Parameter-Uniform Numerical Methods for a Class of Singularly Perturbed Problems with a Neumann Boundary Condition

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1 Parameter-Uniform Numerical Methods for a Class of Singularly Perturbed Problems with a Neumann Boundary Condition P. A. Farrell 1,A.F.Hegarty 2, J. J. H. Miller 3, E. O Riordan 4,andG.I. Shishkin 5 1 Department of Mathematics and Computer Science, Kent State University Kent, Ohio 44242, USA 2 Department of Mathematics and Statistics, University of Limerick, Ireland 3 Department of Mathematics, Trinity College, Dublin, Ireland 4 School of Mathematical Sciences, Dublin City University, Ireland 5 Institute for Mathematics and Mechanics, Russian Academy of Sciences Ekaterinburg, Russia Abstract. The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed modelordinary differentialequations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbation parameter. For the same classicalfinite difference operator on an appropriate piecewise uniform mesh, it is shown that the numerical solutions converge, uniformly with respect to the perturbation parameter, to the exact solution of any problem from this class. 1 Introduction Consider the following class of linear one dimensional convection diffusion problems L ε u ε εu ε + a(xu ε = f(x, x Ω =(, 1, (1a εu ε ( = A, u ε(1 = B, (1b a, f C 2 (Ω, a(x α>, x Ω. (1c Note that a Neumann boundary condition has been specified at x =. We recall the comparison principle for this problem (see [2], for example. Theorem 1. Assume that v C 2 (Ω. Then, if v (, v(1 and L ε v(x for all x Ω, it follows that v(x for all x Ω. This research was supported in part by the Russian Foundation for Basic Research under grant No , by the NationalScience Foundation grant DMS and by the Enterprise Ireland grant SC L. Vulkov, J. Waśniewski, and P. Yalamov (Eds.: NAA 2, LNCS 1988, pp , 21. c Springer-Verlag Berlin Heidelberg 21

2 Parameter-Uniform NumericalMethods 293 From this we can easily establish the following stability bound on the solution u ε (x u ε (1 + ε α u ε ( e αx/ε + 1 f (1 x. α Lemma 1. [2] The derivatives u (k ε of the solution of (1 satisfy the bounds u (k u (3 ε where C depends only on a and a. ε Cε k max{ f, u ε }, k =1, 2 Cε 3 max{ f, f, u ε } Consider the following decomposition of the solution u ε u ε = v ε + w ε,v ε = v + εv 1 + ε 2 v 2 ; w ε = w + εw 1 (2a where the components v,v 1 and v 2 are the solutions of the problems av = f, v (1 = u ε (1, (2b av 1 = v, v 1(1 =, (2c L ε v 2 = v 1, εv 2( =, v 2 (1 = (2d and the components w,w 1 are the solutions of L ε w =, εw ( = εu ε(, w (1 =. (2e L ε w 1 =, εw 1( = v ε(, w 1 (1 =. (2f Thus the components v ε and w ε are the solutions of the problems L ε v ε = f, v ε( = v ( + εv 1(, v ε (1 = u ε (1 L ε w ε =, w ε ( = u ε ( v ε (, w ε(1 =. Also, they satisfy the bounds given in the following lemma. Lemma 2. The components v ε,w ε and their derivatives satisfy v (k ε C(1 + ε 2 k, k =, 1, 2, 3 w (k ε (x C(ε u ε( + εε k e αx/ε, k =, 1, 2, 3. Proof. UseLemma1andthefactthat εψ ε (x = is the exact solution of 1 x e A(t/ε dt, where A(t = t L ε ψ ε =, εψ ε( = 1, ψ ε (1 =. a(sds

3 294 P. A. Farrell et al. 2 Upwinding on a Uniform Mesh In this section we examine the convergence behaviour of standard upwinding on a uniform mesh. The Neumann boundary condition is discretized by the scaled discrete derivative εd + U ε (. L N ε U ε εδ 2 U ε + a(x i D + U ε = f(x i, x i Ω N, (3a εd + U ε ( = εu ε (, U ε(1 = u ε (1, (3b where Ω N is an arbitrary mesh. We nowstate a discrete comparison principle. Theorem 2. [2] LetL N ε be the upwind finite difference operator defined in (3 and let Ω N be an arbitrary mesh of N +1 mesh points. If V is any mesh function defined on this mesh such that D + V (x, V(x N and L N ε V in Ω N, then V (x i for all x i Ω N. Hence, on an arbitrary mesh, the discrete solution U ε satisfies the bound U ε (x i u ε (1 + ε u ε( Φ i + 1 α f (1 x i. where Φ i is the solution of the constant coefficient problem εδ 2 Φ i + αd + Φ i =, εd + Φ = 1, Φ N =. Theorem 3. Let u ε be the continuous solution of (1 andletu ε be the numerical solution generated from the upwind finite difference scheme (3 on a uniform mesh Ωu N.Then, (a if u ε ( C, we have U ε u ε Ω CN 1 where U ε is the linear interpolant of U ε and C is a constant independent of N and ε. Also, (b if ε u ε( = C, then for any fixed N, U ε as ε Proof. (aconsiderfirstthecaseof u ε( C. The discrete solution U ε can be decomposed into the sum U ε = V ε + W ε where V ε and W ε are respectively the solutions of the problems L N ε V ε = f(x i, x i Ωu N, εd + V ε ( = εv ε(, V ε (1 = v ε (1 L N ε W ε =,x i Ωu N, εd+ W ε ( = εw ε (, W ε(1 =.

4 Parameter-Uniform NumericalMethods 295 We estimate the errors V ε v ε and W ε w ε separately. By standard local truncation error estimates, we obtain L N ε (V ε v ε (x i ε 3 (x i+1 x i 1 v ε (3 + a(x i 2 (x i+1 x i v ε (2 CN 1. Note also that D + (V ε v ε ( = v ε( D + v ε ( = 1 h h (s hv ε (s ds CN 1. With the two functions ψ ± (x i =CN 1 (1 x i ± (V ε v ε (x i, and the discrete minimum principle for L N ε we easily derive Note that if u ε ( C then (V ε v ε (x i CN 1. w (k ε (x Cε 1 k e αx/ε, k =, 1, 2, 3. The local truncation error for the layer component is given by and L N ε (W ε w ε (x i Cε 1 (x i+1 x i 1 e αxi 1/ε Cε 1 N 1 e αxi 1/ε εd + (W ε w ε ( = ε h h (s hw ε (s ds CN 1. Introduce the two mesh functions Ψ ± i = Cλ2 γ(α γ N 1 Y i ± (W ε w ε (x i where γ is any constant satisfying <γ<αand Y i = λn i 1 λ N 1, λ =1+γh ε, h =1/N. Note that Y i λ i. It is easy to see that λ 2 D + Y i γ ε e γxi 1/ε and so Y i decreases monotonically with Y i 1. We then have εd + Ψ ±, Ψ ± N =andusing( εδ 2 + γd + Y i =,weobtain L N ε Ψ ± i = Cλ2 γ(α γ N 1 (a(x i γd + Y i ± L N ε (W ε w ε (x i Cε 1 N 1( a(x i γ α γ e γxi 1/ε e αxi 1/ε <.

5 296 P. A. Farrell et al. By the discrete minimum principle we conclude that Ψ ± i andsoforallx i Ω N u ( W ε w ε (xi Cλ2 γ(α γ N 1 Y i Cλ 2 N 1. Thus, we have that ( W ε w ε (xi CN 1, when h ε. From an integral representation of the truncation error, we have xi+1 L N 1 ε (W ε w ε (x i C x i 1 ε e αt/ε dt Ce αxi 1/ε. As before we can establish Hence, for 1 i N and ε h, Note that which implies that ( W ε w ε (xi Cελ 2 i. ( W ε w ε (xi Cελ Ch. D + (W ε w ε ( = 1 h h (s hw (s ds C ( W ε w ε ( ( Wε w ε (x1 + Ch Ch. On the interval [x i,x i+1 ]wehave x (w ε w ε (x = w ε (tdt x x i w ε x i x i+1 x (tdt CN 1 w ε i x i Combining this with the argument in [2] completes part (a. (b If we discretize the Neumann boundary condition εu ε( = C by the standard discrete derivative εd + U ε ( = C on a uniform mesh then For a fixed distance h, ε xi+1 U ε (h =U ε ( + C h ε. lim U ε(h U ε (. ε On a uniform mesh, the discrete solution is not bounded independently of ε.

6 Parameter-Uniform NumericalMethods 297 Remarks. (i We define a a weak boundary layer by u ε (x C, thatis,the derivative is uniformly bounded with respect to ε. In this case, Theorem 3 states that the solution u ε can be approximated ε-uniformly on a uniform mesh. However, the first derivative u ε (x is not approximated ε-uniformly by the discrete derivative D + U ε (x i on a uniform mesh. This can be checked by solving a nontrivial constant coefficient continuous problem and its corresponding discrete problem directly, setting εn = 1 and then taking the limit as N. (ii In the case of the constant coefficient problem (1, we observe that lim ε εd+ U ε (x i εu ε (x i =. Thus, although the discrete solutions are unbounded as ε, the scaleddiscrete derivatives are at least ε uniformly bounded and, moreover, converge as N to εu (x i for each fixed ε. However, the scaled discrete derivatives εd + U ε ( are not ε uniformly convergent to εu ε (. In contrast, for the problem L ε u ε εu ε + a(xu ε = f(x, x Ω, (4a u ε ( = A, u ε (1 = B, (4b with Dirichelet boundary conditions, we have that lim U ε(x i u ε (x i =, ε and that lim ε εd+ U ε ( εu ε ( = O(1. This can be seen easily from the explicit solutions to the constant coefficient continuous problem. u ε (x =u ε ( + f α x (u ε( u ε (1 + f ( 1 e αx/ε α and the discrete problem 1 e α/ε U ε (x i =u ε ( + f α x i (u ε ( u ε (1 + f α ( 1 λ i 1 λ N,λ=1+αh/ε. 3 Upwinding on a Piecewise Uniform Mesh Consider the same upwind finite difference scheme (3 on the piecewise uniform mesh Ω N ε = {x i x i =2iσ/N, i N/2; x i = x i 1 +2(1 σ/n, N/2 <i} (5a where the transition parameter σ is fitted to the boundary layer by taking σ =min{ 1 2, 1 ε ln N}. α (5b The next result shows that upwinding on this mesh produces an ε uniform numerical method.

7 298 P. A. Farrell et al. Theorem 4. Let u ε be the continuous solution of (1 andletu ε be the numerical solution generated from an upwind finite difference scheme (3 on the piecewiseuniform mesh (5. Then, for all N 4, we have U ε u ε N Ω CN 1 ln N ε where C is a constant independent of N and ε. Proof. As for the uniform mesh we derive (V ε v ε (x i CN 1. When σ =1/2, the mesh is uniform and applying the argument of the previous theorem, we get ( W ε w ε (xi CN 1 ε 1 CN 1 ln N. When σ<1/2, the argument is divided between the coarse mesh and fine mesh regions. Consider first the coarse mesh region [σ, 1], where w ε (x Ce ασ/ε CN 1. Using the discrete comparison principle, we get W ε (x i ε w ε( Φ i where Φ i is the solution of the constant coefficient problem εδ 2 Φ i + αd + Φ i =, εd + Φ = 1, Φ N =. From an explicit representation of Φ i one can showthat Φ N/2 CN 1 Hence, for x i σ W ε (x i w ε (x i W ε (x i + w ε (x i CN 1 Consider nowthe fine mesh region, using the same argument as in the previous theorem we get ( W ε w ε (xi Cλ 2 N 1 ln N CN 1 ln N. This completes the proof. In [1], an essentially second order scheme is constructed on a piecewiseuniform mesh, using a more complicated finite difference operator. As in [2], the nodal error estimate for the simpler scheme presented here can easily be extended to a global error estimate by simple linear interpolation. That is, we have Ūε u ε Ω CN 1 ln N where Ūε is the linear interpolant of U ε. Also, using the techniques in [2], one can deduce that ε D + U ε u ε Ω\{1} CN 1 ln N.

8 4 Parabolic Boundary Layers Parameter-Uniform NumericalMethods 299 In this section, we introduce a new class of problems. Let Ω = (, 1,D = Ω (,T]andΓ = Γ l Γ b Γ r where Γ l and Γ r are the left and right sides of the box D and Γ b is its base. Consider the following linear parabolic partial differential equation in D with Dirichlet-Neumann boundary conditions on Γ L ε u ε (x, t ε 2 u ε x 2 + b(x, tu ε + d(x, t u ε t = f(x, t, (x, t D, (6a u ε = ϕ b on Γ b, ε u ε x = ϕ l on Γ l, u ε = ϕ r on Γ r (6b d(x, t >δ> and b(x, t β>, (x, t D (6c ϕ l ( = εϕ b(, ϕ b (1 = ϕ r (. (6d We have the comparison principle Lemma 3. Assume b, d C (D and ψ C 2 (D C 1 (D. Suppose that ψ on Γ b Γ r and ψ x on Γ r.thenl ε ψ in D implies that ψ in D. and the following stability bound Theorem 5. Let v be any function in the domain of the differential operator L ε. Then ε v (1 + αt max{ L ε v, v Γb Γ r } + β v x Γl e βx/ ε where α =max D {, (1 b/d} 1/δ. Assume that the data b, d, f, ϕ satisfy sufficient regularity and compatibility conditions so that the problem has a unique solution u ε and u ε Cλ 4 (D and, furthermore, such that the derivatives of the solution u ε satisfy, for all nonnegative integers i, j, i +2j 4 i+j u ε x i t j Cε i/2 D where the constant C is independent of ε. We write the solution as the sum u ε = v ε + w ε where v ε,w ε are smooth and singular components of u ε defined in the following way. The smooth component is further decomposed into the sum where v,v 1 are defined by v ε = v + εv 1 bv + d v t = f in D, v = u ε on Γ b (7a L ε v 1 = 2 v x 2 in D, v 1 =onγ \ Γ l, v 1 x =onγ l. (7b

9 3 P. A. Farrell et al. The singular component is decomposed into the sum where w l and w r are defined by w ε = w l + w r L ε w r =ind, w r = u ε v on Γ r,w r =onγ b Γ l (8a L ε w l =ind, (8b w l x = u ε x v x w r x on Γ l,w l =onγ r Γ b. (8c It is clear that w l,w r correspond respectively to the boundary layer functions on Γ l and Γ r. Assume that the data satisfy sufficient regularity and compatibility conditions so that v ε,w ε Cλ 4(D. Theorem 6. [3] For all non-negative integers i, j, such that i +2j 4 i+j v ε x i t j C(1 + ε 1 i/2 D and for all (x, t D, i+j w l (x, t x i t j Cε i/2 e x/ ε, i+j w r (x, t x i t j Cε i/2 e (1 x/ ε where C is a constant independent of ε. Problem (6 is discretized using a standard numerical method composed of a standard finite difference operator on a fitted piecewise uniform mesh. L N ε U ε = εδxu 2 ε + bu ε + ddt U ε = f,(x, t Dσ N U ε = u ε on Γb,σ N Γ r,σ N, D+ x U ε = u ε x on Γ l,σ N (9a (9b where Dσ N = ΩNx σ Ω Nt, and Γσ N = DN σ Γ. (9c A uniform mesh Ω Nt with N t mesh elements is used on (,T. A piecewise uniform mesh Ωσ Nx on Ω with N x mesh elements is obtained by putting a uniform mesh with N x /4meshelementsonboth(,σand(1 σ, 1 and one with N x /2 mesh elements on (σ, 1 σ, with the transition parameter { 1 σ =min 4, 2 } ε ln N x. (9d We have the following discrete comparison principle Lemma 4. Assume that the mesh function Ψ satisfies Ψ on Γ N b,σ Γ N r,σ and D + x Ψ on Γ N l,σ.thenln ε Ψ on DN σ implies that Ψ on DN σ.

10 Parameter-Uniform NumericalMethods 31 The ε uniform error estimate is contained in Theorem 7. Let u ε be the continuous solution of (6 andletu ε be the numerical solution generated from (9. Assume that v ε,w ε Cλ 4 (D. Then,forallN 4, we have sup U ε u ε N D CNx 1 ln N x + CNt 1 <ε 1 σ where C is a constant independent of N x,n t and ε. Proof. The argument follows [3]. The discrete solution U ε is the sum U ε = V ε + W ε where V ε and W ε are the obvious discrete counterparts to v ε and w ε.the classical truncation error estimate yields L N ε (V ε v ε C εn 1 x It follows that V ε v ε CN 1 x + CN 1 t and D + x (V ε v ε (,t CN 1 x. + CNt 1. Note also that L N ε (W l w l CNx 1 ln N x + CNt 1 and ε D + x (W l w l (,t CNx 1 ln N x. The proof is completed as in [3]. 5 Numerical Results In this section we present numerical results for the following specific elliptic problem ε u ε x =, (x, y Γ L, ε u ε + u ε x =16x(1 x(1 yy, (x, y (, 12 (1a u ε =1, (x, y Γ R Γ T (1b ε u ε y = 16x2 (1 x 2, (x, y Γ B (1c whose solution has a parabolic boundary layer near Γ B. The nature of the boundary layer function associated with this layer is related to the solutions of the parabolic problems examined in the previous section. In Figure 1 we present the numerical solution generated by applying standard upwinding on a uniform mesh. The numerical solutions are not bounded uniformly with respect to ε as ε. This should be compared with the accurate approximation given in Figure 2, which was generated by applying standard upwinding on the piecewise uniform mesh Ωσ N,N 2 = Ωu N Ωτ N,

11 32 P. A. Farrell et al x y Fig. 1. Numerical solution generated by upwinding on a uniform mesh with N=32, ε=1 8 for problem (1 where { 1 τ =min 2, } ε ln N. Note the significant difference in the vertical scale in these two figures. In Table 1 we present the computed orders of convergence (see [2] generated by applying standard upwinding on this piecewise uniform mesh. These indicate that the method is ε uniformly convergent for problem (1. References 1. Andreyev V. B. and Savin I. A. (1996. The computation of boundary flow with uniform accuracy with respect to a small parameter. Comput. Maths. Math. Phys., 36 ( Farrell, P. A., Hegarty A.F, Miller, J. J. H., O Riordan, E.,Shishkin G. I., Robust Computational techniques for boundary layers, Chapman and Hall/CRC Press, , 293, 294, 296, 298, Miller, J. J. H.,O Riordan, E.,Shishkin G. I.and Shishkina,L. P. (1998 Fitted mesh methods for problems with parabolic boundary layers, Math. Proc. RoyalIrish Academy, 98A ( , 31

12 Parameter-Uniform NumericalMethods x y Fig. 2. Numerical solution generated by upwinding on a piecewise uniform mesh with N=32, ε=1 8 for problem (1 Table 1. Computed orders of convergence generated by upwinding on a piecewise uniform mesh applied to problem (1 Number of intervals N ε p N