A CLASS OF SINGULARLY PERTURBED QUASILINEAR DIFFERENTIAL EQUATIONS WITH INTERIOR LAYERS

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1 MATHEMATICS OF COMPUTATION Volume 78, Number 265, January 2009, Pages S ) Article electronically published on June 27, 2008 A CLASS OF SINGULARLY PERTURBED QUASILINEAR DIFFERENTIAL EQUATIONS WITH INTERIOR LAYERS P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN Abstract. A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform numerical method with respect to the singular perturbation parameter. Numerical results are presented, which support the theoretical results. 1. Introduction Convection-diffusion equations of the form u x ) x +bu)) x = fx), with a nonlinearity of the type bu) =u 2, arise in numerous applications involving fluid dynamics. The Navier-Stokes equations involve such a nonlinearity, as do the driftdiffusion equations for modelling semiconductor devices. Depending on the specified boundary conditions, boundary and/or interior layers can arise in the solutions of such nonlinear equations. In this paper, we examine a class of nonlinear singularly perturbed ordinary differential equations, whose solutions exhibit interior layers. Moving interior layers are often associated with shock waves in gas dynamics. Burgers equation is typically used as an initial mathematical model to study such shock layer phenomena. The nonlinear problem analysed in this paper can be viewed as a step towards understanding such classical nonlinearities. In the case of a linear singularly perturbed ordinary differential equation, classical numerical methods usually give unsatisfactory numerical results, when the singular perturbation parameter is small. A parameter-uniform numerical method [4] is a numerical method for a singularly perturbed problem having an asymptotic error bound in the pointwise maximum norm that is independent of the size of the singular perturbation parameter. Parameter-uniform behaviour may be achieved by fitting the mesh [4] or by fitting the finite difference operator to the boundary/interior layer [13]. In [5] it was proved that for a class of singularly perturbed semilinear two point boundary value problems, parameter-uniform convergence in the maximum norm is not achievable using fitted operator schemes with frozen coefficients on uniform meshes. However, a parameter-uniform numerical method Received by the editor April 2, 2007 and, in revised form, December 7, Mathematics Subject Classification. Primary 65L20; Secondary 65L10, 65L12, 34B15. Key words and phrases. Quasilinear, uniform convergence, layer adapted mesh, interior layer. This research was supported in part by the Mathematics Applications Consortium for Science and Industry in Ireland MACSI) under the Science Foundation Ireland SFI) mathematics initiative. The third author was supported in part by the Russian Foundation for Basic Research under grant No c 2008 American Mathematical Society Reverts to public domain 28 years from publication 103

2 y 104 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN 2 u0) = -0.9, u1) = x Figure 1. A representative set of solutions for different values of for a problem from the class of problems given in 1.1) for such semilinear problems, which are of reaction-diffusion type, that combined a standard finite difference operator with a fitted piecewise-uniform mesh was given in [6]. In this paper, we construct a parameter-uniform method, based on a standard upwind finite difference operator and a fitted piecewise-uniform mesh, for a nonlinear convection-diffusion problem. Farrell et al. [7], Linß et al. [10, 9], and Vulanović [15] examined quasilinear convection-diffusion problems where the problem class was such that only boundary layers occurred in the solutions. In the papers [7, 10, 15], parameter-uniform numerical methods were developed, which were based on Shishkin-type piecewise uniform meshes [14, 12, 4] for the problem. In this paper, we examine the case of a quasilinear convection-diffusion problem, where interior layers can occur in the solutions. An analytical discussion of quasilinear problems with interior layers is given in [2]. In this paper the following class of singularly perturbed quasilinear ordinary differential equations with discontinuous data is considered. Let Ω =0,d), Ω + = d, 1) and Ω =[0, 1]. Find u C 1 Ω) C 2 Ω Ω + ) such that 1.1a) 1.1b) 1.1c) u x)+bx, u)u x) =fx), x Ω Ω +,u 0) = A, u 1) = B, { { b1 u) = 1+cu, x < d, δ1,x<d, bx, u) = fx) = b 2 u) =1+cu, x > d, δ 2,x>d, 1 <u 0) < 0, 0 <u 1) < 1, 0 <c 1, where δ 1,δ 2 are nonnegative constants. Note the strict inequalities in 1.1c). In order to study and analyse monotonically increasing solutions, we impose further conditions on the magnitudes of f Ω, and the boundary values u 0), u 1). These monontonicity-related restrictions are introduced at appropriate locations 3.5) and 6.2)) in this paper. A representative example of the possible solutions to 1.1) is given in Figure 1, which illustrates the presence of an interior layer that steepens as 0. A linear version of 1.1) was studied in [3], where a parameter-uniform numerical method based on a suitably designed piecewise-uniform mesh was shown to be parameter-uniform of essentially first order for a linear convection-diffusion problem with discontinuous data. The methodology in [3] is extended in this paper to the quasilinear problem 1.1). Note that if u Ω, < 1in1.1),thenb 1 u) < 0and

3 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 105 b 2 u) > 0. For this particular sign pattern either side of the point of discontinuity, a strong interior layer will normally be present in the solution. Alternative sign patterns on the coefficient of the first derivative can result in a weak interior layer appearing in the solution. These alternative sign patterns are discussed in [3] in the case of a linear version of problem 1.1). In [8], a parameter-uniform method was analysed for a semilinear singularly perturbed problem with discontinuous data, whose solution contained an interior layer. In this case, it is relatively straightforward to establish the conditions both for existence of the continuous solution and for inverse-monotonicity of a linearization of the discrete finite difference operator. The conditions on the data, under which existence can be established for the quasilinear problem examined in the present paper are more intricate. Moreover, the analysis of the inverse-monotonicity property of a linearized finite difference operator is significantly more complicated in the case of the quasilinear problem. The paper is organized in the following manner. First, we establish the existence of the solution, and its uniqueness and derive aprioriestimates. To do this we utilize an asymptotic approach. We associate a set of left and right problems with problem 1.1). The left problem and right problem are defined to be: find u L x; γ) C 2 Ω )andu R x; γ) C 2 Ω + ) such that 1.2a) 1.2b) u L + b 1u L )u L = f, x Ω =0,d), u L 0) = u 0), u L d) =γ, u R + b 2 u R )u R = f, x Ω + =d, 1), u R d) =γ, u R 1) = u 1). In the next section we identify a natural restiction 2.4) on the data, so that there exist unique regular components see Theorem 2.3) of the solutions to the problems 1.2a) and 1.2b). The left regular component v L x) is defined so that it satisfies the same differential equation as u L x; γ) ontheintervalω, agrees with u L at the left boundary x = 0, and the first two derivatives of v L x) are bounded independently of. Exterior to the interior layer region, the solution of 1.1) approaches the left and right) regular component on Ω and Ω + ). The multi-valued discontinuous regular component v of 1.1) is defined to be the left regular component on Ω and the right regular component on Ω +, respectively. In order for this regular component to be monotonically increasing, we impose a further condition 3.5) on the data. In 3, we first establish existence and uniqueness of u L and u R for certain ranges of γ. We then show in Theorem 3.3 that by assuming 3.5), a value γ for the parameter γ can be chosen so that u R d+,γ )=u L d,γ ). This establishes the existence of a solution to problem 1.1). In 4, the continuous solution u to problem 1.1) is written as a sum of the discontinuous regular component v and a discontinuous singular component w. Parameter-explicit bounds on the first three derivatives of these two components are established in Lemma 4.2. The magnitude of the singular component is negligible outside of a O ln )-neighbourhood of the point x = d. Based on these a priori bounds, a fitted mesh is constructed in 5. A nonlinear finite difference method is introduced and the existence of a discrete solution is established using appropriate choices of discrete lower and upper solutions. The existence of a discrete regular component is also established in this section. In 6 the main result Theorem 6.2) of the paper is given. This shows that the numerical method produces numerical approximations, which converge to the unique solution of the continuous problem 1.1). The rate of convergence is independent of the small parameter. The method of proof requires that a discrete linear operator

4 106 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN associated with the nonlinear difference operator) preserve inverse-monotonicity. This requirement imposes an additional constraint 6.2) on the data for problem 1.1). At the end of 6, the implications of this assumption are discussed. Numerical results are given in 7 and the appendix 8) deals with discrete comparison principles for related linear problems, which are used in 6 in the proof of the main convergence result. Throughout the paper C denotes a generic constant that is independent of and the mesh parameters. We always use the pointwise maximum norm and denote it by z D =max x D zx). For notational convenience, we will omit the subscript when D = Ω and simply write z. 2. Existence and uniqueness of the regular component The regular components v L and v R of any possible solutions to problems 1.2a) and 1.2b) are formally defined to be the solutions of the two boundary value problems 2.1a) v L + b 1 v L )v L = f, x < d, v L 0) = u 0), v L d) =v 0 d )+v 1 d ), 2.1b) v R + b 2 v R )v R = f, x > d, v R d) =v 0 d + )+v 1 d + ),v R 1) = u 1), where v 0 and v 1 are solutions of the following nonlinear first order problems 2.2a) bx, v 0 )v 0 = f, x Ω Ω +,v 0 0) = u 0); v 0 1) = u 1), v 2.2b) 0 + bx, v 0 + v 1 )v 0 + v 1 ) = f, x Ω Ω +, v 1 0) = 0, v 1 1) = 0. By simply integrating, we have that the reduced solution v 0 satisfies the quadratic equations 2.3a) 1 0.5cu 0))u 0) + δ 1 x =1 0.5cv 0 x))v 0 x), x<d, 2.3b) cu 1))u 1) δ 2 1 x) =1+0.5cv 0 x))v 0 x), x>d. If we assume that 2.4) δ 1 d< u 0) + 0.5cu 2 0) and δ 21 d) <u 1) + 0.5cu 2 1), then there exists a unique reduced solution v 0 C 1 Ω ) C 1 Ω + ) with the property that v 0 x) < 0, x Ω and v 0 x) > 0, x Ω +. Note the following additional properties of the reduced solution v 0 when 2.4) is satisfied: b 1 v 0 )x) < 1,x<d, b 2 v 0 )x) > 1,x>d, v 0x) > 0, x d, δ 1 >v 0x) δ 1 > 1 cu 0), x < d, δ 2 v 0x) > 1+cu 1),x>d, v 0 x) = cv 0) 2 bx, v 0 ), From these properties and the fact that v 0 d )= d 0 v 0x)dx + u 0), d x)v 0 x) > 0, x d. v 0 d + )=u 1) 1 d v 0x)dx,

5 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 107 we deduce the following bound on the jump in the reduced solution at x = d, 2.5) [v 0 ]d) :=v 0 d + ) v 0 d δ 1 d ) <u 1) u 0) 1 cu 0) + δ ) 21 d). 1+cu 1) Let us now examine the second term v 1 x) in the expansion of the regular component vx). Note that bx, v 0 + v 1 ) bx, v 0 )=cv 1. Hence, on both Ω and Ω +, 2.6) bx, v 0 + v 1 )v 1x)+cv 0v 1 x) =bx, v 0 )v 1 ) +0.5cv1) 2 = v 0 x). On Ω, integrate this equation from t =0tot = x, andonω +,integratefrom t =1tot = x. This yields a quadratic equation in v 1 of the form x v 0 t) dt < 0, x < d, bx, v 0 )v cv1 2 0 = 1 v 0 t) dt < 0, x > d. On each subdomain, we require to be sufficiently small so that b 2 1v 0 ) > 2c x 0 v 0 t) dt, x Ω and b 2 2v 0 ) > 2c x 1 x v 0 t) dt, x Ω +. With this restriction on, there are two possible solutions v1,v s 1 b with 0 <v1 s < v1 b, x Ω and v1 b <vs 1 < 0, x Ω+. Define v 1 uniquely by setting v 1 = v1 s. Consider a quadratic of the form kx) =0.5lx 2 mx+i, wherel, m, I are positive constants and is sufficiently small so that m 2 > 2lI. Then the minimum of k occurs at x = m/l). Since k2i/m) < 0, we note that the smallest root of kx) = 0 is smaller than 2I/m. Hence, on the intervals Ω and Ω + select the smallest roots such that 2 x 0 v 0 t) dt >v 1 > 0, x Ω, 0 >v 1 > 2 1 x v 0 t) dt, x Ω +. b 1 v 0 ) b 2 v 0 ) This yields a unique v 1 with v 1 x) > 0), which is bounded independently of and from 2.6) it follows that v 1x) C, v 1 x) C. To establish the existence and uniqueness of the regular component, we employ the technique of upper and lower solutions. Definition 2.1. A function α C 1 Ω )isalower solution of problem 1.2a) if 2.7) α + b 1 α)α f, x < d and α0) u 0), αd) γ. An upper solution β is defined in an analogous fashion, with all inequalities reversed. Consider the general quasilinear problem y = gx, y, y ), x J =a 1,a 2 ). Let g C[J R R; R] andα, β C[J, R] withαx) βx),x J. Suppose that for x J, αx) yx) βx), gx, y, y ) Ψ y ), where Ψ C[[0, ), 0, )]]. If λ s Ψs) ds > max βx) min αx), x J x J

6 108 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN where λa 2 a 1 )=max{ αa 1 ) βa 2 ), αa 2 ) βa 1 ) }, thenwesaythatg satisfies a Nagumo condition on J relative to the lower and upper solutions [1]. The nonlinearity in 1.2a) and 1.2b)) satisfies a Nagumo condition for any bounded α and β, since we can take Ψx) = f + γx, γ := sup α y β b 1 y). Thus we can cite the following existence result. Lemma 2.2 [1, page 31]). If α, β C 1 Ω, R) are lower and upper solutions for the problem 1.2a) and αx) βx), x Ω, then there exists a solution u L C 2 Ω, R) to 1.2a) and αx) u L x) βx), x Ω. Hence, to establish existence of a regular component v L on Ω, it suffices to construct lower and upper solutions. Theorem 2.3. Assume 2.4). There is a unique regular solution to 2.1a) and v L C Ω, [u 0), 0)), withv L x) v 0 x). Proof. Note that, by assuming 2.4), v 0 d ) < 0andfor sufficiently small, v 0 d ) v L d) =v 0 d )+v 1 d ) < 0. Since v L d) < 0onΩ, we can use the lower and upper solutions αx) =v 0 x) and dβx) = u 0)x d) to establish existence of a regular solution v L satisfying 2.1a) and 2.4) on Ω. Suppose v 1,v 2 C 2 0,d),[u 0), 0) ) are two regular solutions of 2.1a) and let ψ = v 1 v 2.Then ψ = ψ +0.5cv2 2 v1) 2, ψ0) = ψd) =0. Integrating from x =0tox = t yields ψ t) 1 0.5cv 1 + v 2 )t))ψt) =ψ 0). Since ψd) =0,ψt) 0. Hence the solution of 2.1a) is unique. There is an analogous result for the existence and uniqueness of a solution v R C Ω +, 0,u 1)]) of 2.1b). Define the regular component of any solution to 1.1) to be the multi-valued discontinuous function { vl x), x d, 2.8) v x) := v R x), x d, where v L is the solution of 2.1a) and v R is the solution of 2.1b). 3. Existence and uniqueness of the continuous solution In this section we establish the existence of u L x; γ) andu R x; γ)) for a certain range of γ. Under additional assumptions on the data, we will show that u L x; γ) and u R x; γ) both exist for a common range of γ. To this end, we define the barrier functions χ L x; γ), χ R x; γ) as the respective solutions of the boundary value problems 3.1a) χ L + 1+cγ)χ L = δ 1, χ L 0) = u 0), χ L d) =γ, 3.1b) χ R +1+cγ)χ R = δ 2, χ R d) =γ, χ R 1) = u 1). Lemma 3.1. For all γ [v L d), 1 c ) such that 3.2) δ 1 d 1 cγ)γ u 0)), problem 1.2a), 2.4) has a unique solution u L x; γ) C 2 0,d),[u 0),γ] ) with the property that v L x) u L x; γ) χ L x; γ), x [0,d].

7 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 109 For all γ 1 c,v Rd) ] such that 3.3) δ 2 1 d) 1 + cγ)u 1) γ), problem 1.2b), 2.4) has a unique solution u R x; γ) C 2 0,d),[γ,u 1)] ) with the property that χ R x; γ) u R x; γ) v R x), x [d, 1]. Proof. Note that where χ L x; γ) = δ 1 1 cγ x + u 0) + Kψx; γ), ψ 1 cγ)ψ =0,ψ0) = 0, ψd) =1, K = 1 cγ)γ u 0)) δ 1 d. 1 cγ Also by 3.2) K 0andsoχ L > 0. Hence χ L + b 1χ L )χ L = δ 1 +b 1 χ L ) 1+cγ))χ L = δ 1 + cχ L γ)χ L δ 1. An analagous argument is used to establish the existence of u R x; γ). If ) 1 3.4a) δ 1 d u 0) + cv R d) c + u 0) v R d) and ) 1 3.4b) δ 2 1 d) u 1) cv L d) c u 1) + v L d), then by the previous lemma u L x; v R d)) and u R x; v L d)) both exist. Hence to guarantee the existence of a continuous solution ux; γ) defined over the entire interval [0, 1] for all γ [v L d),v R d)], we are required to restrict the data of problem 1.1). Hence we are led to the following assumption. Assumption 1. Assume that the problem data for problem 1.1) are such that 3.5a) δ 1 d< u 0), δ 2 1 d) <u 1) and 3.5b) { u 1) u 0) < 1/c +min δ 1 d 1 cu 0), δ 21 d) 1+cu 1) Note that 3.5) implies 2.4). By the properties of v 0 x) established in 2, it follows from 3.5) that, for sufficiently small, 0 >v L d) > 1 cu 1) and 0 <v R d) < 1+cu 0). c c The assumption 3.5) suffices for the inequalities 3.4) to hold true and consequently for u L x; γ) andu R x; γ) toexistforallγ [v L d),v R d)]. In the next lemma we establish that u L x; γ) andu R x; γ) depend continuously on the parameter γ. Lemma 3.2. Assuming 3.5), for all γ 1,γ 2 [v L d),v R d)], u Lx; γ 1 ) u Lx; γ 2 ) C γ 1 γ 2, x 0,d), u Rx; γ 1 ) u Rx; γ 2 ) C γ 1 γ 2, x d, 1). }.

8 110 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN Proof. Let Gx; γ 1,γ 2 )=u L x; γ 1 ) u L x; γ 2 ). Note that G + 1+ c ) 2 u Lx; γ 1 )+u L x; γ 2 )) G =0, G0) = 0, Gd) =γ 1 γ 2, and 1+ 2 c ul x; γ 1 )+u L x; γ 2 ) ) 1+0.5cγ 1 +γ 2 ) 1+cv R d) cu 0) < 0. It follows that G C γ 1 γ 2. We now state a central result in this paper. Theorem 3.3. Assuming 3.5), the nonlinear problem 1.1) has a unique solution u C 1 0, 1), u 0),u 1)). Moreover, v L x) u x) χ L x; v R d)), x d, χ R x; v L d)) u x) v R x), x d, where χ L,χ R are defined in 3.1). Proof. For all x Ω, x t=0 f b 1 u L )u L )t) dt =u Lx) u 0)) 1 0.5cu L x)+u 0)) ) δ 1 x. Integrating 1.2a), from 0 to x, yields u Lx; γ) =u L0; γ)+u L x; γ) u 0)) 1 0.5cu L x; γ)+u 0)) ) δ 1 x. By the Mean Value Theorem, for some z 0,), < d, u L z; γ) =u L; γ) u 0). Since cγ < 1 and 3.2), using the lower and upper solutions given in Lemma 3.1, we deduce that, for all γ [v L d),v R d)], 0 u L z; γ) u 0) C, 0 u L0; γ) C. For all x Ω +, 1 t=x f b 2 u R )u R )t) dt = δ 21 x)+u R x) u 1)) 1+0.5cu R x)+u 1)) ). Integrating 1.2b) from x to 1 yields u Rx; γ) =u 1; γ) δ 2 1 x)+u 1) u R x; γ)) 1+0.5cu R x; γ)+u 1)) ). Since cγ > 1, using 3.2) and the lower and upper solutions given in Lemma 3.1, we deduce that, for all γ [v L d),v R d)], 0 u R1; γ) C. We wish to establish the existence of a γ = u L d) =u R d) such that u L d ; γ )= u R d+ ; γ )and 1 <cγ < 1. This is equivalent to finding a γ such that 1 < cγ < 1and u L0; γ )+γ u 0)) 1 0.5cγ + u 0)) ) δ 1 d = u R1; γ ) δ 2 1 d)+u 1) γ ) 1+0.5cγ + u 1)) ). Rearranging, gives 2γ = u R1; γ ) u L0; γ )+δ 1 d+u 0) 0.5cu 2 0)+u 1)+0.5cu 2 1) δ 2 1 d). By 2.3), this further simplifies to 3.6) 2γ =Γ+u R1; γ ) u L0; γ ), where Γ := v 0 d + )+v 0 d )+0.5cv0d 2 + ) 0.5cv0d 2 ). Define the function Hγ) :=Γ+u R1; γ) u L0; γ) 2γ.

9 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 111 For all γ [v L d),v R d)], u R1; γ) u L0; γ) C. Note that v L d) < 0 <v R d) andso,for sufficiently small, Hv L d)) > 0 > Hv R d)). By Lemma 3.2, H is a continuous function of γ, so there exists a γ [v L d),v R d)], where Hγ ) = 0. Hence we have established the existence of asolutionu C 1 0, 1), u 0),u 1)) to problem 1.1), 3.5). Let u +,u be two solutions of problem 1.1). The difference in these two solutions is y := u + u C 1 0, 1), u 0),u 1)) and solves the problem y + bx, u + )u + ) bx, u )u ) =0, y0) = y1) = 0. Integrate over Ω and over Ω + to get y y cu+ + u )y = y 0), x Ω, y + y cu+ + u )y = y 1), x Ω +. Using integrating factors and integrating again, we have that y = y 0)e px) y = y 1)e qx) x 0 1 x e pt) dt, px) = e qt) dt, qx) = x 0 1 x cu+ + u ) dt, x Ω, cu+ + u ) dt, x Ω. Note that y is continuous at x = d, sofromabovey 0)y 1) 0. If y 0) = 0, then y =0for x Ω, which implies that yd) = 0. This, in turn, implies that y =0, x Ω. Without loss of generality, let us assume that y 0) > 0. From the expression for y above, y>0, for x Ω. Also, from the expression for y and the fact that cu+ +u ) > 0, we deduce that y > 0, for x Ω. Also, if y 0) > 0, then y 1) < 0. Repeating the above argument, we deduce that y x) < 0,x Ω +. Hence, the maximum value of y occurs at x = d. Subtracting the two expressions for the derivative of y at x = d, yields 2yd) =y 1) y 0)) < 0, which is a contradiction. Hence y 0) = 0, which implies that y = 0. This establishes the uniqueness of u. Remark 3.4. Note that for the solution to 1.1), 3.5) we have that b 1 u ) b 1 v R d)) 1+cv R d) <cu 0) < 0, x d, b 2 u ) b 2 v L d)) 1+cv L d) >cu 1) > 0, x d. Recall that we also have b 1 u ) > 1 cu 1),x d and b 2 u ) > 1+cu 0),x d. Combining these we get 3.7a) 3.7b) b 1 u ) >θ 1 := max{ cu 0), 1 cu 1)}, x d, b 2 u ) >θ 2 := max{cu 1), 1+cu 0)}, x d. In the next lemma we state parameter-explicit pointwise estimates on the derivatives of the solution to 1.1), 3.5). Lemma 3.5. Let u be the solution of 1.1), 3.5), then,forall1 k 3, u k) x) C k, x Ω Ω +.

10 112 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN Proof. Use the argument from the proof of the previous lemma to establish that u C. Then use the differential equation 1.1a) to get the bounds on the second and third derivatives of u. 4. A priori bounds on the singular component Since the solution u of 1.1) and the regular component v defined in 2.8) are uniquely defined, we can define the discontinuous singular component w implicitly by u = w + v and u 4.1a) + bx, u )u = f, x d, 4.1b) u C 1 0, 1), u 0) = A, u 1) = B. Since u and v are unique, we have that w = u v u + v C, and the singular component w is the solution of w + bx, u )w +cv )w =0, x d, w 0) = w 1) = 0, [w ]d) = [v ]d), [w ]d) = [v ]d), [ω]d) :=ωd+) ωd ). Let L denote the linear differential operator, which is defined as L ω := ω + ax)ω + bx)ω, where ax) α 1 < 0, x<d, ax) α 2 > 0, x>d, and, for α =min{α 1,α 2 }, α 2 4b > 0, x d. The differential operator L satisfies the following comparison principle. Lemma 4.1. Suppose that a function ω C 0 Ω) C 2 Ω Ω + ) satisfies ω0) 0, ω1) 0, [ω ]d) 0, and L ωx) 0, for all x Ω Ω +,thenωx) 0, for all x Ω. Proof. Follow the proof of the corresponding result in [3], but include the zero-order term in the proof. Introduce the function vx), defined by ωx) =e αx) x d /2) vx), where αx) =α 1,x<d,αx) =α 2,x>d. Hence, for x Ω, L ω = e θx) x d /2) v +a + α 1 )v + α aα ) b)v, and for x Ω +, L ω = e αx) x d /2) v +a α 2 )v + α2 2 4 aα ) b)v. Assume that max Ω v = vq) > 0. With the above assumption on the boundary values, either q Ω Ω + or q = d. Ifq Ω,then ) L ωq) =e α 1d q)/2) v q)+b α2 1 4 )vq) < 0, which is a contradiction. If q Ω +, then an analogous argument also leads to a contradiction. The only possibility remaining is that q = d. Notethat [v]d) = [ω]d) =0and[ω ]d) =[v ]d) α 1+α 2 2 vd). Since d is where v takes its maximum value, v d ) 0, v d+) 0, which implies that [v ]d) 0. This implies that [ω ]d) < 0, which is a contradiction.

11 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 113 Lemma 4.2. Assume 3.5). For each integer k, satisfying 1 k 3, the solutions v and w of 2.8) and 4.1), respectively, satisfy the following bounds: v C, v k) Ω Ω + C1 + 2 k ), [v ]d) C, [v ]d) C, [v ]d) C, { w k) C x) k e d x)θ1/, x Ω, C k e x d)θ2/, x Ω +, where C is a constant independent of, andθ 1,θ 2 are given in 3.7). Proof. Define v 2 to be such that v := v 0 + v v 2. From above such a function exists and is unique. Note that v 2 is the solution of v 0 + v v 2 )+bx, v )v 0 + v v 2)=f = bx, v 0 )v 0 ), x d. Hence v + bx, v )v 1 + v 2)= cv 1 + v 2 )v 0, which can be written in the form v + bx, v )v 1 + v 2)= cv 1 + v 2 )v 0. From the definition of v 1,wehavethat bx, v)v 1 = bx, v) bx, v 0 + v 1 ) ) v 1 cv 0v 1 v 0. Inserting this into the equation above and simplifying, shows that v 2 satisfies the following problem: 4.2a) v 2 + bx, v)v 2 + cv 0 + v 1)v 2 = v 1, x d, 4.2b) v 2 0) = v 2 d) =v 2 1) = 0. Note that v 0 + v 1 > 0 and, for sufficiently small, b2 x, v) 4cv 0 + v 1 ) > 0. We rewrite 4.2a) in the form v 2 = gx, v 2 ):= bx, v)v 2 + cv 0 + v 1)v 2 + v 1 ). Define M 1 := v 1 and β 1 := min cv 0 + v 1) > 0. Check that αx),βx) defined by β 1 αx) =β 1 βx) =M 1 are lower and upper solutions. Thus v 2 M 1. β 1 Note that v 2 + bx, v)v 2 = g 1 := v 1 cv 0 + v 1 )v 2, which implies that v 2 Cx g 1,x<d.Thus v 20) C, and using integration we have v 2x) C, x < d. The bounds on the derivatives of v 2 follow. Now we estimate the singular term. Note that u + bx, u )u = f = v + bx, v )v, x d. Hence w +bx, u ) bx, v ))v + bx, u )w = w + bu )w +cv )w =0, x d, and w 0) = 0, w 1) = 0, w d )=u d ) v d ),w d + )=u d + ) v d + ). Choose sufficiently small so that b 2 x, u ) 4cv > 0. Then we can apply the arguments from the linear problem [3] and Lemma 4.1 to get bounds on w and its

12 114 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN derivatives separately on Ω and Ω +. Note that we require sufficiently small so that the barrier function { Ce d x)θ 1 Bx) = /, x Ω, Ce x d)θ2/, x Ω +, satisfies the inequalities B x)+b 1 u )B + cv B 1 {θ 1θ 1 b 1 u ) )+cv )}B <0, x Ω, B x)+b 2 u )B + cv B 1 {θ 2θ 2 b 2 u )) + cv )}B <0, x Ω Existence of discrete solutions The domain Ω is subdivided into four subintervals 5.1a) [0,d σ 1 ] [d σ 1,d] [d, d + σ 2 ] [d + σ 2, 1]. The transition points σ 1 and σ 2 are defined by { d 5.1b) σ 1 =min 2, 2 } { 1 d ln N, σ 2 =min θ 1 2, 2 } ln N, θ 2 where θ 1 =max{ cu 0), 1 cu 1)} and θ 2 =max{cu 1), 1+cu 0)} as defined in 3.7). On each of the four subintervals a uniform mesh with N 4 mesh-intervals is placed. The mesh points are denoted by Ω N = {x i } N 0,wherex 0 =0, x N = 1, x N/2 = d. The fitted mesh method for problem 1.1) is: find a mesh function U such that 5.2a) δ 2 U x i )+bx i,u x i ))DU x i ) = fx i ) for all x i Ω N, 5.2b) U 0) = u 0), U 1) = u 1), 5.2c) D U d) = D + U d), where δ 2 Z i = D+ Z i D { Z i D and DZ i = Z i, i < N/2, x i+1 x i 1 )/2 D + Z i, i > N/2. Here D + and D are the standard forward and backward finite difference operators, respectively. This is a nonlinear finite difference scheme. Let G : R N+1 R N+1 be a mapping associated with this finite difference scheme. For mesh function Y, we have an associated vector Y R N+1,where Y i = Y x i ). Let Y 0), i =0, δ GY ) i = Y i + bx i,y i )DY i, δ Y i, i = N/2, Y 1), i = N. i N/2, 0 <i<n, We also define a vector F by { A, 0,B, i=0,n/2,n, F i = fx i ), otherwise. The finite difference scheme 5.2) can then be written in the form GU = F. Definition 5.1. Given any vector H R N+1,alower mesh solution V for the problem GW = H is a mesh function, which satisfies GV H.

13 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 115 There is an analogous definition for an upper mesh solution to GW = H. Theorem 5.2. If Φ and Ψ are, respectively, lower and upper mesh solutions for the problem GW = H, with the additional properties that 1 <cφ < 1, 1 <cψ < 1, and Φx i ) Ψx i ), x i Ω N, then there exists a solution to GW = H and Φx i ) W x i ) Ψx i ), x i Ω N. Proof. We follow the argument from Lorentz [11]. Let Φ 1, Φ 2 be two lower mesh functions. Define the mesh function Φ 3 by Φ 3 x i ) := max{φ 1 x i ), Φ 2 x i )}. At some point x j, we assume, w.l.o.g., that Φ 3 x j )=Φ 1 x j ). Note that Φ 3 x i ) Φ 1 x i ), x i. For any x j, δ 2 Φ 3 x j )+bx j, Φ 3 )DΦ 3 x j ) δ 2 Φ 1 x j )+bx j, Φ 1 )DΦ 1 x j ) Hx j ), x j d, δ 2 Φ 3 x j ) δ 2 Φ 1 x j ) Hd), x j = d, Φ 3 0) H0), Φ 3 1) H1). Then Φ 3 is also a lower mesh solution. Let L = {φ : Gφ H, Φ φ Ψ} be the set of all possible lower mesh solutions. Define Ux i ):=sup φ L {φx i )}. First note that U L exists and GU H. Assume that we do not have equality, then there exists some j such that GUx j ) >Hx j ). If U Ψ, construct a new vector Y = U + γδ i,j,γ>0. Then γ can be chosen sufficiently small such that GY = GU + cy U)DU H. Hence, Y L, U < Y, which is a contradiction. Define the mesh functions V L and V R to be the solutions of the following discrete nonlinear problems: 5.3a) 5.3b) 5.3c) 5.3d) L N left V L := { δ 2 + b 1 V L )D } V L x i )= δ 1,x i Ω N Ω, V L 0) = v 0), V L d) =v d ), L N rightv R := { δ 2 + b 2 V R )D +} V R x i )=δ 2,x i Ω N Ω +, V R 1) = v 1), V R d) =v d + ). In an analogous fashion to Theorem 5.2 we have the following Theorem 5.3. If Φ and Ψ are two mesh functions such that Φ0) V L 0) Ψ0), Φd) V R d) Ψd), L N leftφ L N leftv L L N leftψ, Φd) V L d) Ψd), Φ1) V R 1) Ψ1), L N rightφ L N rightv R L N rightψ with the additional properties that 1 <cφ < 1, 1 <cψ < 1, Φx i ) Ψx i ), x i Ω N, then there exists a solution to 5.3) and Φx i ) V L x i ) Ψx i ), x i Ω N Ω, Φx i ) V R x i ) Ψx i ), x i Ω N Ω +.

14 116 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN From 2.4) u 0) v d ) 0andu 1) v d + ) 0, we can use the following mesh function: A 1 x i )=u 0), x i d, A 1 x i )= v 1)x i d),x i d, 1 d B 1 x i )=v 0)1 x i d ),x i d, B 1 x i )=u 1), x i d, to show that V L and V R exist using the previous theorem. Hence A 1 V L B 1 0, 0 A 1 V R B 1 and thus b 1 V L ) 1, b 2 V R ) 1. Lemma 5.4. Assume 3.5). Given any V L,V R,solutionsto5.3), we have that V L x i ) v x i ) CN 1 x i, x i Ω N Ω, V R x i ) v x i ) CN 1 1 x i ), x i Ω N Ω +, where v is the unique regular component defined in 2.8). Proof. We outline the proof for the first inequality. An analogous argument will establish the second inequality. For all x i Ω N Ω, { δ 2 + b 1 V L )D } V L v )=v + b 1 v )v { δ 2 + b 1 V L )D } v ) = v δ 2 v )+b 1 V L )v D v )+b 1 v ) b 1 V L ))v = v δ 2 v )+b 1 V L )v D v )+cv V L )v. Introduce the linear difference operator M N V Z := δ 2 + b 1 V L )D + cv ) Z. Note that cv C and so, by Lemma 8.3 in the appendix, this finite difference operator satisfies a discrete comparison principle, provided that is sufficiently small such that 5.4) b 2 1V L ) 4cv > 0, x i Ω N Ω. Using the bounds in Lemma 4.2 and standard local truncation error estimates, we get M V N V L v )x i ) CN 1. With the two functions ψ ± x i )=CN 1 x i ± V L v )x i ), and the discrete comparison principle the proof is completed in the usual way. To establish uniqueness for the discrete regular component V L,wefirstobtain bounds on the discrete derivative of any possible regular component V L. Lemma 5.5. Assume 3.5). For any V L, we have the following -uniform bounds D V L x i ) C, x i d σ 1 and D V L x i ) C 1+ N 1 ),d σ 1 <x i d. Proof. Note that D V L x i )=D V L v )x i )+D v x i ) v x i )+v x i ). We also have v C and, as in [4, page 60], D v x i ) v x i ) CN 1. Hence, D V L x i ) D V L v )x i ) + C.

15 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 117 On 0,d σ 1 ], using the previous bound on V L v )x i ), wegetthat D V L v )x i ) C. As in [4, pp. 61 and 62], D V L v )x i ) CN 1 on d σ 1,d], wherewenotethatweuse b 1 V L )x i ) b 1 V L )x i 1 ) = c V L x i ) V L x i 1 ) V L x i ) v x i ) + v x i ) v x i 1 ) + V L x i 1 ) v x i 1 ) CN 1. This completes the proof. Lemma 5.6. There exist unique solutions V L and V R to the discrete problems 5.3) and 3.5). Proof. Assume the contrary. Let V + L,V L be two mesh solutions, then δ 2 V L + b 1V L )D V L = δ2 V + L + b 1V + L )D V + L,x i <d, V + L V L )0) = V + L V L )d) =0. Thus δ 2 V + L V L )+b 1V + L )D V + L V L )+cd V L V + L V L ) = 0. From the previous lemma and Lemma 8.3, the linear difference operator L N Z := δ 2 Z + b 1 V + L )D Z +cd V L )Z satisfies a discrete comparison principle. This guarantees uniqueness. We are now ready to state the discrete counterpart to Theorem 3.3. First we define the discrete barrier functions Ξ L x i ; γ), Ξ R x i ; γ) asthesolutionsof δ 2 Ξ L + 1+cγ)D Ξ L = δ 1,x i 0,d), Ξ L 0) = u 0), Ξ L d) =γ, δ 2 Ξ R +1+cγ)D + Ξ R = δ 2, x i d, 1), Ξ R d) =γ, Ξ R 1) = u 1). Theorem 5.7. There exists a solution U N to the discrete problem 5.2), 3.5) and V L x i ) U N x i ) Ξ L x i ; v R d)), x i d, Ξ R x i ; v L d)) U N x i ) V R x i ), x i d. Proof. The argument is the discrete analogue of the argument given to establish the existence of the continuous solution. Define U L x i ; γ),u R x i ; γ) tobethesolutions of the problems δ 2 U L + b 1 U L )D U L = δ 1,x i 0,d), U L 0) = u 0), U L d) =γ, δ 2 U R + b 2 U R )D + U R = δ 2,x i d, 1), U R d) =γ, U R 1) = u 1). From assumption 3.5), we have that for all γ [v L d),v R d)] both problems have asolutionu L x i ; γ),u R x i ; γ) and V L x i ) U L x i ; γ) Ξ L x i ; γ), Ξ R x i ; γ) U R x i ; γ) V R x i ). Note the following: D V L d) = v Ld )+D V L d) v Ld)) = O)+ON 1 ), D + Ξ R d; γ) = ξ Rd + ; γ)+d + Ξ R d; γ) ξ Rd + ; γ)) 1 + cγ)u 1) γ) δ 2 1 d) + O)+ON 1 ). 2 Hence, for sufficiently small and N sufficiently large, D + V R d) D Ξ L d; v R d)) and D + Ξ R d; v L d)) D V L d).

16 118 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN Use the following: { { VL x Φ= i ), if x i d, ΞL x Ψ= i ; v R d)), if x i d, Ξ R x i ; v L d)), if x i d, V R x i ), if x i d, as lower and upper mesh solutions to establish the existence of U. Remark 5.8. If there exists a solution U to the discrete problem 5.2), 3.5) with the additional property that DUx i ) C 1+ N 1 ) 5.5a), d σ 1 <x i <d+ σ 2, 5.5b) DUx i ) C, otherwise, then this solution is unique. This follows by observing that if there are two solutions U 1 and U 2 satisfying 5.5), then δ 2 U 2 U 1 )+bx, U 2 )DU 2 U 1 )+cdu 1 U 2 U 1 )=0, and so by Lemma 8.4, U 2 U 1 )=0. Given any discrete solution U of 5.2), 3.5) we can define W L and W R using W L = U V L,x i d, W R = U V R,x i d. These functions W L : Ω N [0,d] R and W R : Ω N [d, 1] R exist, are uniformly bounded, and satisfy the following system of finite difference equations: 5.6a) δ 2 +b 1 W L )+cv L )D + cd V L )W L )=0,x i Ω N Ω, 5.6b) δ 2 +b 2 W R )+cv R )D + + cd + V R )W R )=0,x i Ω N Ω +, 5.6c) W L 0) = 0, W R 1) = 0, 5.6d) W R d)+v R d) =W L d)+v L d), 5.6e) D + W R d)+d + V R d) =D W L d)+d V L d). 6. Error analysis In the next theorem, we show that the discrete layer functions are small in a discrete sense) exterior to the interior layer region. Theorem 6.1. When σ 1 = 2 θ 1 ln N and σ 2 = 2 θ 2 ln N, we have that W L x i ) CN 1, x i d σ 1 ; W R x i ) CN 1, x i d + σ 2, where W L and W R are the solutions of the problems defined in 5.6). Proof. Considerthecaseofx i d. Let Bx i )=Π i j=11 + θ 1h j 2 )j, h i = x i x i 1. Then D + Bx i )= θ 1 2 Bx i), 1+ θ ) 1h i D Bx i )= θ Bx i), and 1+ θ ) 1h i δ 2 Bx i )= θ h ) i+1 Bx i ). h i

17 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 119 Hence 1+ θ 1 h i ) ) 2 δ 2 + b 1 U)D + cd V L Bxi ) < θ 1 2 θ 1 + b 1 U)+ 2 θ 1 cd V L )1 + θ 1h i 2 ) ) Bx i ) < 0. Then W x i ) C Wd) Bx i) Bd), x i d. Thus, for x i d σ 1, W x i ) C 1+ θ ) N/4 1h, h = 4σ 1 2 N. Then, if σ 1 = 2 θ 1 ln N, we have that W L CN 1, x i d σ 1 W R CN 1,x i d + σ 2. Thus, when σ 1 = 2 θ 1 ln N, wehavethat 6.1a) W L x i ) w x i ) W L x i ) + w x i ) CN 1 + Ce θ 1σ 1 / CN 1, x i d σ 1. Similarly, for σ 2 = 2 θ 2 ln N, weobtain 6.1b) W R x i ) w x i ) CN 1, x i d + σ 2. These bounds and the bounds given in Lemma 5.4 together imply that the numerical approximations are essentially first order convergent at the mesh points outside the interior layer region d σ 2,d+σ 2 ). To obtain an error estimate at the mesh points in the interior layer region, we assume the following implicit restriction on the data. Assumption 2. Assume that the problem data for problem 1.1) are such that 6.2) b 2 x i,u x i )) 4cu x i ) > 0, x i d. As in Remark 3.4, from the bounds in Theorem 5.7, we have the strict inequality bx i,u ) >θ>0. Hence, assumption 6.2) can be satisfied for certain problem data. Theorem 6.2. Assume that N is sufficiently large and is sufficiently small, independently of each other. Assume further that 3.5) and 6.2) hold. The continuous solution u of problem 1.1) and any set of discrete solutions U of 5.2) satisfy the following asymptotic error bound U u N Ω CN 1 ln N) 2, where C is a constant independent of N and. Proof. Consider first the case of σ 1 = 2 θ 1 ln N and σ 2 = 2 θ 2 ln N. By Lemma 5.4 and 6.1), the result is valid for mesh points outside the interior layer region d σ 1,d+ σ 2 ). Hence 6.3) U d σ 1 ) u d σ 1 ) CN 1 and U d + σ 2 ) u d + σ 2 ) CN 1.

18 120 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN On the other hand, in the layer region d σ 1,d) d, d + σ 2 ), we have that δ 2 u U ) + bxi,u )D u U ) = δ 2 u u )+bx i,u )Du bx i,u )u = δ 2 u u )+ bx i,u ) bx i,u ) ) u + bx i,u ) Du u ) = δ 2 u u )+ cu u ) ) u + bx i,u ) Du u ). We introduce the linear difference operator MU N Z := δ 2 + bx i,u )D + cu ) Z, xi d, MU N Zd) := D + Z D Z)d). At the mesh point x i = d, [u ]=[DU ] = 0, and so D + D )U u ) = D D + )u )+[u ] u d) D + u d) + u d) D u d) Ch u xi 1,x i+1 ), where h = 4σ N and σ =max{σ 1,σ 2 } is the fine mesh size. Hence, using u = v +w and the bounds in Lemma 4.2, MU N u U) = δ 2 u u )+bx i,u ) Du u ), xi d, MU N u U)x i ) Ch 1+ 1 ) 4lnN e N N/2 i, x 2 i d σ 1,d+ σ 2 ). Note that e 4lnN N 1+ 4lnN ) 1 N and hence, we have a truncation error bound of the form MU N u U)x i ) Ch lnN N ) N/2 i ), x i d σ 1,d+ σ 2 ). The finite difference operator MU N satisfies a discrete comparison principle see Lemma 8.4 in the appendix), provided that 6.2) is assumed. Consider the discrete barrier function x Ψ= CN 1 C N i d σ 1 ) 1 σ 2 σ 1, x i Ω N d σ 1,d], 2 d+σ 2 ) x i σ 2, x i Ω N d, d + σ 2). Form the product λx i )Ψx i ), where bu) >θ=min{θ 1,θ 2 }, and we define 1 + θ 1h 1 2 )i N/2, x i Ω N d σ 1,d), λx i )= 1 + θ 2h 2 2 )N/2 i x i Ω N d, d + σ 2 ). Then λx i+1 )δ 2 Ψ+ãλx i 1 )D Ψ+ bψ, x i Ω N MU N d σ 1,d), λψ)x i )= λx i 1 )δ 2 Ψ+âλx i+1 )D + Ψ+ˆbΨ, x i Ω N d, d + σ 2 )

19 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 121 where, for N sufficiently large, and using the strict inequality bx i,u) >θ, Also, b λx i 1 ) ˆb λx i+1 ) ã = b 1 U)+θ 1 + θ2 1h 1 4 < bu)+θ 1 + CN 1 ln N<0, x i Ω N d σ 1,d), â = b 2 U) θ 2 θ2 2h 2 4 > 0, x i Ω N d, d + σ 2 ), = θ θ 1b 1 U) + cu 1+ θ ) 1h < 1 ) cu θ CN 1 ln N < 0, x i Ω N d σ 1,d), = θ2 2 4 θ 2b 2 U) + cu 1+ θ ) 2h 2 2 < 1 ) cu θ CN 1 ln N < 0, x i Ω N d, d + σ 2 ). 2 D + 1 σ2 Ψx i )=CN, x i >d, 2 D 1 σ2 Ψx i )=CN,x i <d, σ 2 σ 1 and δ 2 Ψx i )=0,x i d. Hence, for x i d, MU N λψ) C ã 1+ θh ) 1 λx i ) N 1 σ 2 2 MU N u U). Noting that θ 1 h 1 = θ 2 h 2, we have the following bound at the point of discontinuity x i = d, MU N λψ)=λd h) D + Ψd) D Ψd) θ ) 1 + θ 2 Ψd) C h 2 2 M U N u U)d). Applying the discrete comparison principle to Ψ ± U u ) over the interval [d σ 1,d+ σ 2 ], we get U x i ) u x i ) CN 1 + C N 1 σ 2 2 CN 1 ln N) 2. We complete the proof by considering the case where at least one of the two transition points σ 1,σ 2 takes the value d 1 d 2 or 2. In all such cases 1 C ln N. Apply the above argument across the entire domain Ω N to complete the proof. Remark 6.3. Note that 6.2) is a restriction on the problem class. In this remark, we show that this restriction is satisfied if is sufficiently small and the data is further restricted. We first examine the restrictions placed on the data when we require that b 2 x, u ) 4cu > 0, x 0, 1). Note that u = v + w and so for sufficently small, u > 0, x 0, 1); u x) > 0,x<d, u x) < 0,x>d.Thus b 2 x, u ) 4cu )x) max{1 cu d)) 2, 1 + cu d)) 2 } 4cu d), x 0, 1).

20 122 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN Note also that d u d) u 0) = δ 1 +1 cu t))u t)) dt 0 = dδ cu d) c 2 u 2 2c d) 2cu 0) + c 2 u 2 0) ) = dδ c 1 cu 0)) 2 1 cu d)) 2), and so 1 cu d)) 2 4cu d) > 4cδ 1d +31 cu d)) 2 21 cu 0)) 2. If η <cu 0) <cu 1) <η,then1 η<1 cu 1) < 1 cu d) < 1 cu 0) < 1+η. This means that 1 cu d)) 2 4cu d) > 4cδ 1 d +1 η) 2 8η =4cδ 1 d η + η 2. Hence, we require the data to be such that 4c max{δ 1 d, δ 2 1 d)} +1 10η + η 2 > 0, where η <cu 0) <cu 1) <ηand δ 1 d< u 0) and δ 2 1 d) <u 1). For example, if η =0.1,c =1,δ 1 d< u 0) < 0.1 andδ 2 1 d) <u 1) < 0.1, then the data constraints 3.5) and 6.2) in Theorem 6.2 are both satisfied. 7. Numerical results To solve the nonlinear difference scheme 5.2) we use the continuation method described in [8]. Table 1 displays the computed rates of convergence p N and the uniform rates of convergence p N, using the double mesh principle see [4] for details on how these quantities are calculated), when the numerical method 5.2) is applied to the problem 1.1) with u0) = 0.5,δ 1 =0.8,u1) = 0.7,δ 2 =1.2,d=0.5,c=1. Note that the conditions in 3.5) are satisfied for this data. The computed rates of convergence are in line with the theoretical rates of convergence established in Theorem 6.2. Table 1. Table of computed orders p N and computed -uniform orders p N for the numerical method 5.2) applied to problem 1.1) with u0) = 0.5,δ 1 =0.8,u1) = 0.7,δ 2 =1.2,d=0.5,c=1. Number of Intervals N p N

21 QUASILINEAR PROBLEM WITH INTERIOR LAYERS Appendix on discrete comparison principles Consider the following linear problem: L 1 u := u + pu + qu = f, x 0, 1), u0) = u 0,u1) = u 1, p α>0, q β, α 2 4β > 0. Lemma 8.1. If w0) 0, w1) 0, L 1 w 0, thenw 0,x [0, 1]. Proof. If β 0, then the standard proof by contradiction argument applies. For β>0, use the transformation w = e α 2 x v. Consider the corresponding discrete problem on an arbitrary mesh Ω N, L N 1 U := δ 2 U + pd + U + qu = f, x i Ω N, U0) = u 0,U1) = u 1, p α>0. If q 0, then the standard discrete comparison principle holds for L N 1. Below we extend this comparison principle to the case of q>0. As in the proof of the continuous operator, consider the transformation W x i )=λx i )V x i ), where λ will be specified below. For any such mesh function λ, D + λx i )V x i )) = λx i+1 )D + V x i )+Vx i )D + λx i ), D λx i )V x i )) = λx i 1 )D V x i )+V x i )D λx i ), δ 2 λx i )V x i )) = V x i )δ 2 λx i )) + 1 hi λxi+1 )D + V x i ) λx i 1 )D V x i ) ), where h i =h i + h i+1 )/2. Hence δ 2 λx i )V x i ))=V x i )δ 2 λx i ))+ λx i+1) λx i 1 )) h i D + V x i )+λx i 1 )δ 2 V x i )). Then, for W x i )=λx i )V x i )wehave where δ 2 W + pd + W + qw)x i )= ˆp = hi 1+ p h i [ λx i 1 )δ 2 V + λx i+1 )ˆpD + V +ˆqV ] x i ), ) λx ) i 1) λx i+1 ) <, ˆq = δ 2 λx i )) + pd + λx i )) + qλx i ). In the following three lemmas, we assume that h i CN 1 ln N and that is sufficiently small independently of N) andn is sufficiently large independently of ). Lemma 8.2. Assume that p α > 0. Under any one of the following three assumptions: 1) qx i ) C 2, x i, 2) p>α>0, α 2 4q > 0 and Ω N is a uniform mesh with h/ CN 1 ln N, 3) Ω N =Ω N is a piecewise uniform mesh which uses a uniform mesh in each of the subintervals [0,σ and [σ, 1] with a fine mesh step h CN 1 ln N andacoarsemeshsteph CN 1 ln N) and qx i ) C N) 1 ),x i <σ, qx i ) C 2,x i σ;

22 124 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN then for any mesh function W,ifW0) 0, W1) 0, L N 1 W 0, thenw x i ) 0, x i Ω N. Proof. We employ functions of the form λx i )=Π i j=1 1 + θ jh j ) 1, which satisfy D + λx i )= θ i+1 λx i+1 ), D λx i )= θ i λx i ), D + λx i ) D λx i )=θ i θ i+1 )+θ i+1 θ i h i+1 )λx i+1 ), 1+ p h i λx i 1) λx i+1 ) = h ) i p hi h i θ i + h i+1 θ i+1 + h i θ i h i+1 θ i+1 ), [ ) )] ˆqx i )=λx i+1 ) 1 + θ i+1 h i+1 ) θ i + qx i ) θ i+1 + px i ). hi hi In each of the three cases, we choose the θ j so that ˆp 0, ˆq <0 and then the normal proof by contradiction argument can be applied. Assume W i > 0andlet V j =maxv i > 0, then D + V j 0,D V j 0, δ 2 V j 0, L N W j 0. Case 1. Take, θ j = θ = 2C 2+1 α. Thenif is sufficiently small independently of N) andn is sufficiently large independently of ), 1+ p h i λx i 1) λx i+1 ) h i α 2θ1 + θh)) 0, h =min{hi,h i+1 }, ˆq λx i+1 ) 2θ 2 ) + q αθ + qθh i+1 < 0. Case 2. Take θ j = α 2. Then, if N is sufficiently large independently of ), 1+ p h i λx i 1) λx i+1 ) = h p α1 + αh ) 2 ) 0, using the strict inequality p>αand ˆq λx i+1 ) θ 2 + q αθ + qθh ) < 0, using the strict inequality 4q < α 2. Case 3. If N 1, then follow the argument in Case 1. Otherwise, take θ i = ζ 1,i N/2, N θ i = ζ 2,i>N/2, where ζ 1 = 2C 1 +1, αζ 2 >q+ ζ 1. α In the layer region, when i<n/2andh CN 1 ln N, 1+ p h i λx i 1) λx i+1 ) h p ζ 1 N 2 + ζ ) 1h N ) 0, for any ζ 1, ˆq λx i+1 ) = 1 N qn pζ 1)+ ζ 1 N qh + ζ 1 N ) < 0, using qn C 1 N +1) 2C 1. Outside the layer region, when i>n/2, for sufficiently small, and 1+ p h i λx i 1) λx i+1 ) = H p ζ2 2 + ζ 2 H)) 0 ˆq λx i+1 ) q αζ 2 + ζ 2 2ζ 2 + qh) < 0.

23 QUASILINEAR PROBLEM WITH INTERIOR LAYERS 125 At the transition point, when i = N/2 andn is sufficiently large, 1+ p h i λx i 1) λx i+1 ) = h p hζ 1 + NHζ 2 + hζ 1 ζ 2 H)) 0, ˆqx i ) λx i+1 ) =1+ζ 2H)qx i )+ζ 1 ) ζ 2 α + N) < 0, using the strict inequality αζ 2 >q+ ζ 1. Inthecaseofanegativeconvectivecoefficientp<0, we employ the operator L N 2 U := δ 2 U + pd U + qu. Lemma 8.3. Assume that p α<0 and any one of the following: 1) qx i ) C 2, x i, 2) p < α < 0, α 2 4q > 0 and Ω N is a uniform mesh with h/ CN 1 ln N, 3) Ω N =Ω N is a piecewise uniform mesh and qx i ) C N) 1 ),x i > 1 σ, qx i ) C 2,x i 1 σ; then for any mesh function W,ifW0) 0, W1) 0, L N 2 W 0, thenw x i) 0, x i Ω N. Proof. The proof is analogous to the case of p > 0. As before, if W x i ) = λx i )V x i ), then δ 2 W + pd W + qw)x i )= [ λx i+1 )δ 2 V + pλx i 1 )D V + qv ] x i ), where p = hi λxi+1 ) λx i 1 ) 1 p h ) i ), q = δ 2 λx i )) + pd λx i )) + qλx i ). Consider functions of the form λx i )=Π i j=1 1 + θ jh j ), which satisfy D + λx i )=θ i+1 λx i ), D λx i )=θ i λx i 1 ), D + λx i ) D λx i = θ i+1 θ i )+θ i+1 θ i h i )λx i 1 ), λx i+1 ) λx i 1 ) 1 p h ) i = h ) i p + hi h i θ i + h i+1 θ i+1 + h i θ i h i+1 θ i+1 ), ) ) 1 λx i 1 ) ˆq 1 + θ ih i ) θ i+1 + q θ i + α. h i hi The proof is analogous to the previous proof. In the case of a discontinuous convective coefficient p< α <0,x < d,and p>α>0,x>d we use the upwind finite difference operator L N 3 U := δ 2 U + pdu + qu. Lemma 8.4. Under any one of the following assumptions: 1) qx i ) C 2, x i, 2) α 2 4q > 0 and Ω N is a uniform mesh with h/ CN 1 ln N, 3) Ω N =Ω N is a piecewise uniform mesh, and qx i ) C N) 1 ),d σ<x i <d+ σ, qx i ) C 2, otherwise;

24 126 P. A. FARRELL, E. O RIORDAN, AND G. I. SHISHKIN then for any mesh function W,ifW 0) 0, W1) 0, L N 3 W 0, D + W d) D W d), we have W x i ) 0, x i Ω N. Proof. Use functions of the form: λx i )=Π i j=11 + θ j h j ),i N/2, λx i )=λd)π i j=i N/2 1 + θ jh j ) 1,i>N/2. In all three cases, using the choices from the previous two lemmas, we get λx N/2 1 ) = λx N/2+1 ). Then δ 2 W d) =λx N/2 1 ) δ 2 V d) 2 θh ) V d). If W>0andmaxV = V d) > 0, then δ 2 W d) < 0, which is a contradiction. References [1] S. R. Bernfield and V. Lakshmikantham, An introduction to Nonlinear Boundary Value Problems. Academic Press, New York, 1974). MR :3393) [2] K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena, Springer- Verlag, New York, 1984). MR e:34090) [3] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O Riordan and G. I. Shishkin, Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Mathematics and Computer Modelling, 40, 2004, MR m:65146) [4] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman and Hall/CRC, Boca Raton, FL 2000). MR c:65003) [5] P. A. Farrell, J. J. H. Miller, E. O Riordan and G. I. Shishkin, On the non-existence of - uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems, Math. Comp., 67, 222), 1998, MR g:65072) [6] P. A. Farrell, J. J. H. Miller, E. O Riordan and G. I. Shishkin, A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation, SIAM J. Num. Anal., 33, 3), 1996, MR b:65086) [7] P. A. Farrell, J. J. H. Miller, E. O Riordan and G. I. Shishkin, Parameter-uniform fitted mesh method for quasilinear differential equation with boundary layers, Computational Methods in Applied Mathematics, Vol. 1, No. 2, 2001, MR f:65102) [8] P. A. Farrell, E. O Riordan and G. I. Shishkin, A class of singularly perturbed semilinear differential equations with interior layers, Math. Comp., 74, 2005, MR f:65067) [9] N. Kopteva and T. Linß, Uniform second order pointwise convergence of a central difference approximation for a quasilinear convection-diffusion problem, J. Comput. Appl. Math., 137, no. 2, 2001, MR h:65109) [10] T. Linß, H.-G. Roos and R. Vulanović Uniform pointwise convergence on Shishkin type meshes for quasilinear convection diffusion problems, SIAM J. Numer. Anal., 383), 2000, MR h:65082) [11] J. Lorenz, Nonlinear singular perturbation problems and the Engquist-Osher difference scheme, Report 8115, University of Nijmegen, [12] J. J. H. Miller, E. O Riordan and G. I. Shishkin, Fitted numerical methods for singular perturbation problems, World Scientific, Singapore), 1996). MR c:65002) [13] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection Diffusion and Flow Problems, Springer-Verlag, New York, 1996). MR a:65134) [14] G. I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic equations, Russian Academy of Sciences, Ural section, Ekaterinburg, 1992). in Russian) [15] R. Vulanović, A priori meshes for singularly perturbed quasilinear two-point boundary value problems, IMAJ.Numer.Anal.,21, 2001, MR i:65073)