Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem

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1 Numer. Math. Theor. Meth. Appl. Vol. 10, No. 1, pp doi: /nmtma.017.y1306 February 017 Analysis of a Streamline-Diffusion Finite Element Method on Bakhvalov-Shishkin Mesh for Singularly Perturbed Problem Yunhui Yin 1,, Peng Zhu 1 and Bin Wang 1 College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang , China. College of Mechanical and Electrical Engineering, Jiaxing University, Jiaxing, Zhejiang , China. Received 16 September 013; Accepted 19 October 016 Abstract. In this paper, a bilinear Streamline-Diffusion finite element method on Bakhvalov-Shishkin mesh for singularly perturbed convection diffusion problem is analyzed. The method is shown to be convergent uniformly in the perturbation parameter provided only that N 1. An ON lnn) 1/ ) convergent rate in a discrete streamline-diffusion norm is established under certain regularity assumptions. Finally, through numerical experiments, we verified the theoretical results. AMS subject classifications: 65N15; 65N30 ey words: singularly perturbed problem, Streamline-Diffusion finite element method, Bakhvalov-Shishkin mesh, error estimate. 1. Introduction In this paper, we consider a Streamline-Diffusion finite element method SDFEM) for the singularly perturbed boundary value problem Lu := u+b u+cu = f on Ω = 0,1), u = 0 on Ω, 1.1) where 0 < 1 is a small positive parameter, b,c and f are sufficiently smooth functions satisfying bx,y) = b 1 x,y),b x,y)) β 1,β ) > 0,0), x,y) Ω, cx,y) 0, cx,y) 1 divbx,y) c 0 > 0, x,y) Ω, 1.a) 1.b) Corresponding author. addresses: yunhui.yin@163.com Y. Yin), zhupeng.hnu@gmail.com P. Zhu), wangbin.70s@aliyun.com B. Wang) c 017 Global-Science Press

2 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 45 where β 1,β and c 0 are some constants. These hypotheses ensure that 1.1) has a unique solution in H0 1Ω) H Ω) for all f L Ω). Note that for sufficiently small, the other hypotheses imply that 1.b) can always be ensured by the simple change of variable vx,y) = e σx ux,y) where σ is chosen suitably. With the above assumptions, the solution of 1.1) typically has boundary layers of width Oln 1 ) at the outflow boundary x = 1 and y = 1. For small values of, standard Galerkin discretisation for 1.1) exhibits spurious oscillations and fails to catch the rapid change of the solution in boundary layers, see the numerical results in [15]. Many methods have been developed to overcome the numerical difficulty caused by the boundary layers. One of the most successful methods is the use of layer-adapted meshes. Provided that some information on the structure of the layers was available, a piecewise uniform Shishkin meshs-mesh) could be chosen a priori, see [1, 3]. Linß [8, 9] introduced Bakhvalov-Shishkin meshb-s-mesh) which is a modification of S-mesh by using a uniform coarse mesh and a graded fine mesh with Shishkin s simple choice of the transition point. The optimal convergence order ON 1 ) on a B-S-mesh had been proved, while on S-mesh it was only convergent of ON 1 lnn). Zhang [5] investigated the superconvergence of order ON lnn) ) in a discrete -weighted energy norm on a S-mesh. A powerful method for stabilising convection-diffusion problems is the streamlinediffusion finite element method which was proposed by Hughes and Brooks [16]. This method was known to provide good stability properties and high accuracy in boundary layers. The convergence properties of the SDFEM had been widely studied[3,10-13]. In [13], the error between the SDFEM solution and the interpolation of the solution of 1.1) on S-mesh was of order ON 3/ lnn) in the streamline-diffusion normsd norm). In [10], a more careful analysis was performed by using interpolation error identities of Lin, and this error was improved to ON lnn) ). In order to achieve estimates for the interpolation error in SD norm, Stynes and Tobiska [10] firstly introduced the discrete streamline-diffusion norm, and estimated an error bound of order ON lnn) ) on S-mesh. Here we shall analyze a SDFEM on B-S-mesh, and it will give more accurate results than on S-mesh. There are three main results in this paper. First, the interpolation error in discrete SD norm is presented to be convergent of ON ). Second, the error between the solution of the discrete problem and the interpolation of the solution of the continuous problem is shown to be bounded in discrete SD norm by ON lnn) 1/ ), uniformly in. Third, we prove that the error between the solution of the discrete problem and the solution of the continuous problem itself can be estimated in discrete SD norm by ON lnn) 1/ ). An outline of the paper is as follows. In Section we describe the B-S-mesh and the SDFEM. A decomposition of the solution u and some important preliminaries to the analysis are presented in Section 3, and in Section 4 we analyze the convergence properties of the method. In order to validate our theoretical results, numerical results are presented in Section 5. We end in Section 6 with some concluding remarks.

3 46 Y.-H. Yin, P. Zhu and B. Wang Notation: Throughout the paper, C will denote a generic positive constant that independent of and the mesh. Note that C is not necessarily the same at each occurrence. The standard notation for the Sobolev spaces Wp k D) and norms will be used for nonnegative integers k and 1 p. An index will be attached to indicate an inner product or a norm on a subdomain, for example,, ) D, k,p,d and k,p,d. When D = Ω, we drop the D from notation for simplicity. We will also simplify the notation in the case p = by setting k,d = k,,d and k,d = k,,d.. The Bakhvalov-Shishkin mesh and the SDFEM Let N be an even positive integer. We let λ 1 and λ denote two mesh transition parameters that will be used to specify where the changes form coarse to fine; these are defined by 1 λ 1 = min,.5 ) lnn β 1 1 and λ = min,.5 ) lnn..1) β In fact we make the very mild assumption that λ 1 =.5 β 1 lnn and λ =.5 β lnn, as otherwise N 1 is exponentially small compared with. We shall also assume throughout the paper that N 1 as is generally the case in practice. We divide the domain Ω as in Figure 1: Ω = Ω 11 Ω 1 Ω 1 Ω, where Ω 11 = [0,1 λ 1 ] [0,1 λ ],Ω 1 = [0,1 λ 1 ] [1 λ,1],ω 1 = [1 λ 1,1] [0,1 λ ],Ω = [1 λ 1,1] [1 λ,1]. Figure 1: Division of Ω left) and a corresponding B-S-mesh right). The interval [0,1 λ 1 ] is uniformly dissected into N/ subintervals, while [1 λ 1,1] is partitioned into the same number of mesh intervals by inverting the function exp β 1 1 x)/.5)). We specify thex i, fori = N/,,N, so that{e β 11 x i )/.5) } i

4 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 47 is a linear function in i, i.e., we set e β 11 x i )/.5) = Ai+B, and choose the unknowns A and B so that x N/ = 1 λ 1 and x N = 1. An analogous formula can be given for the mesh points y j, for j = 0,,N. The mesh points Ω N = {x i,y j ) Ω : i,j = 0,1,,N} are the rectangular lattices defined by { 1.5 β 1 lnn) i N i = 0,, N x i = ), 1+.5 β 1 ln N N i)n 1) i = N N +1,,N, { 1.5 β y j = lnn) j N j = 0,, ) N, 1+.5 β ln N N j)n 1) j = N N +1,,N. Our mesh is constructed by drawing lines parallel to the coordinate axes through these mesh points. This divides Ω into a set T N of mesh rectangles whose sides are parallel to the axes see Figure 1). Given an element = x i 1,x i ) y j 1,y j ), its dimensions are written as h x,i h x, ) = x i x i 1,h y,j h y, ) = y j y j 1, and its barycenter is denoted by x,y ). We now describe the SDFEM on the rectangular mesh. The bilinear form B SD, ) used in the SDFEM is defined by where B SD u,v) = B GAL u,v)+b STAB u,v), B GAL u,v) = u, v)+b u+cu,v), B STAB u,v) = ρ u+b u+cu,b v). Ω 11 The weak formulation of the model 1.1) is: Find u H0 1 Ω) such that B SD u,v) = f,v)+ ρ f,b v), v H0Ω). 1 Ω 11 Let V N H0 1 Ω) be the continuous piecewise bilinear finite element space on the B-S-mesh: } V N = {v C Ω) : v Ω = 0, v Q 1 ) T N. Then the SDFEM is defined as follows: Find u h V N such that B SD u h,v h ) = f,v h )+ Ω 11 ρ f,b v h ), v h V N..)

5 48 Y.-H. Yin, P. Zhu and B. Wang The orthogonality property holds clearly B SD u u h,v h ) = 0, v h V N. For each T N, set h = min{h x,,h y, }. Let C inv be a constant such that the inverse inequality v h C inv h 1 v h, v h V N, Ω 11 is valid. Similarly to [3], we set ρ = { C1 N 1 Ω 11 0 otherwise, where the positive constant C 1 is chosen independent of ) such that 0 ρ 1 min c 0 h ) max cx,y), Cinv, Ω 11. Then the argument of [3] shows that the inequality holds, where SD norm SD is given by B SD v h,v h ) 1 v h SD, v h V N v SD = c 0 v 0 + v 1 + ρ b v Ω 11 It follows that.) has a unique solution u h V N. We shall use the discrete SD norm SD,d defined by 1/. v SD,d { = c 0 v 0 + area) vx,y ) + } 1/ ρ area) b vx,y ). Ω Ω Solution decomposition and preliminary Our subsequent analysis will rely on the precise knowledge of the behaviour of the solution u of the convection-diffusion problem 1.1). The typical behaviour of u is given in the following assumption. Assumption 3.1. Assume that u = S +E 1 +E +E 1,

6 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 49 where there exists a constant C such that for all x,y) Ω we have i+j S x i y jx,y) C 3.1) for 0 i+j 3 and i+j E 1 x i y jx,y) C i β 11 x) e, 3.) i+j E x i y jx,y) C j β 1 y) e, 3.3) i+j E 1 x i y j x,y) C i+j) β 11 x)+β 1 y) e 3.4) for 0 i+j 3. Remark 3.1. In [4], a proof was given that under certain compatibility conditions on the date f of problem 1.1), the bounds 3.)-3.4) and 0 i + j in 3.1) of Assumption 3.1 hold true. The extension of this result to the case 0 i+j 3 in 3.1) as needed in our case seems to be possible but tedious. The number of these sufficient conditions will increase rapidly with increasing differentiation order. Next we introduce some equalities and inequalities that will be used in the analysis. inverse inequalities: for v V N, ) v dxdy C v dxdy, 3.5a) x ) dxdy C v y h x, h y, v dxdy. 3.5b) Let v I be the piecewise bilinear interpolation of function v W Ω), 1 then [5] v I x x,y ) = 1 xi v h x, x i 1 x x,y j 1)+ v ) x x,y j) dx, 3.6) v I y x,y ) = 1 yj v h y, y j 1 y x i 1,y)+ v ) y x i,y) dy. 3.7) and if v W 3 Ω), we have v I v) x,y ) 3.8) x = 1 [ ) hx,/ t 3 v h x, h x,/ x 3 th y, 3 v x y + h y, 3 ) v h y, 8 x y x +s 1 t,y s 1 ) t 3 v + x 3 + th y, 3 v x y + h y, 3 ) ] v h y, 8 x y x +s t,y +s dt,

7 50 Y.-H. Yin, P. Zhu and B. Wang v I v) x,y ) 3.9) y = 1 [ ) hy,/ h x, 3 v 8 x y th x, 3 v x y + t 3 ) v h x, y 3 x s 3,y +s 3 t h y, + h y,/ h x, 8 3 v x y + th x, 3 v x y + t where 0 < s i < 1 for i = 1,,3,4; see [5]. 3 v y 3 ) x +s 4 h x,,y +s 4 t ) ] dt, Lemma 3.1. [6]) Let T N andp [1,+ ]. Assume that v W p), We denote by v I the bilinear function that interpolations to v. Then { } v v I 0,p, C h v v x, x +h x, h y, 0,p, x y +h v y, 0,p, y, 3.10) 0,p, { } v v I ) v x C h x, v 0,p, x +h y, 0,p, x y, 3.11) 0,p, { } v v I ) v y C h x, v 0,p, x y +h y, 0,p, y. 3.1) 0,p, Lemma 3.. Lin identities) [7]) Let be a mesh rectangle, and we denote the south, east, north, west edges of by l i, for i = 1,,3,4, respectively. Let v H 3 ), and let v I Q 1 ) be its bilinear interpolation. Then for each v N Q 1 ), we have v v I ) v N x x dxdy = 3 v vn x y Fy) x ) 3 y y ) v N dxdy, 3.13) x y v v I ) v N y y dxdy = 3 v x y Gx) vn y ) 3 x x ) v N dxdy, 3.14) x y v v I ) v N dxdy = Rv,v N )dxdy + h x, x 1 l, l 4, where [ Gx) = 1 ) ] [ x x ) hx,, Fy) = 1 y y ) Rv,v N ) = 1 3 Gx)x x ) 3 v v N x 3 x h x, 3 v 1 x 3v N +Fy) 3 v x y v N ) v x v Ndy, 3.15) hy, ) ] x x ) v N x 3 y y ) v N y + 3 x x )y y ) v N x y, ). 3.16)

8 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 51 Lemma 3.3. [8]) Let γ 1 = i N,γ = j N, then the step sizes of the mesh T N satisfy: for i,γ 1 = 1,,N/, and for j,γ = 1,,N/, h x,i N 1, h y,j N 1, h x,γ1 5 β 1 γ 1 CN 1 ; 3.17) h y,γ 5 β γ CN ) Finally, we list some inequalities regarding the exponential boundary layer function which will be used in the next section: for i = N/ + 1,,N and for j = N/ + 1,,N, N/ N/ j=1 xi e x i 1 yj e y j 1 h x,i e β 1 1 x i ) CN 5 +N 1 ), 3.19) h y,j e β 1 y j ) CN 5 +N 1 ), 3.0) β 1 1 x) dx.5 N 1 e 8β 1 1 x i ) 5 ; 3.1) β 1 β 1 y) dy.5 N 1 e 8β 1 y j ) 5. 3.) β 4. Analysis and main results Lemma 4.1. [9]) If Assumption 3.1 holds, then we have the interpolation error estimates u u I 0, CN. 4.1) Lemma 4.. [10]) There exists a positive constant C such that v N SD,d C v N SD, v N V N. Lemma 4.3. Let E = E 1 + E + E 1 satisfy the regularity 3.)-3.4). Then there is a constant C, such that area) E E I )x,y ) CN 4. Ω Proof. Based on the boundary layer behaviour of E 1, we separate the discussion into the case of Ω 1 Ω and Ω 11 Ω1. a) Ω 1 Ω. Applying the regularity results 3.) to 3.8),3.9), we derive E 1 E 1,I )x,y )

9 5 Y.-H. Yin, P. Zhu and B. Wang ) Ce β 1 1 x i ) h x, )+ h x,h y, + 1 )+ h y, ) Ce β 1 1 x i ) h x, 3 +h x, h y, +h y, 1). Adding all elements on Ω 1 Ω yields area) E 1 E 1,I )x,y ) Ω 1 Ω C C C N j=1 N h y,j i=n/+1 +CN 4 N/ γ 1 =1 CN 4, 5 N i=n/+1 h x,i e β 1 1 x i ) h 5 x,i 5 e β 1 1 x i ) +CN N β 1 γ 1 N i=n/+1 N/ +CN h x,i 1 e β 1 1 x i ) ) 5 5 γ1 N 1)+N γ 1 =1 N/ +CN 4 γ 1 =1 5 β 1 γ 1 5 β 1 γ 1 N h x,i 3 +h x,i h y,j +h y,j 1) i=n/+1 ) 5 ) 3 3 γ1 N 1)+N N h 3 x,i 3 e β 1 1 x i ) ) 5 ) ) 1 γ1 N 1)+N 5 N 4.) where we used Lemma 3.3 and the expression of x i. b) Ω 11 Ω1. By the regularity 3.), we have area) E 1 x,y ) Ch x, h y, +1)e β 1 1 x ). Summing up all elements on Ω 11 Ω1 yields Ω 11 Ω1 area) E 1 x,y ) C 1 +) N j=1 C 1 e β 1 h x, N/ h y,j N/ h x,i e β 1 1 x ) h x,i e β 1 1 x i )

10 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 53 CN 5 e β 1 N 1+N) 1) CN 5, 4.3) where we used 3.19) and the boundedness of e t 1 + t) on R +. It remains only to estimate E 1,I x,y ). Invoking the Cauchy-Schwarz inequality and 3.6), 3.7), we have E 1,I x x,y ) C xi h x, E 1,I y x,y ) C h y, x i 1 1 e yj y j 1 1/ β 1 1 x) dx C 1 xi e β 1 1 x) dx), hx, x i 1 e β 1 1 x i ) +e β 1 1 x i 1 ) ) dy Ce β 1 1 x i ). Summing up, we obtain Ω 11 Ω1 area) E 1,I x,y ) C 1 N CN 5, j=1 N/ xi h y,j x i 1 e β 1 1 x) dx+c N j=1 N/ h y,j h x,i e β 1 1 x i ) 4.4) where we used 3.19). Combining 4.3) with 4.4), we get Ω 11 Ω1 area) E 1 E 1,I )x,y ) CN 5. This, combined with 4.), established the conclusion area) E 1 E 1,I )x,y ) CN ) Ω The argument for E is similar. The proof for E 1 is separated into the case of Ω, Ω 11 Ω1 and Ω 1. Using the similar techniques, we have area) E 1 E 1,I )x,y ) CN ) Ω Thus, collecting 4.5) and 4.6), we get the statement of the Lemma. Now we have a look at the interpolation error. Theorem 4.1. If Assumption 3.1 holds, then we have the interpolation error estimates u u I SD,d CN.

11 54 Y.-H. Yin, P. Zhu and B. Wang Proof. By Lemma 4.1, we obtain Next, we now analyze Ω u u I 0 CN. 4.7) area) u u I )x,y ). S S I ) x,y ) x h x, 3 S 8 x 3 + h x,h y, 3 S 0, 4 x y + h y, 3 S 0, 8 x y ) h x, C + h x,h y, + h y, CN, S S I ) x k,y k ) y h x, 3 S 8 x y + h x,h y, 3 S 0, 4 x y + h y, 3 S 0, 8 y 3 ) h x, C + h x,h y, + h y, CN, , 0, where we used Lemma 3.3 and 3.1),3.8),3.9). So area) S S I )x,y ) CN ) Ω Moving on to the layer part E of u, it is given in Lemma 4.3. Finally, we analyze Ω 11 ρ area) b u u I )x,y ). It is shown in [10, Lemma 5.] that Ω 11 ρ area) b u u I )x,y ) CN ) Combining 4.7)-4.9) and Lemma 4.3, Theorem 4.1 is proved. Remark 4.1. It is difficult to bound the term Ω 11 ρ b u u I ) independent of. In order to avoid this dilemma, we estimate the interpolation error in discrete SD norm instead of SD norm. Next, we discuss the error bound foru I u h. In order to obtain the bound, We firstly give Lemma 4.4, Lemma 4.5 and Lemma 4.6 by invoking the sharp superconvergence results of Lin [7].

12 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 55 Lemma 4.4. Let S satisfy the regularity 3.1). Then there exists a constant C, such that b S S I ),v N ) CN v N SD, v N V N. Proof. We define a piecewise constant approximation b of b by 1 b = bdxdy, T N. area Then we decompose b S S I ),v N ) = b b) S S I ),v N )+ b S S I ),v N ) The first term can be bounded using standard interpolation error estimate and the property of b: b b) S SI ),v N ) CN b 1, S v N 0 CN v N SD. 4.10) For the second term, we write b S S I ),v N ) S S I ) b1 v N dxdy x + b Ω Ω S S I ) y v N dxdy. 4.11) Through the inverse inequalities 3.5) and the expression of RS,v N ), we are able to get b1 RS,v N )dxdy Ω CN S 3, v N CN v N SD. 4.1) Ω Set b1 x,y) = b i,j for x i 1 < x < x i and y j 1 < y < y j. Since v N 0,y) = v N 1,y) = 0, we have h ) S S x, b1 Ω l, x v Ndy b1 l 4, x v Ndy CN N N yj [ ) S ) ] S b i,j y j 1 x v N x i,y) x v N x i 1,y) dy j=1

13 56 Y.-H. Yin, P. Zhu and B. Wang CN N N 1 yj ) S b i,j b i+1,j ) j=1 y j 1 x v N x i,y)dy CN 3 N j=1 CN 3 N j=1 N 1 N 1 yj v N x i,y) dy y j 1 xi yj x i 1 y j 1 v N x,y) dxdy CN v N SD. 4.13) Substituting 4.1) and 4.13) into 3.15), we finish the estimate for the first term on the right-hand side of 4.11). The estimate of the second term on the right-hand side of 4.11) is similar. Hence, the proof of Lemma 4.4 is completed. Lemma 4.5. Let E = E 1 + E +E 1 satisfy the regularity 3.)-3.4). Then there is a constantc, such that E E I ), v N ) CN v N SD, v N V N. Proof. Using the expression of Gx) and Fy), the identity 3.13) is estimated E E I ) v N x x dxdy 3 E x y Fy) v N x + ) 3 y y ) v N x y dxdy Ch 3 E v N y, x y x, 4.14) where we used the inverse inequality 3.5) and the Cauchy-Schwarz inequality in the last step. In the y-direction, we have E E I ) v N y y dxdy 3 E x y Gx) v N y + ) 3 x x ) v N x y dxdy Ch 3 E v N x, x y y. 4.15) Now, we estimate E 1. a) Ω 1 Ω. Recall the regularity 3.), apply 4.14) to E 1, summing over, we have E 1 E 1,I ) v N x x dxdy Ω 1 Ω

14 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 57 C 1/ N N C 1/ N v N x i=n/+1 j=1 N yj xi e β 1 1 x) y j 1 x i 1 1/ dxdy) v N x CN v N SD, 4.16) 0 where we used Lemma 3 and the Cauchy-Schwarz inequality. Applying 4.15) to E 1, and summing over, we have E 1 E 1,I ) v N y y dxdy Ω 1 Ω C 1 C 1 N N i=n/+1 j=1 N i=n/+1 C 1.5 N v N y yj xi h 4 x,i e β 1 1 x) y j 1 x i 1 1/ dxdy) v N y ) 1/ 5 4 N 1 e 8β 1 1 x i ) v N 5 β 1 γ 1 y 0 CN 3 v N SD. 4.17) 0 Here we used 3.1) and the Cauchy-Schwarz inequality. Putting together the above two estimates yields E 1 E 1,I ) v N dxdy Ω 1 Ω CN v N SD. 4.18) b) Ω 11 Ω1. From 3.), by integration, we obtain E 1 x C 1 N 5. 0,Ω 11 Ω1 On the other hand, apply the inverse inequality 3.5) to E 1,I, E 1,I x CN E 1,I 0,Ω 11 Ω1 0,Ω 11 Ω1 N/ CN h x,i e β 1 1 x i ) N h y,j CN 4, where we used 3.19). Summing up all Ω 11 Ω1, applying the triangle inequality and the Cauchy-Schwarz inequality, we have E 1 E 1,I ) v N x x dxdy Ω 11 Ω1 j=1

15 58 Y.-H. Yin, P. Zhu and B. Wang ) E 1 C x + E 1,I v N 0,Ω11 Ω1 x 0,Ω11 Ω1 x 0,Ω11 Ω1 C 0.5 N.5 +N ) v N x 0,Ω11 Ω1 CN.5 v N SD. 4.19) Furthmore, by 3.1) in Lemma 3.1 with p =, we get E 1 E 1,I ) y 0,Ω 11 Ω1 C h E 1 x, x y Ω 11 Ω1 xi N/ C N x i 1 e +h y, E 1 y β 1 1 x) dx C 1 N 7. Thus, by the Cauchy-Schwarz inequality, we obtain E 1 E 1,I ) v N Ω 11 Ω1 y y dxdy C E 1 E 1,I ) v N y 0,Ω11 Ω1 y C 0.5 N 3.5 v N y 0,Ω11 Ω1 0,Ω11 Ω1 CN 3.5 v N SD. 4.0) ) Collecting 4.19), 4.0) and 4.18) yields E 1 E 1,I ) v N dxdy CN v N SD. Ω Using similar arguments, we can show that E E,I ) v N dxdy CN v N SD, Ω E 1 E 1,I ) v N dxdy CN.5 v N SD. The proof is complete. Ω Lemma 4.6. Let E = E 1 + E +E 1 satisfy the regularity 3.)-3.4). Then there is a constantc, such that E E I ),b v N ) CN v N SD, v N V N. 4.1)

16 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 59 Proof. We estimate E 1 which is discussed into the case of Ω 11 Ω1 and Ω 1 Ω. a) Ω 1 Ω. Using 3.10) in Lemma 3.1 with p =, we get E 1 E 1,I 0 4.) { C h 4 E 1 x, x +h x,h E 1 y, x y +h 4 } E 1 y, y. Ω 1 Ω By the bound 3.), the first term is estimated C Ω 1 Ω h 4 x, N j=1 CN 1 h y,j N i=n/+1 N i=n/+1 E 1 x 5 β 1 γ 1 ) 4 xi x i 1 4 e β 1 1 x) dx γ 4 1 e 8β 11 x i) 5 CN 4, 4.3) where we used 3.1) and Lemma 3.3. For the second term in 4.), we obtain by using Lemma 3.3 C Ω 1 Ω h x, h y, N N h 3 y,j j=1 i=n/+1 CN 3 N i=n/+1 E 1 x y 5 β 1 γ 1 By integration and Lemma 3.3, we calculate Ω 1 Ω h 4 y, CN 4 1 E 1 y e β 1 1 x) 1 λ 1 ) xi x i 1 e β 1 1 x) dx γ 1 e 8β 11 x i) 5 CN ) C N dx CN 4, which, combined 4.3) and 4.4), proves E 1 E 1,I ),b v N ) Ω1 Ω N h 5 y,j j=1 i=n/+1 xi x i 1 e β 1 1 x) dx C E 1 E 1,I 0,Ω1 Ω v N 0,Ω1 Ω CN v N SD, 4.5)

17 60 Y.-H. Yin, P. Zhu and B. Wang where we used the Cauchy-Schwarz inequality and the continuity of b. b) Ω 11 Ω1. By the bound 3.), we obtain by integrating E 1 0,Ω 11 Ω1 C 1 λ1 0 e β 1 1 x) dx CN 6. We notice that on an element, E 1,I E 1 Ce β 1 1 x i ). By using 3.19), we have E 1,I 0,Ω 11 Ω1 C N j=1 N/ h y,j Therefore, by the inverse inequality 3.5), we have E 1 E 1,I ),b v N ) Ω11 Ω1 h x,i e β 1 1 x i ) CN 6. C E 1 E 1,I 0,Ω11 Ω1 v N 0,Ω11 Ω1 CN v N SD. This, together with the estimate 4.5), proves E 1 E 1,I ),b v N ) CN v N SD. The estimates for E and E 1 are similar. Hence, 4.1) is obtained. We will estimate the term B GAL. Lemma 4.7. Let u be the solution of the continuous problems 1.1) and let u I be the bilinear interpolation of u on the B-S-mesh. If Assumption 3.1 holds, then there is a constantc, such that B GAL u u I,v N ) CN v N SD, v N V N. Proof. We rewrite the bilinear form for w H 1 0 Ω): B GAL w,v) = w, v)+b w,v)+cw,v) = w, v) w,b v)+c divb)w,v). We shall use whichever of these expressions is more convenient. B GAL u u I,v N ) B GAL E E I,v N ) + B GAL S S I,v N ). Now we estimate E-term. In the light of Lemma 4.5 and Lemma 4.6, for any v N V N we have E E I ), v N ) CN v N SD, E E I ),b v N ) CN v N SD, c divb)e E I ),v N ) C E E I 0 v N 0 CN v N SD,

18 Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 61 where we used the Cauchy-Schwarz inequality and 4.7). Hence, B GAL E E I,v N ) CN v N SD. 4.6) Next, we analyze S-term. Applying 4.14) and 4.15) to S, we get S S I ) v N x x dxdy 3 S CN v N x y x, S S I ) v N y y dxdy 3 S CN v N x y y. These, together with the Cauchy-Schwarz inequality, prove S S I ), v N ) CN Ω 3 S x y + 3 S x y CN.5 v N SD. ) v N The second term of B GAL S S I,v N ) is handled in Lemma 4.4 that b S S I ),v N ) CN v N SD. Furthermore, standard approximation theory gives us Hence, we obtain cs S I ),v N ) C S S I 0 v N 0 CN v N SD. B GAL S S I,v N ) CN v N SD by collecting the above three bounds. Recalling 4.6), Lemma 10 is proved. It is now straightforward to prove the second main result. Theorem 4.. Letu h be the solution of the discrete problem.) and letu I be the bilinear interpolation of u on the B-S-mesh. If Assumption 3.1 holds, then there is a constant C, such that u I u h SD CN lnn) 1/. Proof. From the inequality 1 u I u h SD B SDu I u h,u I u h ) B GAL u I u,u I u h )+B STAB u I u,u I u h ). Setting v N = u I u h, the second part is shown in [10, Lemma 4.4] that B STAB u I u,v N ) CN lnn) 1/ v N SD.

19 6 Y.-H. Yin, P. Zhu and B. Wang Table 1: Error in the discrete SD norm and maximum norm. = 1.0e 05 = 1.0e 06 = 1.0e 07 N u u h SD,d u u h 0, u u h SD,d u u h 0, u u h SD,d u u h 0, 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 Invoking Lemma 4.7, we get v N SD C B SDu I u,v N ) CN lnn) 1/ v N SD. This proves the statement of the theorem. The combination of Theorem 4.1, Theorem 4. and Lemma 4. leads to our main result directly, i.e., Theorem 4.3. Let u be the solution of the continuous problems 1.1) and let u h be the solution of the discrete problem.). If Assumption 3.1 holds, then u u h SD,d CN lnn) 1/. 5. Numerical results We study the performance of the method when applied to the test problem u+u x +u y +u = f in Ω = 0,1), u = 0 on Ω, where the right-hand side f is chosen such that ux,y) = xy 1 e 1 x)/) 1 e 1 y)/). This function exhibits typical boundary layer behaviour. For our tests we take = 10 5,,10 7 and N =,, 9. The computation was performed by Matlab 7 and a five-points Gauss-Legendre formula was used to estimate the error. Table 1 lists the error in the discrete SD norm and maximum norm. When = 10 7, the errors in the discrete SD norm are plotted on log-log chart in Figure. Another two reference curves with different decay rates have been plotted in Figure. From this Figure, the rate of convergence can be observed directly. From Table 1 and Figure, they are clear illustrations of the convergence results of Theorem 4.3. Table displays the errors for the S-mesh and B-S-mesh. It is obvious that the method on B-S-mesh outperforms the S-mesh for all N.

20 10 0 ON ln 1/ N) Analysis of a SDFEM on Bakhvalov-Shishkin Mesh 63 Table : Error on the B-S-mesh and S-mesh. = 1.0e 05 = 1.0e 06 = 1.0e 07 N B S mesh S mesh B S mesh S mesh B S mesh S mesh 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 ON 3/ ) ON ln N ) Figure : Error: the discrete SD norm, = Conclusion In this paper, the Streamline-Diffusion finite element method is applied to a singularly perturbed convection-diffusion problem posed on the unit square, using a Bakhvalov- Shishkin mesh with piecewise bilinear trial function. The method is shown to be convergent, uniformly in the perturbation parameter, of convergent rateon lnn) 1/ ) in a discrete Streamline-Diffusion norm. It is obvious that the method on Bakhvalov- Shishkin mesh yields more accurate results than on Shishkin s mesh. Acknowledgments This work was supported by Zhejiang Provincial Natural Science Foundation of China Grant No. LY15A010018) and Zhejiang Provincial Department of Education Grant No. Y ). References [1] J. MILLER, E. O RIORDAN, AND G. SHISHIN, Fitted Numerical Methods for Singular Per-

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