A Laguerre-Legendre Spectral-Element Method for the …...spectral methods [3]. As in the...

ELSEVIER

Available online at www.sciencedirect.com MATHEMATICAL AND

SC,ENCE ~ D , R B C T " COMPUTER MODELLING

Mathematical and Computer Modelling 41 (2005) 1171-1192 www.elsevier.com/locate/mcm

A Laguerre-Legendre Spectra l -Element M e t h o d for the Solut ion of Partial Differential

Equat ions on Infinite Domains: Appl icat ion to the Diffusion of Tumour Angiogenes i s Factors

J. VALENCIANO AND M. A. J . CHAPLAIN Division of Mathematics, University of Dundee

Dundee DD1 4HN, Scotland, U.K. chaplain©maths, dundee, ac. uk

A b s t r a c t - - I n this paper, the spectral-element method formulation is extended to deal with semi- infinite and infinite domains without any prior knowledge of the asymptotic behaviour of the solution. A general spectral-element method which combines finite elements with basis functions as Lagrangian interpolants of Legendre polynomials and infinite elements with basis functions as Lagrangian interpolants of Laguerre functions, whilst preserving the properties of spectral-element discretizations: diagonality of the mass matrix, conformity, sparsity, exponential convergence, generality, and flexibility is presented. The Laguerre-Legendre spectral-element method of lines is applied to an evolutionary reaction-diffusion equation describing the early stages of the diffusion of turnout angiogenesis factors into the surrounding host tissue. Q 2005 Elsevier Ltd. All rights reserved.

K e y w o r d s - - Spectral-element method, Semi-infinite, Infinite domains, Tumour angiogenesis.

1. I N T R O D U C T I O N

Many physical and biological processes involve the solution of partial differential equations over a semi-infinite or infinite domain. An example, in cancer biology, is the diffusion of turnout angiogenesis factors (TAF) in the human body. Biological evidence suggests that tumours grow up to a size of a few millimetres before secreting this chemical which might be found not only in the vicinity of the tumour but in distant parts of the human body as well [1]. The small size of the tumour compared to the human body can be modelled by considering the tumour immersed in an infinite domain. Partial differential equations describing the diffusion of TAF [1] might be nonlinear and solutions over complex geometries might only be attainable by numerical methods. However, there is a shortage of numerical methods for the solution of partial differential equations over semi-infinite and infinite domains. These have been based traditionally on spectral methods [2,3] which for complex geometries present severe limitations.

On the other hand, spectral elements [4] are domain decomposition methods for the solution of boundary value problems in complex domains which combine the generality and flexibility of

This work was supported by an EU Research Training Network HPRN-CT-2000-00105 "Using mathematical modelling and computer simulation to improve cancer therapy". The authors would like to thank professor Philip Maini for granting them access to some of the computer facilities at the Centre for Mathematical Biology at Oxford University.

0895-7177/05/$ - see front mat te r (~) 2005 Elsevier Ltd. All rights reserved. Typeset by ~4A/~S-TEX doi: 10.1016/j .mcm. 2005.05.010

1172 J. VALENCIANO AND M. A. J. CHAPLAIN

the finite-element method [5] and the high accuracy and rapid asymptotic rate of convergence of spectral methods [3]. As in the finite-element method, a weak statement is constructed and the domain is decomposed into finite elements. The approximation within each element is spectral. The method can be extended to treat evolutionary problems when combined with a time-stepping procedure giving rise to the spectral-element method of lines [4,6]. Due to their hybrid character, spectral-element discretizations enjoy the advantages of its both predecessors and they present advantages over both of them [4]. The main two advantages over the low-order finite-element method are first the higher accuracy provided by the spectral basis and second the diagonal mass matrix which results from the fact that in the spectral-element method the elemental Lagrangian bases are taken at the points that form the basis of a quadrature rule, usually Gauss-Lobatto. This quadrature rule is used to evaluate the integrals appearing after the weak formulation. The main three advantages of the spectral-element method over spectral methods are first in the greater flexibility when dealing with complex geometries. Second in the computational cost when coupled with a time-integrator for evolutionary problems. Due to the fact that, the domain is decomposed into elements the polynomial orders are not as high in the spectral-element method giving a more competitive computational cost which is of the order of the polynomial to the cube for two- dimensional reaction-diffusion-advection equations [4]. A third advantage is that while spectral methods provide exponential rates of convergence for problems where the solution is smooth, their performance might deteriorate in the presence of lack of regularity of the solution [3]. However, spectral elements (similarly to h - p finite elements) guided by an appropriate adaptive strategy might provide exponential rates of convergence even in the event of lack of regularity of the solution [7]. Furthermore, in the spectral-element method the domain decomposition approach provides sparsity in the matrices obtained after the discretization and it is this property together with the direct product formulation which contributes to reduce storage for large matrices and provides economical ways of performing matrix-vector products in a completely elemental fashion without the need of forming large matrices. For discretizations using quadrilateral elements in two or three dimensions only one-dimensional matrices need to be stored [4].

The spectral-element method for finite domains is normally based on Legendre polynomials and the Gauss-Lobatto.Legendre quadrature rule [4]. This is because Legendre polynomials are orthogonal in [-i, I] with unitary weight which appears naturally when a weak statement corresponding to a boundary value problem is constructed. The Gauss-Lobatto-Legendre quadrature rule is of closed type [8] which means that it includes both end points of the interval of integration [-i, I]. Since the Lagrangian interpolants of Legendre polynomials are taken at the Gauss-Lobatto-Legendre quadrature points, boundary conditions and interelemental continuity might be enforced. With this philosophy in mind, Laguerre polynomials (in conjunction with Legendre polynomials) would seem to be the natural choice for a spectral-element method for infinite domains since Laguerre polynomials are orthogonal in [0, c~). However, the weight function is not unity but e -~. The Gauss-Radau-Laguerre quadrature rule is of semi-open type [8] which means that it excludes the upper end point of the interval of integration at ¢~ although it includes the lower end point 0 allowing for enforcement of boundary conditions or interelemental continuity at this lower point. Moreover, this quadrature rule distributes the quadrature points in such a way that there is a higher concentration of points new the lower end point of the interval of integration and a progressively lower concentrations near the end point of the interval of integration. First attempts to incorporate Laguerre polynomials [9] together with Legendre polynomials as the basis for a one-dimensional spectral-element method for a semi-infinite domain were based on constructing two separate weak statements one with weight function unity for the Legendre elements and another one with the measure e -~ for the Laguerre elements. Although this strategy was successful for the simple one-dimensional problem treated in [9] it has the potential disadvantage of introducing problems when matching Laguerre and Legendre elements. Another deficiency in using Laguerre polynomials is that unless properly rescaled, the discrete Gauss-Radau-Laguerre quadrature weights tend to zero exponentially [9]. This intro-

A Laguerre-Legendre Spectral-Element Method 1173

duces a problem for codes based on this family of orthogonal polynomials. These disadvantages can be overcome by using Laguerre functions instead as shown in [10], since they are orthogonal in a standard nonweighted space, can be computed in a stable way and the discrete GRL weights when expressed in terms of Laguerre functions remain under control.

Another point is the combination with a time-integrator to treat evolutionary equations. Al- though, using explicit time-integrators is a viable alternative when the polynomial orders or representation amongst elements are moderate (less or equal to 8) [11,12] in the current work we will consider implicit time-stepping schemes only. This is because a stiff initial value problem is obtained due to the usage of different bases functions within different elements and the fact that, the polynomial orders of representation within Laguerre elements are rather high. Placing espe- cial emphasis in robustness we use the fully implicit Crank-Nicolson finite difference scheme [13] for the solution of one-dimensional problems and the semi-implicit splitting backward/forward Euler finite difference scheme [6] for the two-dimensional problem.

The outline of the paper is as follows: in Section 2, we present the basics of a Laguerre-Legendre spectral-element method for semi-infinite and infinite domains. In Section 3, we apply it to several elliptic test problems from [10] with known analytical solutions. We prove that exponential convergence towards these solutions is obtained by the proposed method. In Section 4, we describe one and two-dimensionai simple models for the diffusion of tumour angiogenesis factors into the surrounding tissue in a semi-infinite and infinite domain, respectively. In Section 5, these models are solved using the spectral-element method of lines and in Section 6 some conclusions are discussed.

2. L A G U E R R E F U N C T I O N S , L E G E N D R E P O L Y N O M I A L S , A N D

T H E S P E C T R A L - E L E M E N T M E T H O D

In what follows, we use standard Sobolev space notation and we refer to the weight w(~) = e -~ for the weighted spaces and R + = [0, co). First of all, we consider the Gauss-Radau-Laugerre (GRL) quadrature rule (see, for example [8], p. 224) for f(~) e C2~p+I(R +)

~o c~ P p!(p + 1)! f(2p+l)(i. ~ R+ (1) j=o

where ~j are the GIlL quadrature points which are the zeroes of the equation 5Z:~+:(~) = 0, and /:p(5) represents the Laguerre polynomial of order p. The discrete weights Wj are given by

1 Wj = (p + 1)Lp2(¢j ) . (2)

The GRL is a semi-open type quadrature rule since it includes the point ~ = 0 which is the lower extreme of the interval of integration R +. The rest of the GRL quadrature points can be obtained through the recurrence relation satisfied by the derivative of Laguerre polynomials (or associated Laguerre polynomials of order one) [8]

t:; (~) = O,

Z:i(~) = -1 , (3)

P

As mentioned in the introduction, Laguerre polynomials are orthogonal in L~ (R+). The presence of the weight e-e which does not appear naturally when a weak statement corresponding to a boundary value problem is constructed would seem to be a first drawback for the use of Laguerre polynomials as the basis of spectral-element method. Moreover, under the point of view of prac- tical numerical computations they also present two important drawbacks: Laguerre polynomials


satisfy a recurrence relation which becomes numerically unstable for moderately high polynomial orders, and consequently, the discrete weights (2) become exponentially small. A final negative point concerning Laguerre polynomials is that they do not seem to approximate well functions which decay to infinity algebraically [9,10,14].

These disadvantages seem to be overcome (see [10]) by defining Laguerre functions as

Lp(~) = e-¢/2Ev(~). (4)

They are orthogonal in the nonweighted space L2(R +)

o ° L~(~)Lj(~) d~ = 5~j, (5)

which means that they share the same weight w = 1 as Legendre polynomials. It is important to highlight that Laguerre functions can be generated in a stable way using their definition (4) and the recurrence relation of Laguerre polynomials

L0(~) = e -~/2,

LI(~) = e-Z/2(1 - ~), (6)

Lp(~) - 2p - 1 - x Lp-l(~) - p - 1 Lp-2(~), p > 2. p P

This represents a considerably much easier alternative to the cumbersome and unnecessary method of obtention proposed in [10].

It is therefore possible to rewrite the Gauss-Radau-Laguerre quadrature rule (1) for a function e +)

~o ~ P p!(p + 1)[ f(~) d~ = E f (~j)w j + {[f(~)e~](2P+l)}~=¢, ( e R +, (7) j=o (2p + 1)!

where ~j are again the zeroes of {£p = 0 and the discrete weights are rewritten in terms of Laguerre functions as

1 wj = Wie ~j = (p + 1)n2p([j ) . (8)

These discrete weights grow moderately when the polynomial order is increased. The quadrature rule (7) would allow for the evaluation of functions f in C2p+I([k, 0c))

oo P

fk f ( x ) dx ~ E f (~ + k)wi, (9) i=0

where the change of variables ~ -- x - k has been performed. And in C2P+I((-oc, k])

f (x ) dx = f ( - x ) dx ~ f ( - ~ + k)w~, (10) oo k i=0

where the change of variables is now ~ = x + k. Using (9) and (10) semi-infinite and infinite domains can be treated.

We can construct Lagrangian interpolants of Laguerre functions of order p at the Gauss-Radau- Laguerre quadrature points in R + (see [10])

e - - ( / 2 ~ ; q - 1 (11)

hi(~) = (p + 1)Lp+l(~i)(~ - ~i)'

which satisfy the property hi(~j) = 5ij, (12)

as can be proved using L'HSpital's rule and Laguerre's equation

(~E;+l) ' = ~gp+l - (P + 1)Ep+l. (13)

The Lagrangian interpolants of Laguerre functions are given in (11) in terms of Laguerre polynomials and Laguerre functions. However, they can also be rewritten exclusively in terms of Laguerre functions as shown in the following proposition.


PROPOSITION 1. Expression (11) is equivalent to

P

h~(~) : wi E Lr(~i)L~(~). (14) r~O

PROOF. Defining the linear space Pp(R +) = {y : y = ve -~/2 : v E PB(R+)} where Pv(R +) represents the lineax space of polynomials of degree p. The set of Laguerre functions {Li(~)}i=oP axe linearly independent and therefore form a basis in Pp(R+), so we can expand the function h~(~) e/~p(R +) given by equation (11) as

P

h~(~) = E a ~ L r ( ~ ) , (15) r = 0

where a~ are some unknown real numbers. Multiplying equation (15) by Lj(¢), integrating over R + and using the orthogonality of Laguerre functions (5), the Gauss-Radau-Laguerre quadrature rule (7) and property (12)

oo P

aj = f0 hi(¢)Lj(~) d~ = E hi(¢k)Lj(¢k)wk = Lj(~i)wi. (16) k = 0

Notice that, the quadrature error in equation (7) in the evaluation of the integral in (16) disap- pears due to the fact that h~ and Lj belong to Pp(R+). |

The derivative of the Lagrangian interpolants of Laguerre functions hg(~) given by (11) evaluated as the jth GRL point might be found by expressing equation (11) as

e-~/2 = ( 1 7 )

where h~(~) is the Lagrangian interpolant of Laguerre polynomials. Then, using L'HSpital's rule and Laguerre's equation (13) and the properties of Laguerre polynomials £v(0) = 1 and E~(0) = - p to give (see [10])

Lp+l (~j) if j ¢ i, Lp+ 1 (~i) (~j - ~i)'

Zo" = 0, if j = i ~ 0,

- (p + 1______)) if j = i = 0. 2 '

Analogously, the second derivative of hi(C) evaluated at Cj is obtained by first calculating

~i - (P + 1) 3~i ' if i = j # 0,

p(p - 1_____~) if i = j = 0, ,,, 6 '

£p+l(~i)(~j-~i) 2 ' i f i # j a n d j # 0 ,

-[2 +p(~j -~i)] i f i # j and j = 0, -

and then using the fact that

( 1 8 )

(19)

~tt(~.~ _ 15 e-~/2 ," ~ , , , - ~ ~ j - Z ~ j + ~ h ~ (~), (9.0)


to finally obtain

i \ 3 1 ~

42 - 4 p - 4 12~j ' i f i = j ~ = O ,

2p 2 + 4p + 3 12 ' if i = j = 0,

-2Lp+l(~j) if i ¢ j and j ¢ 0,

-[(~j - ~i)(P + 1) + 2] if i # j and j = 0. n ~ + ~ ( ¢ ~ ) ( ¢ j - ¢~)~ '

(21)

PROOF. expand the polynomial/~i(~) as

where 2 r + 1 if 0 < r < p - 1 ,

2 ' (28) "Y~ = P if r = p.

2'

The set of Legendre polynomials P {Li(~)}~= 0 form a basis in Pp(~). Therefore, we can

P

/2/~(~) = E fl~L~(~), (29) r = O

To complete our description of the basis functions used for the proposed Laguerre-Legendre spectral-element method we now turn our attention to the standard Lagrangian interpolants of Legendre polynomials bases functions. The Gauss-Lobatto-Legendre quadrature rule is a closed type quadrature rule which includes both ends of the interval of integration ~ = [-1, 1]. For

P

f f(~) d~ = ~ f(~)wj + E, (22) 5 = 0

where {~}j are the p + 1 zeroes of the equation (1 - ~2)(dLp/d~) = 0 where Lp(~) denotes the Legendre polynomials of order p in ~. The quadrature error E [8] is given by

E = , ~ e ft. (2~)

The quadrature weights are given by

2 i = 0 , . . . , p . (24) w , = p(p + 1 ) [ L ~ ( 5 ) ] : '

The spectral-element method is normally based on Lagrangian interpolants of Legendre polynomials at the Gauss-Lobatto-Legendre quadrature points

- 1 ( 1 - ~ 2 ) ( d L p / d ~ ) i ---- O, . , p . (25) H~(~) = p (p + 1 ) L ~ ( ~ ) ( ~ - ~ ) ' '

These Lagrangian interpolants evaluated at the jth Gauss-Lobatto-Legendre quadrature point satisfy the property

[Ii(~j) = 5ij, i , j = 0 , . . . ,p. (26)

The Lagrangian interpolants can also be written in terms of Legendre polynomials only, as shown in the following proposition.

PROPOSITION 2. Expression (25) is equivalent to

P

[-Ii(~) = w, ~ ~.L.~(~i)Lr(~), (27) r ~ 0


where f~r are some unknown real numbers• Multiplying both sides of equation (29) by the Legendre polynomial Lj(~) for j --- 0, . . . ,p - 1 integrating over ~ and using the orthogonality relation of Legendre polynomials

f ~ 2 (30) L~(~)Lj(~) = 5~ 2i + 1'

L~(~i) if i # j, (~j - ~i)Lp(~i)' 0, i f i = j ¢ O , p ,

= (39) Zi3 -p(p + 1) if i = j = 0, 4 '

P(P + I_____A) if i = j = p. 4 '

we obtain

/~j = ( ~ - ) / _ 1 1 L j (~)/:/i(~) . (31)

As Lj(~)/~i(~) is a polynomial of maximum order 2 p - 1 we can apply the GLL quadrature rule to give exact evaluation• Thus,

~j = wi ( ~ - - ) Lj(,~), j = O, .. . , p - 1 , (32)

where the property (26) has been used• To obtain/3p we now multiply both sides of equation (29) by Lp(~), integrate over ~t and use orthogonality of the Legendre polynomials we obtain

/_: [-I~(~)Lp(~) d ~ = ~ p ~ : Lp(~)Lp(~)d~. (33)

Applying the GLL quadrature rule to both sides of the previous equation p

Lp(~i)w, + E1 = tip E wkL~(~k) + flpE2, (34) k=0

where the property (26) has been used and E1 and E2 represent the quadrature errors, given by

E1 = (p + 1)pa22p+l[(P - 1)!]4 f d(2V)[[-I~(~)nP(~)] I ~, (35) (2p + 1)[(2p)!] 3 [ ~ J , ¢1 e

E~ = (p+ 1)p32~P+l[(P- 1)!]4 ~ d(~P)[r~(~)] ~ ~ ~ j , ~ E ~ • (36)

If we further notice that,

d(2p)[[Ii(~)LP(~)] ~" '~ d(2")[Lk(~)LP(~)] d(2")[L2(~)] = f~pC, (37) d~(2p) = A. , pk d - ~ = f~p d~(2P)

k=0

where C is the derivative of a polynomial of order 2p and therefore a constant giving a constant value whether evaluated at ~1 or @. Since E1 = ~pE2 both quadrature errors in equation (34) cancel out and using equation (24) for the quadrature weights we obtain

P /~p = ~wiLp(~), (38)

which completes the proof of the proposition. |

To finalize the description of the Legendre basis we denote by Z~. the derivative of the La- grangian interpolant (25) at the node i th GLL quadrature point evaluated at the j t h GAUSS- Lobatto-Legendre point given by (see for example [2])


Due to the common unitary weight for Laguerre functions and Legendre polynomials, it seems possible to devise a spectral-element method based on combining finite elements where Lagrangian interpolants of Legendre polynomials would be the basis functions with infinite elements where Lagrangian interpolants of Laguerre functions would be used as basis functions. The important point to consider is whether or not the Laguerre-Legendre global basis preserves the properties of spectral-element discretizations which are sparsity, C°-conformity and diagonality of the mass matrix. Sparsity is ensured as it is a consequence of the domain decomposition approach and the Lagrangian basis. To prove graphically the C°-conformity of the global basis we first plot some examples of Laguerre and Legendre basis functions in R + and ~, respectively. Using equation [14] we can easily display some of the Lagrangian interpolants of Laguerre functions in R + as shown in Figure 1 where the Lagrangian interpolants of order 15, /~0(~),/h2(~) and h15(~) are plotted. In Figure 2 the Lagrangian interpolant of order 50 at ~24, h24(~), is also plotted. In these two pictures we can observe that property (12) is satisfied. Using equation (27) some of the Lagrangian interpolants of Legendre polynomials of order 10 are plotted in Figure 3. Property (26) is satisfied. We observe that, while the GLL points in Figure 3 are distributed over ~ with a clustering of points near both extremes of the interval, the GRL points in Figures 1 and 2 are clustered towards 0 in R + with a progressively growing distance between the final points.

In Figure 4, we show an example of the C°-continuous Laguerre-Legendre global basis made up of a Lagrangian interpolant of Legendre polynomials of order 10 (left) and a Lagrangian interpolant of Laguerre functions of order 20 (right) at an arbitrary point x = 4. The figure is obtained by mapping ~ and R + to two contiguous elements f~k = [%,1 ek ]2 and f~k, = [eL, , c~) in a global domain a with e~ = e~, -- 4 using the linear mappings ~ = (2/(e~ - elk))(x -- e~) -- 1

1 respectively, and using the identities/2/10(~ ) = H l o ( x ) and ho(~) = h o ( x ) . It is and ~ = x - e k , ,

clear that even matching bases of different spectral orders is possible as long as the end points

¢-.

0.8

0.6

0.4

0.2

- 0 . 2

--0.4 0

, N I

I l l l l l I

t l I l l

l I /

l I l l |

! l

/ I I , ~ l

l I l I l i

%1 \ /

\ /

× / 'l

I I I I l I I

5 10 15 20 25 30 35

! P /

I

I

I

I

i i

/

I .

i I

i I

I i

\ .

~N P t % S

l"

_ t" . _ . ho(~ ) [

- - h 1 2 ( ~ )

I I

40 45 50

Figure 1. Lagrangian interpolants h0(~), h12(~) and hl~(~) of Laguerre functions of order 15 and GRL points in R +.


e-.

0.8

0.6

0.4

0.2

-0.2

-0.4 0

I I I [ l I

I I I I I I I I I

20 40 60 80 1 O0 120 140 160 180

Figure 2. Lagrangian interpolants h24(~) of Laguerre functions of order 50 and GRL points in R +.

200

0 , 8

0.6

0.4

A =.t j r , "--'t'._ -1-

= ~ = = / r - \ , , , =

I I

I I I I

I I I I

I t I t

I I I I

I I I I

I t I I

°-2rI , , i- / / . - . ,' ', . - . I - ~ ' / - ~ " , ~ /.,." " ~ - -,

. . . . . : . . ._. . ' . " , , , ,. / -,,. /.

\ I ~ / ~. / / \

/ %,...

I I I I I

~ . 8 ~ . 6 ~ . 4 ~ , 2 0 -0 .4

-1

\ / % /

_ _ H e ( ~ )

. _ , HIO(~)

I I I I

0.2 0.4 0.6 0.8

Figure 3. Lagrangian interpolants/:/o(~),/?/6(~) and/?/lO(~) of Legendre polynomials of order 10 and GLL points in ~.


1.2

0.8

0.6

0.4

0.2

- - 0 . 2 I i I I I I I I I

0 1 2 3 4 5 6 7 8 9

Figure 4. C°-global Laguerre (right p -= 20)-Legendre (left p = 10) basis at x = 4.

expanded as

Pk Pk

i = 0 j = 0

PK PK K K

i~O j=O

k = l , . . . , K - 1 ,

(42)

of both elements coincide. The C°-continuity of the global basis ensures the diagonality of the

mass matr ix provided that the GRL quadrature rule is used for the Laguerre elements and the GLL quadrature rule is used for the Legendre elements.

3. L A G U E R R E - L E G E N D R E SPECTRAL ELEMENTS

In this section, we address the issue of exponential convergence of the proposed Laguerre- Legendre spectral-elements method. For that purpose and for the sake of simplicity and in order to compare our results with those obtained in [10] we apply the method to the one-dimensional Helmholtz equation over the domain gt = R +.

Find u E C 2 (~) such that

- u " + )~u = f , u(x = 0) = 0, lim u(x) = 0. (40)

The spectral-element approximation to (40) is: find u ¢ VP(~) = {y E C°(~) [ Yk ¢ Ppk(~k) V Yk e /~pk (ilk) A y(0) = 0 A lim~-~o~ y(x) = 0} such that

A(u p, v p) + A2B(uP, v p) = F(v p) Vv p e VP(~), (41)

where A(u, v) = f~ u'v' dx, B(u, v) = f~ uv dx and F(v) = f~ f v dx. The domain is decomposed g

into I4 nonoverlapping spectral elements in such a way that ~t = ~Jk=l gtk and there are K - 1

Legendre spectral elements and one Laguerre spectral element. The admissible functions are


--5 I I I I I

- 10

-15 " C "

0

- 2 0

-25

- - Hi.error _ _ L-error

,%

I I I I - 3 0 ' 100 150 200 250 300 350 400

ndf

Figure 5. Convergence of t h e Laguer re -Legendre spec t ra l e l ement for Tes t 1.

Introducing (42) into (41) and using the GLL quadrature rule and the GRL quadrature rule to evaluate the integrals in the Legendre elements and Laguerre element, respectively, we obtain the following system of linear equations

K Pk K Pk

= F_, (431 k----1 i=0 k : l i=0

where Pk

k dki gl E k k k k k : Z]rZ r r, : (44) ",'~0

where gkl = 2 / h k , gk = h k / 2 , k = 1, . . . , K - 1, and gl g = gK = 1 (where hk is the length of the Legendre spectral-element ~k) are the geometric factors associated with the respective mappings for Legendre or Laguerre elements.

Since for a one-dimensional problem the sizes of the matrices involved are expected to be relatively small, the linear system of equations (43) is assembled to obtain a global linear system of equations which is posteriorly condensed which means that global nodes corresponding to Dirichlet boundary conditions are eliminated and the matrices are reduced accordingly. The condensed linear system is solved using Gaussian elimination with pivoting.

We consider the three cases proposed and analyzed in [10] and for all of them we consider for simplicity A 2 = 1 and the domain g~ initially decomposed into two elements: one Legendre element [0, 5] plus one Laguerre element [5, co). Defining the error in the H 1- and L~-norms in

the usual way as

)112 = I]u - uPllHl(n) = [(u' -- uP') 2 + (u -- uP) 2] d x


= I l u k - t ~ ) l = e~ , (45) k=l ) k=l

Since exponential convergence will only be achieved if an appropriate strategy guides the discretization we perform adaptive h-refinements and/or p-enrichments. In an attempt to equidis- tribute the globM error ~ we stop refining or enriching an element whose elemental error 0k falls below a certain tolerance level which we take to be of the order of 10 -9.

Test 1. Exponential Decay at Infinity

Following [10] we consider the solution to equation (40) with exact solution u ( x ) = e - ~ sin(kx) with k = 4. The initial polynomial orders are taken to be 50 in each element. Since the function decays to zero exponentially and is smooth in the totality of the domain, p-adaptive enrichments are carried out. After the solution over the initial mesh, the error in the Legendre element falls below the tolerance. Therefore, we only p-enrich the Laguerre element. We perform five successive p-enrichments in the Laguerre element incrementing its spectral order by 50 in each one of them. The final mesh has N = 351 degrees of freedom. The convergence rate of the logarithm of the error versus the final number of degrees of freedom is shown in Figure 5. As expected, and in agreement with the results obtained in [10] for a single Laguerre element, exponential convergence is observed in the H 1- and Loo-norms. The errors seem to behave as e - cy .

Test 2. Algebraic Decay without Essential Singularity at Infinity

Following [10] we consider the solution of equation (40) with exact solution u ( x ) = x / ( 1 -t- x ) h

with h = 3.5. Since the function is still smooth we only perform p-adaptive enrichments. The initial mesh has spectral orders 25 in each element. After a solution over this initial mesh we

=o

0

o

-5.5

-6

-6.5

-7

-7.5

- 8

-8.5

- 9

-9 .5

-10

-10.5 7

I I I I I I I I I

~ HLerr0r

I _ _ k -error

%

% .

I I L I I I I I I

8 9 10 11 12 13 14 15 16 sqrt(ndf)

17

Figure 6. Convergence of the Laguerre-Legendre spectral element for Test 2.


enrich both elements to orders 35. The error in the Legendre element falls below the tolerance so we continue refining the Laguerre element only to a spectral order 50 and further four successive enrichments by 50. The final mesh consists of the two elements with spectral orders 35 and 250 for the Legendre and Laguerre elements, respectively. In Figure 6, we have plotted the rate of convergence and the errors in the H 1- and L~-norms seem to behave as e -c~f~.

Test 3. Algebraic Decay wi th Essential Singularity at Infinity

Following [10] we consider the solution of equation (40) with exact solution u(x) = (sin(kx))/(1 + x ) h with k = 3 and h = 3. The initial polynomial orders of approximation in each element are taken to be 40. After an initial solution the Legendre element error falls below the tolerance. Due to the presence of the singularity at c~ we h-refine the Laguerre element introducing new Legendre elements of polynomial orders 40 and keeping the polynomial order 40 in the Laguerre element as well. We perform seven successive h-refinements. The final mesh consists of nine spectral elements with spectral orders 40: eight Legendre elements of length 5 and one last Laguerre element. The logarithm of the error in the H 1- and L~-norms versus the square root of the final number of degrees of freedom is plotted in Figure 7. The errors seem to behave as e - c a . This seems to be an advantage over the single Laguerre element used in [10] where the convergence rate obtained was only algebraic.

-2 I I I I I I I

Hi.norm L=-norm

-2 .5

- 3

0

o

-3 .5

- 4

-4.5 9 19

I I [ I I I I I I

10 11 12 13 14 15 16 17 18 sqrt(ndf)

Figure 7. Convergence of the Laguerre-Legendre spectral element for Test 3.

4. T H E D I F F U S I O N OF T U M O U R A N G I O G E N E S I S F A C T O R S

In this section, we describe one- and two-dimensional models for the early stages in the diffusion of tumour angiogenesis factors (TAF) into the surrounding host tissue. We first describe the two-dimensional model in an infinite domain shown in Figure 8 and then specialize into a one-dimensional model in a semi-infinite domain. A full description of the biological processes


\

/ '

. e ° ~ " . . . . . ~ .

• TAF "

EC

Figure 8. Schemat ic d i a g r a m of t he diffusion of T A F f rom a solid t u m o u r in t he presence of a ne ighbour ing blood vessel into an infinite two-d imens ionM domain ,

governing TAF secretion and its role in tumour growth may be found in [1,11,12,15]. We denote by u(x, y, t) the concentration of TAF at the point (x, y) at t ime t and we assume that a solid circular tumour of radius r0 is located at the point (0, 0) which is secreting the TAF into the surrounding tissue. The boundary of the tumour is kept at a constant concentration ub and the TAF concentration at infinity is zero. We assume that initially, the tumour has already secreted some TAF whose concentration is given by

u0(x, y) = u b e -(~2+y2-r~)/e. (47)

The blood vessel containing endothelial cells is located at x = 1. Consequently~ the EC concentration is given by

n(x , y) -- e-((~- l )2) /~H(y) , (48)

where the function H(y) indicates a finite extension of the blood vessel

1, i f y f f [-1,11, H(y) = 0, if y ~t [ - , 1, 1]. (49)

The surrounding host tissue a distributed according to a concentration given by

g ( x , y) = 1 - y ) . (50)

We assume that the TAF evolves in a much faster time-scale than the concentrations of EC and tissue [15] which are assumed to be in a steady state and not to evolve in time. We also consider that , the TAF is degraded by the EC upon contact according to a logarithmic law [1] and by the surrounding host tissue linearly upon contact. The diffusion of the TAF over the infinite domain

= {(x, y) : x 2 + y2 > ro} might be formulated mathematically as a reaction-diffusion equation

Find u(x, t) E C2(f~ x (0, T]) such that

Ou dV2u Ot

u(x, y, t) = Ub,

X u n ( X , y) -- ~/g(x, y)u, 1

on x 2 + y 2 _= ro 2,

lira u(x ,y , t ) -~- lim u ( x , y , t ) = O , x--*4-c~ y--*:t:oo

u ( x , y , 0 ) =

(51)

where d is the constant diffusion coefficient, X the chemotactic response of the EC and 7 the uptake rate of TAF by the surrounding tissue. Inspired by [15], we consider the values of the parameters ro = 0.25, d = Ub = 1, X ---- 10, 3, ----- 0.5, 5 = 1 × 10 -3 and t~ -- 0.45.

In the one-dimensional model over [0, co), the tumour is located at x = 0 and the Dirichlet boundary conditions in equation (51) are replaced by u(x = O,t) = ub, lim~__.~ u(x , t ) = O, and the initial condition by u(x,O) = uo(x) = e -x2/e. The EC and tissue concentrations are n(x) : e -((~-D2)/~ and g(x) : 1 - n(x), respectively.


5. L A G U E R R E - L E G E N D R E S P E C T R A L - E L E M E N T M E T H O D

OF LINES

In this section, in order to illustrate the use of the spatial semi-discretization based on the Laguerre-Legendre spectral-element method in conjunction with a time-stepping scheme we solve the one- and two-dimensional equations for the diffusion of the TAF. In order, to validate the results obtained we first solve a one-dimensional problem or Burger's equation with known analytical solution. This test case is taken from [16] with the difference that we account for nonhomogeneous Dirichlet boundary conditions.

O n e - D i m e n s i o n a l C a s e

We consider the solution of the initial boundary value problem. Find u(x , t ) E C2(R + × [0,T]) such that

Ou 02u 10u ~ - d -7 2 2 a T + f ( x , t ) ,

1 u(0, t) = Ub = 2' ~-~lim u(z , t) = 0,

1 x 0) = 0(x) : -2 e - ,

(52 )

where f ( x , t) is a given function in such a way that, the exact solution to (52) is

u(x, t) -- x sin(kt) 1 z (1 + x) h + ~ e - ,

and we take k = 2 and h : 3.5. The spectral-element spatial semi-discretization proceeds in a similar fashion as the approximation of equation (40) with the differences that now the pointwise values of the solution depend on time and the new tensorial form C(u, v) = fR+ u2v 'dx has

k k been introduced. This tensorial form is discretized to give ck~ = Z~iw ~ . After the spatial semi- discretization a system of ordinary differential equations is obtained. This system is assembled forming global matrices since the expected total size of the matrices for a one-dimensional problem are relatively small. The system is condensed by eliminating rows and columns corresponding to Dirichlet boundary conditions. The assembled and condensed final system of ordinary differential equations is

Bit : - d A u + 2 Cu 2 + f ( t ) + g, (53) u(O) = uo.

The function g{ : --dA{bub + (1/2)C{bU[ is introduced to take account of the nonhomogeneous Dirichlet boundary conditions. The index i runs through all the nodes where the values of the function u are not prescribed by Dirichlet boundary conditions (i # b). Due to the fact that, the polynomial orders in both elements are rather high and different bases are used in both of them, the initial value problem is expected to be extremely stiff. Therefore, we use the fully implicit unconditionally stable Crank-Nicolson finite difference scheme as a time integrator. The temporal domain (0, T] is discretized into n constant intervals (t~, tr+l), r = 0, 1 , . . . , n such that to : 0 and tn+l : T and At : t~+l - t~ is a constant time-step. Equations (53) are integrated over a time step to give the following system of second order Volterra integral equations.

ftr+l ~]~'+ 1 /tr+l B u ~+1 : B u ~ - d A u d t + 1 0 u 2 d t + B f d t + g A t .

J ~r 2 J t.r (54)

1186 J. VALENCIANO AND M. A. J. CHAPLAlN

0.9

0.8

0.7

0,6

= 0.5

0.4

0.3

0.2

o.1

I

l

l

I

l

k . , I___

I

r ~" i T r T T - -

2 3 4 5 6 7 8 9 X

Figure 9. Initial and steady state ld-concentrations of TAF.

Applying the trapezoidal rule to the integrals appearing in the right-hand side of (54) we obtain the final system of nonlinear equations to be solved at each time step by Newton's method

( Aid . _~ ) A t ( f r + l - b f r ) A4t Cu r+l~ = B - --~--A + Cu r u r + B 2 + gat. (55)

The initial guess for Newton's method is taken as the solution at the previous time step. This seems to accelerate the convergence from four iterations when the initial guess is taken to be zero to 2 iterations or even 1 iteration when the solution is close to the steady state. The linear system of equations at each Newton iteration at each time step is solved by Gaussian elimination with pivoting. The spatial semi-discretization is applied using a coarse mesh with two spectral elements: a Legendre element [0, 5] and a Laguerre element [5, oc) with polynomial orders of 35 and 100, respectively. Taking a time step At = 10 -2 and a tolerance for Newton's method of 10 -6 the maximum pointwise error obtained is of the order of 10 -5.

Following the same approach with the required changes we solve the one-dimensional equation for the diffusion of TAF. The mesh used is composed of one Legendre element [0, 5] and one Laguerre element [5, oc), the polynomial orders are 50 in each element, the time step is At = 10 -2 and the tolerance for Newton's method is 10 -6. A blow-up (magnification) showing the initial and steady state concentrations (at T = 6) are shown in Figure 9. The TAF has diffused over the tissue up to approximately 8 times the relative distance between the tumour and the blood vessel.

T w o - D i m e n s i o n a l C a s e

In higher dimensions and especially for nonlinear problems, the size of the matrices obtained after the spatial semi-discretization make it almost impossible to form global matrices. Storage is achieved using sparsity and the direct product formulation of the spectral-element method [4].


Iterative solvers [17], such as the conjugate gradient substi tute direct solvers because iterative solvers are based on matrix-vector products which can be performed in an element by element fashion without forming global matrices.

The semi-discrete spectral-element approximations of equation (51) is the following. Find uP(x, t) e VP(~]) such that VvP(x) • V?(~) and Vt • (0, T]

B \ Ot ' vp = -dd(uP ' "Up) - - x B 1-~uV' vp - 7B(guP' vp)' (56)

B(uP(x, O), vP) = B(u0(×),'UP),

where the tensorial forms A and B are the two-dimensional version of those defined in Section 2 for the solution of equation (40) and the spaces of approximation for the trial and test functions in (56) are defined as

vP(~) = {y • c°(~) I yk • Pp~(~k) x Pp~(~]k) v yk • Pp,(ilk) x Pp~(ftk)

A y ( x , y , t ) = u b , onx2 +y2=r2oA lim y ( x , y , t ) = lim y ( x , y , t ) = O } , x --* :k oo y -~ q- oo

and

yop(n) = {y e c ° ( ~ ) I y~ e p p ~ ( ~ ) × p ~ ( ~ ) v ~ e pp~(nk) × Pp~(a~)

Ay(x, y, t) = 0, on x 2 + y2 = r 2 A 1~2oo y(x, y, t) ---- lim y(x, y, t) = 0}, y--*=koo

respectively. The mesh used for the solution of (56) is made up of 2180 elements and is presented in Figure 10. This mesh is constructed in several steps. First, a rectangular region [-1, 1] x [-1, 1] is placed around the tumour. Meshing the region [-1, 1] x [-1, 1] - {(x,y) : x 2 + y2 < r02} is achieved as described in [12] using a blending mapping function to a rectangular domain [0, 1] x [0, 1]. Legendre isoparametric spectral elements are used in that region. Therefore, the geometry is approximated as [18]

2 .5 [ , ,

]]!///] I] ]11]] 1.5

0.51 : •

--0.5

- 1 . 5

- 2 : i i

- 2 . 5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 X

Figure 10. Spectral-element mesh for infinite domain.


= ( 5 7 ) ~=o j = 0

Where the basis functions are direct product of the one-dimensional Lagrangian interpolants of Legendre polynomials given by (25). Then, semi-infinite Laguerre-Legendre rectangular spectral elements are constructed around the perimeter of the rectangle [-1, 1] x [-1, 1]. Finally, the four Laguerre-Legendre infinite corner elements are constructed. For these semi-infinite and infinite elements the geometry is handled via one-dimensional linear mappings of the Legendre or Laguerre type as appropriate. The test and trial functions are expanded as

p~ Y

i=0 j = 0 /=0 rn=0

where 72/(~) and 72/(r/) are given by equations (25) or (11) depending on whether the element is finite, semi-infinite or infinite. The spectral-element approximation is conforming in the sense that, the global basis is continuous since the points at inter-elemental boundaries match. We now introduce the approximations given by equations (57) and (58) into a diseretized version of equation (56) and applying the G L L (22) and GRL (9), (10) quadrature rules as appropriate we obtain the following system of nonlinear ordinary differential equations in autonomous form

aImijUij k = l k = l j = 0

K K

k = l k = l

1 + Ukm ~blrnglmUlm ' (59)

for I = 0, . . . ,p~ and m = 0, . . . ,p~. Where the elemental diffusion matrix-vector is simplified by using equations (12) and (26) to give

k k k Gl(~r,r~m)(:D ~kr w ~k k k E E almijUiJ = E 1 J ( ~ r , ~ r n ) l t x J l r~ yJrn (Zx)irUim i=o j=o r=o \ ~=o /

V ° [ p~ ) P~ G2(~l,rlq) k k k k + ~-" 'J(~l,nq)' (~Y)~q(w~)I |~_,(Zy)jqul~

q =° = ( 6 0 )

- ( ~ ) ~ j ( ~ ) ~ ) j=O

- ( ~ ) ~ ( ~ , ~ ) , ~ ( z ~ ) j . . ~ , j .

i=0 "=

The Jacobian J and geometric factors G1, G2 and G3 are obtained according to the way the geometry has been treated for a particular element (via isoparametric or linear mapping). The discretized derivatives Z~ and Zy are given by equations (18) or (39) as appropriate. The diagonal elements of the two-dimensional mass matrix are given by

w k w k b~ = IJ(~,v~)l( ~)~( ~)m, (61)

and the one-dimensional discretized diffusion is

k k k (62) 79~j = Z i j w j .


Since expressions in equations (60) and (59) involve only two discrete indices, the storage re-

= 1 u 1). The discrete equations (59) are not quirement is of O ( K p 2 ) , where P max(p~ + , Pk + assembled. Its solution is obtained using the conjugate gradient method [17] based on matrix- vector products. The matrix-vector product is performed in three steps: first the local vector u~m is assembled and disassembled. Second, the local matrix-vector (60) is performed and the second and third terms in the right-hand side of the first equation in (59) added. And third, the resulting vector is finally assembled. In this way, only globM vectors are formed but the global matrix A

k associated to the local matrix aijlm is never formed. However, for notational purposes we might write the global assembled and condensed (the nodes corresponding to the Dirichlet boundary conditions are eliminated) system of ordinary differential equations as

Bit = - d A u - 7Bgu X B n + f , 1 + u (63)

u(0) = u0,

where the function f is introduced due to the nonhomogeneous Dirichlet boundary conditions at the perimeter of the turnout and given by

nb d E Aibub, i ---- 1 , . . . , N, (64) fi = --"fBibgbub -- x B i b 1 + Ub b

where the sum extends to all the nodes associated to the nonhomogeneous Dirichlet boundary condition, and N is the total number of degrees of freedom where u is no prescribed by Dirichlet boundary conditions. Since i 7~ b in equation (64) and B is the diagonal mass matrix the first two contributions to f~ are zero. The initial value problem (63) is expected to be extremely stiff. This is because of the mixture of different basis functions and the usage of smaller elements with lower polynomial orders due to the Legendre contribution (of order 4) in conjunction with larger elements with higher polynomial orders due to the Laguerre contribution (of order 40) introduces an ill-conditioning. Therefore, in view of stability concerns we march forward in time using the backward/forward Euler finite difference splitting scheme (6). The time derivative in (63) is substituted by an Euler backward difference scheme, the diffusion and linear reaction terms are treated implicitly and the nonlinear reaction term is treated explicitly. This gives rise to a linear system of equations to be solved at each time step.

n (B 4- d A A t 4- 7 A t B g ) u r+l = B u r - x B A t l--T~u~ 4- f a t . (65)

This semi-implicit scheme is equivalent to a single Picard iteration for the solution of the nonlinear system of equations every time step. The splitting avoids the solution of a nonlinear system of equations every time step and also gives rise to a symmetric positive definite matrix. Therefore, the conjugate gradient method might be used since it is guaranteed to converge for these type of matrices [17]. The stability restriction in the finite difference scheme has been translated into a convergence problem for the iterative method due to the stiffness of the initial value problem. The slow convergence of the conjugate gradient is overcome using a Jacobi or diagonal preconditioning [17], given in disassembled form as

m~m= E b~m+~/Atb~mg~m+dAt Gl(~r,~l'~)~l ) , k , w ~ k t Z ,k

p~ k G2( l,Vq) k k k ,k krZ

4- E [Z(~t,r/q)l (Du)"~q(W~)t (ZY)mq [j-"ff~/,/.i'll uJ.~mk ~/t ~ ~m (66) q=O

a3(6, (wy) -IZ(~l' r/m)l ,/1

D


The initiM guess for the conjugate gradient is t~ken as the solution at the previous time step. The Jacobi preconditioned conjugate gradient algorithm (see [17, p. 247]) requires one matrix-vector product for the initial guess plus another matrix-vector product per iteration. Therefore, the computational cost per time step can be estimated as

C ,~ c~(I + 1)KPP~P~ + c~(I + 1 )KP ~ + c~(I + 1)KP, (67)

. .¢-

1.4 • , • • ~.~ • • ~ *

1.2]..J ..,. -~ . * ~

1,J "~-- "$ "$" * "~ ~ ~ * "~ "~ * "~ ~ "~ ~ "~

0 .4 - .~*~...~.,~** ..,~ ~' , ~ , ~', ~ ~,~ **_ '~'~.-

. . " "~1~ ' ' ~ Z " ~ . . ~ A ' ~ " ~ . ~ ~ * ~ * ~ .~.:'*- ~.~""-~,,a,~'-~.,~ ~ ~ . "~..~5 "~ .~

" ~ ' ~ . .-~-.~ ~ ."-'~..,,. ~_ ~" ~.,..-,,~v'-~- . , ~ , . . O ( a ~ ' ~ d ~ r " . . . . ~

- 1 0 - 1 0 Y x

Figure 11. Magnification showing the initial concent ra t ion of TAF.

¢- *

* * *

* * * *

1.4 ... -~ '~ ~ "$"

/

1.2- .J . . ¢"

O . 8 . . . L . ~ ~ r ~ , ~ * ,

04T, :.,

- 1 0 - 1 0

* * * * _

10

Figure 12. Concent ra t ion of TAF at T = 8 in t he ' infinite discrete ' domain.

A L a g u e r r e - L e g e n d r e S p e c t r a l - E l e m e n t M e t h o d 1 1 9 1

0.8,~ . * * . * * * * i * * * * , * .

0 . 6 ~ + + + + + + + / + , . + .~ +

i ' ' ~ + ' + - , + + a l I ~ M I ~ +X( +'~ ~ ' ~ " ~ ~ + ~ "~++I

u.=~ I L++ +,+ .,,+-,%+" + , + , ~ ' - + ~ + , ~ . . ~ L . , , , , . ~ . + . , : , . , , + . ~ : ~ . . ~ . - I + -" .,..'x+ + + . m = , ' , ~ ..,~+.-~..,..+~,+m=.:'~l~, ~+~.+ ' " *

+I-; ,̀ + +<+ ; + ~ - : ' - ~ ' l + ~ - - ~ + . + + , + . . ' . ~ ~ +'+,-++'+

,,+ +-: +. +. + + .,,+ + +,+

%,,,...%"~IZ,,,.:"%.."-+.,+ -++ +.+ .+ : .,t++"++'.,...,+.,+~-~...,,.~ +.*- "+ ~+""~++:.+++,+,+,+,+,+_+~+ ..: + + ~ , + ~ . + +,,++ ++y:'-+,++,~ ++-..+,+ + +'+.+ +,++ - - , . . + ~ ; +<+ --+ ++ -,, + ' + " + ; . , + ~ + "

-10 -10 Y x

Figure 13. Magnification showing the concentration of TAF at T -- 8.

where P= = maxp~, Py = maxp~. The constants cl, c2 and c3 are integers, independent of K and Pk, and I is the number of conjugate gradient iterations to convergence. The first term in equation (67) represents the evaluation of the discretized diffusion operator (60), the second one is the cost of the preconditioning step and the last term represents the cost of the assembly and disassembly processes involved in the way the matrix-vector is performed in an element by element fashion.

The details of the discretization are: the mesh consists of 2180 elements as mentioned before and pictured in Figure 10. The polynomial orders are 4 x 4 for the Legendre isoparametric elements, 4 x 40 for the Legendre-Laguerre elements and 40 x 40 for the Laguerre-Legendre elements. The time step is At ---- 10 -2 the tolerance for the conjugate gradient iteration is 10 -6 and the size of

the matrices or number of degrees of freedom is 60320. In Figure 11 a blow-up of the vicinity of the origin shows the radially symmetric initial concentration of TAF (equation (47)). In Figure 12 the steady state concentration of TAF at T = 8 is shown in the 'infinite discrete' domain. As can be seen, the semi-open nature of the GRL quadrature rule truncates the infinite domain into this 'infinite discrete' domain. A blow-up (magnification) of the vicinity of the tumour is pictured in Figure 13, where it can be observed that, the steady state concentration of TAF at T = 8 has lost the radial symmetry, as expected, due to the presence of the blood vessel. The final extent of TAF is more than ten times the relative distance tumour-blood vessel.

6 . C O N C L U S I O N S

In this paper, we have described and applied a general Laguerre-Legendre spectral-element method for semi-infinite and infinite domains. In agreement with [10] we conclude that La- grangian interpolants of Laguerre functions seem to handle infinite and semi-infinite elements (domains) in a simple and consistent way. The method can be applied without any prior knowledge of the asymptotic behaviour of the solution while other methods based on mappings [14] require parameters which are related to this asymptotic behaviour. For an optimal performance, the method does not require any more knowledge of the solution than a standard Legendre spectral-element method for finite domains would need: location of elements where there exists


a lack of regular i ty of the solution which is sometimes obta ined th rough error indicators [19].

Er ror indicators of the type described in [19] would require equat ion (21) for the second order

derivative of the Lagrangian interpolants at the GRL points and might be ob ta ined provided tha t

an interpolat ion result in the nonweighted space L 2 using Laguerre functions is obtained.

The backward / fo rward Euler tempora l scheme used to solve the two-dimensional equat ion of

the T A F represents a ra ther crude level of approximat ion if in termediate values of the solution

are sought. A higher order split t ing [6] could be used provided t h a t s tabil i ty is no compromised.

In this case, since more than two time-levels would be coupled it might be difficult to find s tar t ing

values of the desired level of accuracy.

R E F E R E N C E S 1. M.A.J. Chaplain and A.M. Stuart, A mathematical model for the diffusion of tumour angiogenesis factor into

the surrounding host tissue, IMA Jour. o/ Maths. Appl. in Medic. and Biol. 8, 191-220, (1991). 2. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-

Verlag, Berlin, (1983). 3. D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-

CBMS, Philadelphia, PA, (1977). 4. Y. Maday and A.T. Patera, Spectral element methods for the incompressible Navier-Stokes equations, State

of the Art Surveys on Computational Mechanics, (Edited by A.K. Noor and J.T. Oden), ASME, New York, 71-143, (1987).

5. G. Strang and G.J. Fix, An Analysis of the Finite Element Method, McGraw-Hill International Editions, New Jersey, (1973).

6. G. Karniadakis, M. Israeli and S. Orszag, High-order splitting methods for the incompressible Navier-Stokes equations, J. of Comput. Phys. 97, 414-443, (1991).

7. I. Babu~ka and M. Suri, The p and h - p versions of the finite element method, an overview, Comput. Methods Appl. Mech. Engrg. 62, 5-26, (1990).

8. P.J. Davies and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, (1984). 9. C. Mavriplis, Laguerre polynomials for infinite-domain spectral elements, J. Comput. Phys. 80, 480-488,

(1989), 10. J. Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J.

Numer. Anal. 38, 1113-1133, (2000). 11. J. Valenciano and M.A.J. Chaplain, Computing highly accurate solutions of a tumour angiogenesis model,

Math. Models Meth. Appl. Sei. 13, 747-766, (2003). 12. J. Valenciano and M.A.J. Chaplain, An explicit subparametric spectral element method of lines applied to

a tumour engiogenesis system of partial differential equations, Math. Models Meth. Appl. Sci. 14, 47-68, (2004).

13. J. Dorman, Numerical Methods for Differential Equations. A Computational Approach, CRC Press Inc., New York, (1996).

I4. K. Black, Spectral elements on infinite domains, SIAM J. Sci. Comput. 19, 1667-1681, (1998). 15. A.R.A. Anderson and M.A.J. Chaplain, Continuous and discret mathematicM models of tumour-induced

angiogenesis, Bull. Math. Biol. 60, 555-597, (1998). 16. B. Guo and J. Shen, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite

interval, Numer. Math. 86, 635-654, (2000). 17. Y. Saad, Iterative Methods for Sparse Linear Systems~ PWS Publishing Company, Boston, MA, (1996). 18. K.Z. Korczak and A.T. Patera, An isoparametric spectral element method for solution of the Navier-Stokes

equations in complex geometry~ J. of Comput. Phys. 62, 361-382, (1986). 19. C. Bernardi, Indicateurs d'erreur en h - N version des 616ments spectraux, Math. Modelling Numer. Anal.

30, 1-38, (1996).

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