Finite Element Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems

Transcription

1 Finite Element Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems Zhimin Zhang Department of Mathematics Wayne State University Detroit, MI Received 19 July 2000; accepted 24 February 2001 Superconvergence approximations of singularly perturbed two-point boundary value problems of reactiondiffusion type and convection-diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of ( 1 ln( +1)) p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. umerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. c 2002 Wiley Periodicals, Inc. umer Methods Partial Differential Eq 18: , 2002; Published online in Wiley InterScience ( DOI /num Keywords: convection-diffusion; reaction-diffusion; finite element method; superconvergence; singular perturbation; Shishkin mesh I. ITRODUCTIO Singularly perturbed problems are important in practice. umerical methods for singularly perturbed differential equations have attracted much attention in the scientific community. A 1996 book by Roos, Stynes, and Tobiska [1] provides an extensive list of literature on this topic. Two other 1996 books [2, 3] also give some literature from different perspectives. One of the computational difficulties in numerically solving singularly perturbed problems comes from the so-called boundary layer behavior, the solution varies rapidly in a very thin layer near the boundary. umerical solutions by traditional finite difference and finite element methods are oscillatory and inaccurate in the presence of boundary layers. Special cares must be taken. Correspondence to: Zhimin Zhang, Department of Mathematics, Wayne State University, Detroit, MI ( zzhang@math.wayne.edu) Contract grant sponsor: ational Science Foundation Contract grant number: DMS , DMS , and IT c 2002 Wiley Periodicals, Inc.

2 SIGULARLY PERTURBED PROBLEMS 375 Model problems of this article are singularly perturbed reaction-diffusion equations and convection-diffusion equations in the one-dimensional setting. They will be solved by the finite element method of any fixed order p. It is well known that straightforward approximation of singularly perturbed problems by the finite element method on a quasi-uniform mesh will produce oscillations when boundary layers are present. There are mainly three strategies to handle the boundary layer. The first one is to use stabilizations (streamline-diffusion) or modifications (up-winding) as in the finite difference method; the second one is to choose special test/trial functions, usually the exponentially fitted L-splines; and the third one is to use anisotropic meshes. The reader is referred to [1] [16] for the literature. Because the exponentially fitted L-splines are not likely to be generated to higher dimensions, recent researches are concentrated on streamlinediffusion type schemes combined with adaptive anisotropic meshes. One of this kind of meshes is the Shishkin mesh, a piecewisely uniform mesh with much smaller mesh size in the boundary layer region. The Shishkin mesh was first introduced in finite difference methods and has been discussed in detail in a 1996 book by Miller, O Riordan, and Shishkin [2]. The reader is also referred to a more recent survey article by Roos [11]. In this work, we consider the finite element method of any fixed order p on a modified Shishkin mesh. We establish a uniform superconvergence rate ( 1 ln ) p+1 in a discrete energy norm for both reaction-diffusion and the convection-diffusion problems under certain conditions. ote that the optimal rate is p. As a direct consequence, a near optimal convergent rate in the L 2 -norm is automatically established. umerical tests indicate that the theoretical analysis is sharp. The logarithm factors are not removable. As a byproduct, error estimates at the nodal points in the boundary layer region are also obtained. Here uniform convergent means that the convergence is uniformly valid with respect to the singular perturbation parameter. For problems considered in this article, regularity of the solution depends on the singular perturbation parameter. Hence, the traditional finite element analysis will result in error bounds depending on, even for a correct method and mesh. Special treatment must be applied in order to obtain -uniform error bounds. Shishkin meshes are simple, but they require a priori information in order to determine the thickness of the boundary layer(s). However, for more general situations such as linearization of nonlinear problems, equations with discontinuous coefficients that differ substantially in various parts of the domain, corner singularities, curved interior, or boundary layers, etc. in the higher dimensional cases, Shishkin meshes, and any other fixed meshes that rely on a priori information will be less attractive compared with the adaptive grid refinement generated by a posteriori error estimates. A realistic approach in practice may be starting with certain up-winding scheme, followed by an adaptive procedure to refine the mesh, and eventually resolving the boundary layer, and maybe locating some possible internal layers. We have seen some recent work in this direction [17, 18]. After the boundary layers have been resolved by adaptive procedure, our analysis in this article can be applied to the final mesh solution. We shall see in the forthcoming article that the result in the current work can be generalized to the two-dimensional problems when dealing with the smooth part of a (straight) boundary layer. The mesh design and the boundary layer function approximation behave very similar to the one dimensional case. The analysis in this work does provide us some insight on what happening in certain higher dimensional settings. Throughout the article, we use standard notation for Sobolev spaces, and we use C as a generic constant that is independent of and (or h). We also use sub-index to indicate a particular norm or inner product that is only used for a sub-domain. For example, v Ωi, (v, w) Ωi, etc.

3 376 ZHAG II. BASIC ASSUMPTIOS AD MAI RESULTS Our model problems are singularly perturbed two-point boundary value problems with homogeneous Dirichlet boundary conditions. Both reaction-diffusion and convection-diffusion problems are considered. The corresponding abstract variational problem is as follows: Find u H 1 0 (Ω) such that B (u, v) =(f,v) v H 1 0 (Ω), (2.1) where > 0 is the singular perturbation parameter. Usually, 1. The finite element method is to find u V,p H1 0 (Ω) such that B (u,v)=(f,v) v V,p, (2.2) where V,p is the space of standard C0 piecewise polynomials of degree p on a modified Shishkin mesh, a mesh that will be specified later in each case. We need the following conditions for convergence: (C1) B (v, v) v 2 for an energy norm (see (2.9) and (2.18)). (C2) B (u u,v)=0for all v V,p. (C3) There is a special interpolation I u V,p of u and a constant C independent of and, such that B (u Iu, v) Cg 1 () v v V,p. (C4) There is a discrete energy norm, (see (2.10) and (2.23)) such that u Iu, Cg 2 (), v, = v v V,p. Among the above conditions, (C2) is automatically satisfied by all conforming finite element methods, and (C1) is the coercivity that can be satisfied under certain assumptions. The other two conditions are not conventional and they should be uniformly valid with respect to with lim g l() =0, l =1, 2. ote that the conventional finite element analysis requires that B (u, v) C u v, which does not hold in the convection-diffusion case. However, a weaker condition (C3) is valid for both the reaction-diffusion and the convection-diffusion equations. The significants of (C3) and (C4) can be seen from the following abstract theorem. Theorem 2.1. Let u and u be solutions of problems (2.1) and (2.2), respectively, and let Conditions (C1) (C4) be satisfied. Then the following error estimate holds with the same constant C as in (C3) and (C4). u u, C(g 1 ()+g 2 ()). Proof. By (C1) (C3), there goes u I u 2 B (u Iu, u Iu) = B (u Iu, u Iu) Cg 1 () u Iu. Canceling u I u on both ends, we have u I u Cg 1 ().

4 This, combined with the triangle inequality and (C4), yields SIGULARLY PERTURBED PROBLEMS 377 u u, u I u, + I u u, = u I u, + I u u C(g 1 ()+g 2 ()). In light of Theorem 2.1, the task is then to establish optimal g l (). A. The Reaction-Diffusion Equation 2 u (x)+b(x)u(x) =f(x) in Ω=(0, 1), (2.3) with b(x) β 2 and β>0. Regularity Result. The solution u can be decomposed into u =ū + u,0 + u,1 (2.4) with the following regularity ū (k) L (I) C, u (k),0 (x) C k e βx/, u (k),1 (x) C k e β(1 x)/. (2.5) Here k is a fixed nonnegative integer, and C is a constant independent of. We may further assume that u,0 (1)=0, u,1 (0)=0, without destroying the regularity property of the decomposition. In fact, we may use ũ,0 (x) =u,0 (x) u,0 (1) and ũ,1 (x) =u,1 (x) u,1 (0), to replace u,0 (x) and u,1 (x), respectively, and absorb u,0 (1) and u,1 (0) into ū. The above regularity implies that b and f are sufficiently smooth. The regularity analysis is omitted here. For the detail, the reader is referred to reference [2, Chapter 6] where (2.5) is established for k 3. However, the result can be extended to any fixed k by the same technique. Because there are boundary layers at x =0and x =1, the mesh should be condensed in a neighborhood of each of these points. The construction is first to choose a positive τ 1 4 and to locate the transition points at τ and 1 τ. ext, the intervals (0,τ) and (1 τ,1) are each divided into equal subintervals, whereas the interval (τ,1 τ) is divided into 2 equal subintervals. Here we slightly modify the original Shishkin mesh by choosing ( 1 τ = min 4, ) (p +1.5) ln( +1). (2.6) β Hence, the element length in (0,τ) and (1 τ,1) is h i = h = τ/, whereas in (τ,1 τ) is h i = h =(1 2τ)/(2). Therefore, 1 4 h = max h i = 1 2τ i 2 < 1 2. In this article, we shall only consider the case when β (p +1.5) ln( +1) 1 4, (2.7)

5 378 ZHAG because otherwise, the problem is either regular (when is not small) or p and are large enough to catch the boundary layer. In either cases, the traditional analysis will work. Indeed, in practical situation, (2.7) does not post any serious restriction. For the reaction-diffusion problem, the bilinear form is and the energy norm is defined by B (u, v) = 2 (u,v )+(bu, v), (2.8) v 2 = v 2 + β 2 v 2, v 2 = 2 (v,v ), v 2 =(v, v). (2.9) Since b(x) β 2, (C1) is satisfied. The discrete energy norm, is defined by v 2, = v 2, + β 2 v 2, v 2, = 2 4 h i i=1 j=1 p w j v (x ij ) 2 (2.10) where x ij are Gaussian points in element Ω i =(x i 1,x i ) and w j > 0 are weights for the p-point Gaussian quadrature rule. Because the p-point Gaussian quadrature rule is exact for a polynomial of degree 2p 1, therefore, v 2,h = v 2 for v V,p and hence v, = v for v V,p. Theorem 2.2. Let u solve the reaction-diffusion problem (2.3) and satisfy the regularity (2.5). Let V,p be the C0 finite element space with piecewise polynomials of degree p on the modified Shishkin mesh. When (2.7) is satisfied, there exists a special interpolation I u V,p, such that ( ( ) ) p+1 ln( +1) B (u Iu, v) C + 1 p+1 v, (2.11) if ū V,p, then u I u, C ( ( ln( +1) ) ) p p+1 ; (2.12) B (u Iu, v) C ( ) p+1 ln( +1) v, (2.13) u Iu, C ( ) p+1 ln( +1), (2.14) where the constant C is independent of and. The proof of this theorem is postponed to the next section. Corollary 2.1. Assume the same hypothesis as in Theorem 2.2. Let u V,p be the finite element approximation of u. Then ( ( ) ) p+1 ln( +1) u u, C + 1 p+1 ; if ū V,p, then u u, C ( ) p+1 ln( +1),

6 SIGULARLY PERTURBED PROBLEMS 379 where the constant C is independent of and. Proof. In light of Theorem 2.2, (C3) and (C4) are both satisfied with g 1 () =g 2 () = ( ) p+1 ln( +1) + 1 p+1. By Theorem 2.1, the conclusion follows. If ū V,p, Theorem 2.2 ensures the satisfaction of (C3) and (C4) with g 1 () =g 2 () = ( ) p+1 ln( +1). The desired result then follows from Theorem 2.1. Corollary 2.2. Under the same assumption as in Corollary 2.1, if ū V,p, then for a nodal point x i in the boundary layer region, there holds (u u )(x i ) C ( ) p+1 ln( +1) ln( +1), where the constant C is independent of and. Proof. We only prove the case when x i (1 τ,1). The proof is similar for x i (0,τ).By the triangle inequality, we have 1 (u u )(x i ) = (Iu u )(x i ) = (Iu u ) (s) ds x i ( 1 1 ) 1/2 1 ) 1/ ds ( 2 (Iu u ) (s) 2 ds x i x i τ 2 I u u C ( ) p+1 ln( +1) ln( +1). Here we have used Iu u C ( ) p+1 ln( +1), which is a direct consequence of Theorem 2.2 (see the proof of Theorem 2.1). Remark 2.1. The condition ū V,p is introduced to isolate the approximation of the finite element space to the boundary layer term without seeing the influence from the regular part of the solution. Indeed, the approximation property for the regular part is well known. Remark 2.2. Although the pointwise error estimate is only obtained at nodal points in the boundary layer region, it is possible to use local analysis of Schatz and Wahlbin (see [12, Theorem 7.1] and [14, Chapter 4]) to develop error bound outside the boundary layer region. However, since the maximum error almost always appears in the boundary layer region, the result in Corollary 2.2 is enough for the practical purpose. B. The Convection-Diffusion Equation u (x)+a(x)u (x)+b(x)u(x) =f(x) in Ω=(0, 1), u(0)=0=u(1) (2.15)

7 380 ZHAG with a(x) α>0, In this case, the bilinear form is b(x) a (x) 2 > 0 x Ω. (2.16) B (u, v) =(u,v )+(au,v)+(bu, v). (2.17) It has been shown in [2, Chapter 9] that there is no essential loss of generality in assuming (2.16) rather than a(x) >α>0, b(x) β x Ω. Under the homogeneous Dirichlet boundary condition, we can show that Therefore, where (av,v)= 1 2 (a v, v). B (v, v) =(v,v )+((b 1 2 a )v, v) (v,v )+c 0 (v, v) = v 2 (2.18) c 0 = min(b(x) 1 x Ω 2 a (x)). Here we have the definition of the energy norm in this case and (C1) is satisfied. Regularity Result. The solution u can be decomposed into u =ū + u, (2.19) with the following regularity: ū (k) L (I) C, u (k) (x) C k e α(1 x)/. (2.20) Here k is a fixed nonnegative integer, and C is a constant independent of. Similarly as discussed in the reaction-diffusion case, we may further assume that u (0)=0. The above regularity implies that a, b, and f are sufficiently smooth. The regularity analysis is omitted here. For the detail, the reader is referred to [2, Chapter 8] where a regularity is proven for a simpler case when b =0and k 3. However, the technique can be extended to derive the regularity for any fixed k and b 0. In this case, there is only one boundary layer at x =1. The transition points is therefore locate at 1 τ, and (0, 1 τ) and (1 τ,1) are each divided into equal subintervals. Here ( 1 τ = min 2, ) (p +1.5) ln( +1). (2.21) α Hence, the element length in (1 τ,1) is h i = h = τ/, whereas in (0, 1 τ) is h i = h = (1 τ)/. Therefore, 1 2 h = max h i = 1 τ i < 1.

8 SIGULARLY PERTURBED PROBLEMS 381 For the same reason as explained in the reaction-diffusion case, we shall only consider the case when α (p +1.5) ln( +1) 1 2. (2.22) The discrete energy norm, is defined by v 2, = v 2,h + c 0 v 2, v 2,h = 2 h i i=1 j=1 p w j v (x ij ) 2. (2.23) Clearly, v, = v for v V,p. Theorem 2.3. Let u solve the convection-diffusion problem (2.15) and satisfy the regularity (2.20). Let V,p be the C0 finite element space with piecewise polynomial of degree p on the modified Shishkin mesh. When (2.22) is satisfied, there exists a special interpolation I u V,p such that ( (ln( ) ) p+1 +1) B (u Iu, v) C + 1 p v, (2.24), ( ) p+1 ln( +1) u Iu, C ; (2.25) if ū V,p, then ( ) p+1 ln( +1) B (u Iu, v) C v, (2.26) where the constant C is independent of and. The proof of Theorem 2.3 is postponed to the next section. Corollary 2.3. Assume the same hypothesis as in Theorem 2.3. Let u V,p be the finite element approximation of u. Then ( (ln( ) ) p+1 +1) u u, C + 1 p ; if ū V,p, then ( ) p+1 ln( +1) u u, C, where the constant C is independent of and. Proof. In light of Theorem 2.3, (C3) and (C4) are both satisfied with ( ) p+1 ln( +1) g 1 () = + 1 ( ) p+1 ln( +1) p, g 2() =. By Theorem 2.1, the conclusion follows. If ū V,p, Theorem 2.3 ensures the satisfaction of (C3) and (C4) with ( ) p+1 ln( +1) g 1 () =g 2 () =. The desired result then follows from Theorem 2.1.

9 382 ZHAG Corollary 2.4. Under the same assumption as in Corollary 2.3, then for a nodal point x i in the boundary layer region, there holds (u u )(x i ) C ( ) p+1 ln( +1) ln( +1), where the constant C is independent of and. The proof of Corollary 2.4 is very similar to that of Corollary 2.2 and hence is omitted. Remark 2.3. Our analysis shows that, an -uniform bound is valid for the discrete -weighted energy norm. In this sense, the superconvergence feature in the current work is different from Zhou and Rannacher s work [16], where the boundary layer is not the main concern and the error bounds depend on the higher order derivatives of the solution. Remark 2.4. Recall that the optimal rate of convergence in the energy norm is of order p. Therefore, most of the error bounds in Corollary 2.1 and Corollary 2.3 are superconvergent. From estimates in the discrete energy norm, we automatically obtain error estimates in the L 2 -norm, since,. Furthermore, error estimates at nodal points in Corollary 2.2 and 2.4 are near optimal since the optimal one is of order p 1. III. PROOF OF THEOREMS 2.2 AD 2.3 We shall apply some superconvergence analysis technique to estimate the approximation error. The reader is referred to Wahlbin s book [19] for general information regarding the finite element superconvergence. Throughout this section, conditions in either Theorem 2.2 or Theorem 2.3 are valid. We present a unified approach to analyze both cases. Although there are two boundary layers for the reactiondiffusion case, we only discuss the one at x =1because the discussion for the other one is similar. Therefore, without loss of generality, we may consider for both cases, the boundary layer term u. In the following, we shall estimate the errors involving the Gauss-Lobatto interpolant. To be more precise, let x i 1 = t (i) 1 < t (i) 2 < < t (i) p+1 = x i be the Gauss-Lobatto points on Ω i =[x i 1,x i ], i.e., t (i) 2 <t (i) 3 < <t p (i) are zeros of the derivative of the Legendre polynomial of degree p on Ω i, and let w I be a polynomial of degree p that interpolates w at those p +1points. Then the remainder of the interpolation can be expressed as (w w I )(x) =ψ p+1 (x)w[t (i) 1,t(i) 2,...,t(i) p+1,x] (3.1) where w[t (i) 1,t(i) 2,...,t(i) ] is the kth-order ewton divided difference [20, 3.2] and k ψ k (x) =(x t (i) 1 )(x t(i) 2 ) (x t(i) k ). ote that ψ p+1(x) is a multiple of the Legendre polynomial of degree p on Ω i ; therefore, (ψ p+1,v ) Ωi =0 v V,p. (3.2) When w is a polynomial of degree p+1on Ω i, the remainder w w I = cψ p+1 (x) with the constant c = w (p+1) /(p + 1)!. Therefore, (w w I,v ) Ωi =0for any v V,p. By the Bremble-Hilbert Lemma [21, Chapter 4] and a scaling argument, we derive (w w I,v ) Ωi Ch p+1 i w p+2,ωi v 1,Ωi. (3.3)

10 In the sequel, the following fact will be frequently used: e α(1 x)/ 2 (1 τ,1) = 1 1 τ SIGULARLY PERTURBED PROBLEMS 383 e 2α(1 x)/ dx = 1 τ e α(1 x)/ 2 (0,1 τ) = e 2α(1 x)/ dx = 0 2α (1 2p 3) ) < 2α, (3.4) ( ) 1 e 2α/ < 2α 2p+3 Similar results hold for the reaction-diffusion case in changing α to β.. (3.5) 2α 2p+3 Lemma 3.1. Let u satisfy regularity (2.20) (or (2.5)). Then (u u,i,v ) (1 τ,1) C ( ) p+1 ln( +1) v 1 ; u (0,1 τ) C p+1.5. Proof. We have, from (3.3), the Cauchy-Schwarz inequality, regularity (2.20), and (3.5), (u u,i,v ) (1 τ,1) Ch p+1 u p+2,(1 τ,1) v 1,(1 τ,1) C 1 h p+1 p 2 ( 1 1 τ = C 1 (h/) p+1 ( 1 1 e 2α(1 x)/ dx) 1/2 v 1,(1 τ,1) 1 τ C ( ) p+1 p ln( +1) v 1,(1 τ,1); e 2α(1 x)/ dx) 1/2 v 1,(1 τ,1) u (0,1 τ) C 1 e α(1 x)/ (0,1 τ) C 2α p+1.5. Lemma 3.2. Let u satisfy regularity (2.20) (or (2.5)). Then u u,i (1 τ,1) C ( ) p+1 ln( +1) ; u (0,1 τ) C p+1.5. Proof. Consider Ω i (1 τ,1), we have, from the standard approximation theory and regularity (2.20), u u,i 2 Ω i Ch 2(p+1) i u 2 p+1,ω i C ( hi ) 2(p+1) h i e 2α(1 xi)/ = C( +1) (2p+3)/ ( hi ) 2(p+1) h i e 2α(1 xi 1)/.

11 384 ZHAG Summing up, we have u u,i 2 (1 τ,1) ( ) 2(p+1) ln( +1) 1 C C ( ) 2(p+1) ln( +1) 2α. 1 τ e 2α(1 x)/ dx ote that ( +1) (2p+3)/ is bounded for any fixed p and xi h i e 2α(1 xi 1)/ < e 2α(1 x)/ dx. x i 1 Finally, by the regularity (2.20) and (3.5), we have 1 τ u 2 (0,1 τ) C e 2α(1 x)/ dx C 0 2α 2p+3. The conclusion then follows by taking the square root. Lemma 3.3. Let u satisfy regularity (2.5), the reaction-diffusion case. Then u u,i,,(1 τ,1) C ( ) p+1 ln( +1), u,,(0,1 τ) C p+1.5. Let u satisfy regularity (2.20), the convection-diffusion case. Then ( ) p+1 ln( +1) u u,i,,(1 τ,1) C, u,,(0,1 τ) C p+1.5. Proof. We only prove the reaction-diffusion case because the argument for the convectiondiffusion case is the same with in place of 2 in front of the summation. Because ψ p+1(x) is a multiple of the Legendre polynomial of degree p on Ω i, it vanishes at all Gaussian points x ij,j =1,...,p. Therefore, u u,i 2,,Ω i = u u,i cψ p+1 2,,Ω i for any c R. ote that w j > 0, j w i =1, one obtains since by choosing u u,i 2,,Ω i 2 h i inf c R (u u,i cψ p+1 ) 2 L (Ω i) C 2 h i h 2(p+1) i u (p+2) 2 L (Ω, i) c = (u,ψ p+1) Ωi ψ p+1, L 2 (Ω i) the operator w w I + cψ p+1 reproduces w if w is a polynomial of degree p +1. Recall the regularity (2.5), (3.4) (3.5), and we have u u,i 2,,(1 τ,1) C 4 i=3+1 ( ) 2(p+1) hi h i e 2β(1 xi)/

12 Here we have used the estimate ext, SIGULARLY PERTURBED PROBLEMS 385 ( ) 2(p+1) ln( +1) 1 C p C ( ) 2(p+1) p ln( +1). 2β h i e 2β(1 xi)/ < ( +1) (2p+3)/ xi u 2,,(0,1 τ) = 2 3 C C h i i=1 j=1 3 p h i i=1 j=1 1 τ 0 1 τ e 2β(1 x)/ dx x i 1 e 2β(1 x)/ dx. p w j u (x ij ) 2 w j e 2β(1 xij)/ e 2β(1 x)/ dx C 2p+3. Here we have used the remainder for the Gauss-Legendre quadrature [22, p. 98], xi x i 1 e 2β(1 x)/ dx h i = p w j e 2β(1 xij)/ j=1 h 2p+1 i (p!) 4 (2p + 1)[(2p)!] 3 The conclusion follows by taking the square roots. ( ) 2p 2β e 2β(1 ηi)/ > 0, i =1, 2,...,3. A. The Reaction-Diffusion Case We choose a special interpolation of the solution u as I u =ū I + u,i where w I is the Gauss- Lobatto interpolation of w and { u,i l u,i = τ, 1 τ x 1 0, 0 x 1 τ with { u l τ (x) = (1 τ) 1 τ x+h h, 1 τ x 1 τ + h 0, 1 τ + h x 1. We see that l τ is only defined in the boundary layer region. otice that l τ (1 τ + h) =0, l τ (1 τ) =u (1 τ). The idea is to make u,i essentially the Gauss-Lobatto interpolation of u in the boundary layer region and zero outside the boundary layer region. This may result in a discontinuity at the transition point 1 τ. By introducing the linear function l τ, we preserve the continuity of u,i. ote that u,i (1 τ) =0, u,i (1 τ + h) =u (1 τ + h).

13 386 ZHAG A direct calculation shows that l τ 2, = 2 h l τ 2 = hu (1 τ) 2 /3 l τ 2 = u (1 τ) 2 /h j=1 Cln( +1) 2(p+2), C 2(p+1) ln( +1) ; p w j (l τ ) 2 = 2 h u(1 C τ)2 2(p+1) ln( +1). Comparing these estimates with results in Section 2, we see that the existence of l τ does not influence the global error analysis. It is worth to point out that the introduction of l τ is technically necessary, because the traditional Lagrangian interpolation of u over the whole domain does not result in the expected error bounds. Proof. Proof of Theorem 2.2 Recall the bound of l τ, we obtain B (l τ,v) 2 l τ v + c l τ v C 3/2 p+1 ln 1/2 ( +1) v + C1/2 ln 1/2 ( +1) p+2 v C1/2 p+1 v. In light of Lemmas 3.1 and 3.2, we derive B (u,v) (0,1 τ) 2 u (0,1 τ) v + c u (0,1 τ) v C3/2 p+1.5 v + C1/2 C 2 v p+1.5 p+1.5 v ; B (u u,i,v) (1 τ,1) 2 (u u,i,v ) (1 τ,1) + c u u,i (1 τ,1) v ( ) p+1 ln( +1) C 3/2 v 1 + C ( ) p+1 ln( +1) v C ( ) p+1 ln( +1) 2 v. ote that B (u u,i,v)=b (u u,i,v) (1 τ,1) + B (l τ,v)+b (u,v) (0,1 τ). Therefore, B (u u,i,v) C ( ) p+1 ln( +1) v. This establishes the error bound (2.13) in Theorem 2.2 when the regular part ū is in the finite element space. When the regular part of the solution is not in the finite element space, we have from (3.3) and (2.5), 2 (ū ū I,v 1 ) C p+1 v. Recall the standard estimate (ū ū I,bv) C 1 v, p+1

14 we have ote that SIGULARLY PERTURBED PROBLEMS 387 B (ū ū I,v) C 1 p+1 ( v + v ) C 1 p+1 v. B (u I u, v) =B (u u,i,v)+b (ū ū I,v). We then finish the proof of (2.11) in Theorem 2.2. Recall Lemmas 3.2, 3.3 and the bounds for l τ, we derive u u,i 2, = u u,i + l τ 2,,(1 τ,1) + u 2,,(0,1 τ) ( ) 2(p+1) ln( +1) C C + 2(p+1) ln( +1) + C 2p+3 ( ) 2(p+1) ln( +1) 2C. This establishes (2.14) in Theorem 2.2. For the regular part of the solution, we use the similar argument as in the proof of Lemma 3.3 to derive Therefore, In addition, we have ū ū I 2, C 2 i h i h 2(p+1) i ū (p+2) 2 L (Ω i) C 2 h 2(p+1). 1 ū ū I, C p+1. ū ū I C 1 p+1 by standard interpolation result. Combine these with the above estimate for the boundary layer term, and we derive ( ( ) ) p+1 ln( +1) u Iu, C + 1 p+1. This finishes the proof of (2.12) in Theorem 2.2. B. The Convection-Diffusion Case The special interpolation I u =ū I + u,i of the solution u is defined slightly different from the reaction-diffusion case in that u,i, 1 τ x 1 u,i = l τ, 1 τ h x 1 τ 0, 0 x 1 τ h with l τ (x) =u (1 τ) x 1+τ + h. h

15 388 ZHAG otice that l τ (1 τ h) =0, A direct calculation shows that l τ 2 = hu (1 τ) 2 /3 l τ 2 = u (1 τ) 2 / h l τ (1 τ) =u (1 τ). C ln( +1) 2(p+2), C 2(p+1) ln( +1) ; l τ 2, = hu(1 τ) 2 C 2(p+1) ln( +1). Proof. Proof of Theorem 2.3 Recall the bound of l τ, we obtain B (l τ,v) l τ ( v + C 1 v )+C 2 l τ v C p+1 ln 1/2 ( +1) ( v + v )+ C ln1/2 ( +1) p+2 v 2C p+1 v. In light of Lemmas 3.1 and 3.2, we derive B (u,v) (0,1 τ) (u,v ) (0,1 τ) + c u (0,1 τ) ( v + v ) C p+1.5 v + C1/2 p+1.5 v 1 2C p+1.5 v ; B (u u,i,v) (1 τ,1) (u u,i,v ) (1 τ,1) + c u u,i (1 τ,1) ( v + v ) ( ) p+1 ln( +1) C v + C ( ) p+1 ln( +1) v 1 ( ) p+1 ln( +1) 2C v. ote that B (u u,i,v)=b (u u,i,v) (1 τ,1) B (l τ,v)+b (u,v) (0,1 τ). Therefore, ( ) p+1 ln( +1) B (u u,i,v) C v. This establishes the error bound (2.26) in Theorem 2.3 when the regular part ū is in the finite element space. To prove (2.24) in Theorem 2.3, we left to estimate the approximation for the regular part. B (ū ū I,v) (ū ū I,v ) + c ū ū I ( v + v ) C p+1 v + C p+1 v 1 C p+1 v + C 1 p v.

16 SIGULARLY PERTURBED PROBLEMS 389 Here we have used the inverse inequality. Together with the error bound for the boundary layer term, we have proved (2.24) in Theorem 2.3. For the discrete energy norm, recall Lemmas 3.2, 3.3, and the bounds for l τ, and we derive u u,i 2, = u u,i 2,,(1 τ,1) + u l τ 2,,(0,1 τ) ( ) 2(p+1) ln( +1) C + C 2p+3 + C 2(p+1) ln( +1) ( ) 2(p+1) ln( +1) 2C. The estimate in the discrete norm for the regular part is the same as for the reaction-diffusion case where ū ū I, C p+1. Therefore, ( ) p+1 ln( +1) u Iu, C, which establishes (2.25) and hence finishes the proof of Theorem 2.3. IV. UMERICAL RESULTS Two numerical examples are presented in this section to illustrate that the obtained error bounds are sharp. In all figures, R-D stands for reaction-diffusion and C-D, convection-diffusion. 1. Consider the reaction-diffusion equation: 2 u(x) + u(x) =x, u(0) = u(1)=0. The exact solution is u = x e(x 1)/ e (x+1)/. 1 e 2/ In this special case, the solution has one boundary layer at x =1. It is solved by linear (p =1) and quadratic (p =2)finite element methods under the modified Shishkin mesh. The transition point is 1 τ where τ = (p + 1) ln( +1). Each interval, (0, 1 τ) and (1 τ,1), is equally divided into =2 j subintervals, j =0, 1,...,10 when p =1, and for j =0, 1,...,9 when p =2. ote that the regular part of the solution is a linear function x, which is contained in the finite element spaces. Therefore, the second estimate in Corollary 2.1 applies. Listed in Table I and Table II are errors in the discrete energy norm u u, and in the maximum norm at nodal points u u for three different values of d = =0.01, 0.001, for cases p =1and p =2, respectively. Plotted in Fig. 1 and Fig. 2 are convergent curves in the discrete energy norm u u, for the three different values of d = in cases p =1and p =2, respectively. They clearly indicate a rate of ln p+1 ( +1)/ p+1 as predicted by Corollary 2.1. ote that the gaps between curves illustrate the influence of. Therefore, the theoretical error estimates are optimal.

17 390 ZHAG TABLE I. Reaction-diffusion, p = 1. = 10 2 = 10 3 = 10 4 u u, u u u u, u u u u, u u e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 06 Plotted in Fig. 3 and Fig. 4 are convergent curves in the maximum norm at nodal points for the three different values of d = for p =1and p =2, respectively. In case p =1, the convergence is quite insensitive to as the three curves are very close with a rate of ln 2 ( +1)/ 2. Therefore, the error predicted by Corollary 2.2 is nearly optimal up to a factor ln( +1), which is negligible. However, in case p =2, the numerical data seems to indicate a rate of ln 4 ( +1)/ 4, which is one order better than the error bound given by Corollary 2.2. Hence, the nodal superconvergence phenomenon exists for singularly perturbed reaction-diffusion problems. 2. Consider the convection-diffusion equation: u(x) + u (x) =x, u(0) = u(1)=0. The exact solution is ( x ) ( 1 u = x )( e (x 1)/ e 1/ 1 e 1/ The solution has one boundary layer at x =1. It is solved by a linear (p =1)finite element method. The mesh is the same as in the reaction-diffusion equation. ote that the regular part x(x/2+) of the solution is not included in the finite element space. ). TABLE II. Reaction-diffusion, p = 2. = 10 2 = 10 3 = 10 4 u u, u u u u, u u u u, u u e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 09

18 SIGULARLY PERTURBED PROBLEMS 391 FIG. 1. R-D, energy norm, p = 1. Listed in Table III are errors in the discrete energy norm u u, and in the maximum norm at nodal points u u for three different values of d = =10 2, 10 4, 10 8 in case p =1. Plotted in Fig. 5 are convergent curves in the discrete energy norm u u, for the three different values of d =. They clearly indicate a rate of ln 2 ( +1)/ 2, which is predicted FIG. 2. R-D, energy norm, p = 2.

19 392 ZHAG FIG. 3. R D, nodal error, p = 1. in Corollary 2.3. Different from the reaction diffusion equaiton, there are no gaps between the three curves here. It seems that the term 1/ p in Theorem 2.3 and Corollary 2.3 may be removable but the term ln p+1 ( +1)/ p+1 is optimal. FIG. 4. R-D, nodal error, p = 2.

20 SIGULARLY PERTURBED PROBLEMS 393 TABLE III. Convection-diffusion, p = 1. = 10 2 = 10 4 = 10 8 u u, u u u u, u u u u, u u e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 06 Plotted in Fig. 6 are convergent curves in the maximum norm at nodal points for the three different values of d =. Again, we see that the error predicted by Corollary 2.4 is nearly optimal up to a factor ln( +1), which is negligible. V. COCLUSIO AD FIAL REMARKS 1. We have established, for the first time, superconvergence error estimates for singularly perturbed one dimensional reaction-diffusion problems and convection-diffusion problems. The main difficulty in the analysis which differs from the traditional superconvergence analysis is FIG. 5. C-D, energy norm, p = 1.

21 394 ZHAG FIG. 6. C-D, nodal error, p = 1. in that the error bounds are -uniform. Our results basically state that the approximated derivatives at the Gaussian points are superconvergent, just like in nonsingularly perturbed problems. Furthermore, our numerical results for the reaction-diffusion equation when p =2suggests that nodal-point superconvergence exists at least in some special cases also like in nonsingularly perturbed problems. Therefore, for problems with boundary layers, numerical superconvergence can be achieved as long as the boundary layer is properly resolved. In order to resolve the boundary layer, the size of the mesh in the boundary layer region should be in the range O( 1 ln ). 2. Comparing with the work of Kellogg and Stynes [8] on optimal approximability to singularly perturbed two-point boundary problems, error bounds in the L 2 -norm obtained here (as a byproduct of error bounds in the discrete energy norm) are nearly optimal up to a factor ln p+1. Therefore, the standard finite element method (polynomial test/trial functions) combined with the Shishkin type mesh is sufficient to resolve the boundary layer for singularly perturbed reactiondiffusion and convection-diffusion problems and to achieve an almost optimal rate of convergence in the one-dimensional setting. Generalization of the idea to higher dimensions is feasible. 3. From a practical point of view, the Shishkin type mesh is simpler than an exponentially graded mesh, and the method discussed in this article is simpler than the Petrov-Galerkin method, where exponentially fitted test/trial functions are introduced. 4. In this work, the finite element space contains piecewise polynomials with a fixed degree p, the so-called h-version method. By increasing p on the Shishkin type meshes, we have the p version and the hp-version methods, in which case an exponential rate of convergence can be achieved if a, b, and f are analytic (see [15]). In this direction, the reader is referred to the work by Schwab and Suri [13], where an exponentially convergent rate is established for the reaction-diffusion equation with a constant coefficient b, and more recent work by Melenk [9] and by Melenk and Schwab [10], where robust exponential convergence are proved for general one-dimensional reaction-diffusion and convection-diffusion equations.

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