partial differential equation, and imposing certain restrictions on te computational parameters and, we can guarantee tat te coefficient matrix arising from te discretization is an M-matrix. As a consequence of tis, we sowed tat under nonnegative boundary and initial conditions, te numerical approximations remain nonnegative for all time. Moreover, we proved tat using te stated restrictions for and, te discrete approximations remain bounded as,. Convergence of te numerical approximations was establised in te error analysis, and terein it is sown tat te metod is O( + ). Our numerical results corroborate our findings and illustrate by means of an example ow te violation of one of te assumptions on te mes can lead to negative numerical approximations for nonnegative problems. REFERENCES [] M. Ablowitz and B. Fucssteiner. Topics in soliton teory and exactly solvable nonlinear equations. World Scientific Publising Co., Singapore, 987. [] R.A. Adams and J.F. Fournier. Sobolev Spaces, volume 4 of Pure and Applied Matematics. Elsevier, 3. [3] D. Braess. Finite Elements. Cambridge University Press, tird edition, 7. Cambridge Books Online. [4] P.G. Ciarlet. Finite Element Metod for Elliptic Problems. Society for Industrial and Applied Matematics, Piladelpia, PA, USA,. [5] P. De Leeneer, J. Gopalakrisnan, and E. Zur. Nonnegativity of exact and numerical solutions of some cemotactic models. Comput. Mat. Appl., 66(3): , 3. 7

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