Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension

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1 University of ebraska at Omaha Mathematics Faculty Publications Department of Mathematics 2016 Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension Mahboub Baccouch University of ebraska at Omaha, Helmi Temimi Gulf University for Science & Technology Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Baccouch, Mahboub and Temimi, Helmi, "Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension" (2016). Mathematics Faculty Publications This Article is brought to you for free and open access by the Department of Mathematics at It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of For more information, please contact

2 ITERATIOAL JOURAL OF UMERICAL AALYSIS AD MODELIG Volume 13, umber 3, Pages c 2016 Institute for Scientific Computing and Information AALYSIS OF OPTIMAL ERROR ESTIMATES AD SUPERCOVERGECE OF THE DISCOTIUOUS GALERKI METHOD FOR COVECTIO-DIFFUSIO PROBLEMS I OE SPACE DIMESIO MAHBOUB BACCOUCH AD HELMI TEMIMI Abstract. In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(h p+1 ) and O(h p ) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p 1 are used. We further prove that the p-degree DG solution and its derivative are O(h 2p ) superconvergent at the downwind and upwind points, respectively. umerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used. Key words. Discontinuous Galerkin method, convection-diffusion problems, singularly perturbed problems, superconvergence, upwind and downwind points, Shishkin meshes. 1. Introduction Problems involving convection and diffusion arise in several important applications throughout science and engineering, including fluid flow, heat transfer, among many others. Their typical solutions exhibit boundary and/or interior layers. It is wellknown that the standard continuous Galerkin finite element method exhibits poor stability properties for singularly perturbed problems. One of the difficulties in numerically computing the solution of singularly perturbed problems lays in the so-called boundary layer behavior. In the presence of sharp boundary or interior layers, nonphysical oscillations pollute the numerical solution throughout the computational domain. In other words, the solution varies very rapidly in a very thin layer near the boundary. Consult [49, 59, 58, 40, 55, 43] and the references cited therein for a detailed discussion on the topic of singularly perturbed problems. The discontinuous Galerkin (DG) methods have become very popular numerical techniques for solving ordinary and partial differential equations. They have been successfully applied to hyperbolic, elliptic, and parabolic problems arising from a wide range of applications. Over the last years, there has been much interest in applying the DG schemes to problems where the diffusion is not negligible and to convection-diffusion problems. The DG method considered here is a class of finite element methods using completely discontinuous piecewise polynomials for the numerical solution and the test functions. DG method combines many attractive features of the classical Received by the editors August 28, 2014 and, in revised form, October 6, Mathematics Subject Classification. 65M15, 65M60,

3 404 M. BACCOUCH AD H. TEMIMI finite element and finite volume methods. It is a powerful tool for approximating some differential equations which model problems in physics, especially in fluid dynamics or electrodynamics. Comparing with the standard finite element method, the DG method has a compact formulation, i.e., the solution within each element is weakly connected to neighboring elements. DG method was initially introduced by Reed and Hill in 1973 as a technique to solve neutron transport problems [46]. In 1974, LaSaint and Raviart [42] presented the first numerical analysis of the method for a linear advection equation. Since then, DG methods have been used to solve ordinary differential equations [7, 23, 41, 42], hyperbolic [19, 20, 21, 22, 34, 35, 45, 38, 39, 30, 57, 44, 2, 3, 16, 6] and diffusion and convection-diffusion [17, 18, 53, 36] partial differential equations. The proceedings of Cockburn et al. [33] an Shu [51] contain a more complete and current survey of the DG method and its applications. In recent years, the study of superconvergence of numerical methods has been an active research field in numerical analysis. Superconvergence properties for finite element and DG methods have been extensively studied in [7, 11, 37, 42, 56, 52] for ordinary differential equations, [2, 3, 16, 6, 4, 15, 13, 7, 10] for hyperbolic problems and [14, 5, 9, 10, 16, 24, 27, 30] for diffusion and convection-diffusion problems, just to mention a few citations. A knowledge of superconvergence properties can be used to (i) construct simple and asymptotically exact a posteriori estimates of discretization errors and (ii) help detect discontinuities to find elements needing limiting, stabilization and/or refinement. Typically, a posteriori error estimators employ the known numerical solution to derive estimates of the actual solution errors. They are also used to steer adaptive schemes where either the mesh is locally refined (h-refinement) or the polynomial degree is raised (p-refinement). For an introduction to the subject of a posteriori error estimation see the monograph of Ainsworth and Oden [12]. The first superconvergence result for standard DG solutions of hyperbolic PDEs appeared in Adjerid et al. [7]. The authors showed that standard DG solutions of one-dimensional hyperbolic problems using p-degree polynomial approximations exhibit an O(h p+2 ) superconvergence rate at the roots of (p + 1)-degree Radau polynomial. They further established a strong O(h 2p+1 ) superconvergence at the downwind end of every element. Recent work on other numerical methods for convection-diffusion and for pure diffusion problems has been reviewed by Cockburn et al. [32]. In particular, Baumann and Oden [18] presented a new numerical method which exhibits the best features of both finite volume and finite element techniques. Rivière and Wheeler[47] introduced and analyzed a locally conservative DG formulation for nonlinear parabolic equations. They derived optimal error estimates for the method. Rivière et al. [48] analyzed several versions of the Baumann and Oden method for elliptic problems. Wihler and Schwab [54] proved robust exponential rates of convergence of DG methods for stationary convection-diffusion problems in one space dimension. We also mention the work of Castillo, Cockburn, Houston, Süli, Schötzau and Schwab [50, 25, 26] in which optimal a priori error estimates for the hp-version of the local DG (LDG) method for convection-diffusion problems are investigated. Later Adjerid et al. [8, 9] investigated the superconvergence of the LDG method applied to diffusion and transient convection-diffusion problems. More recently, Celiker and Cockburn [27] proved a new superconvergence property of a large class of finite element methods for one-dimensional steady state convection-diffusion problems. We also mention the recent work of Shu et al.

4 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 405 [29, 30] in which the superconvergence property of the LDG scheme for convectiondiffusion equations in one space dimension are proven. Finally, Baccouch [14] analyzed the superconvergence properties of the LDG formulation applied to transient convection-diffusion problems in one space dimension. The author proved that the leading error term on each element for the solution is proportional to a(p+1)-degree right Radau polynomial while the leading error term for the solution s derivative is proportional to a (p + 1)-degree left Radau polynomial, when polynomials of degree at most p are used. He further analyzed the convergence of a posteriori error estimates and proved that these error estimates are globally asymptotically exact under mesh refinement. Cheng and Shu [28] developed a new DG finite element method for solving time dependent partial differential equations with higher order spatial derivatives including the generalized KdV equation, the convection-diffusion equation, and other types of nonlinear equation with fifth order derivatives. Unlike the classical LDG method which was first introduced by Cockburn and Shu in [36] for solving convectiondiffusion problems, their method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. They designed numerical fluxes to ensure the stability of the schemes. Furthermore, they proved sub-optimal p-th order of accuracy when using piecewise p-th degree polynomials, while computational results show the optimal (p + 1)-th order of accuracy, under the condition that p+1 is greater than or equal to the order of the equation. In this work, we study the convergence and superconvergence of the DG method applied to a linear convection-diffusion problem(1). We prove that the p-degree DG solution and its derivative exhibit optimal O(h p+1 ) and O(h p ) convergence rates in the L 2 -norm, respectively. We further prove that the p-degree DG solution and its derivatives are O(h 2p ) superconvergent at the downwind and upwind points, respectively. Ourproofsarevalidfor arbitraryregularmeshes andforp p polynomials with p 1, and for periodic, Dirichlet, and mixed Dirichlet-eumann boundary conditions. We present several numerical examples to validate the theoretical results. To the best knowledge of the authors, this work presents the first analysis of optimal error estimates and superconvergence at the downwind and upwind points. Thispaperisorganizedasfollows: Insection2wepresenttheDGschemeforsolving the convection-diffusion problem and we introduce some notation and definitions. We also present some preliminary results which will be needed in our error analysis. In section 3, we present the DG error analysis and prove our main superconvergence results. In section 4, we present several numerical examples to validate the global superconvergence results. We conclude and discuss our results in section A model problem In this paper, we study the superconvergence properties for the DG method for solving the following one-dimensional convection-diffusion problem (1a) ǫu +cu = f(x), a < x < b,

5 406 M. BACCOUCH AD H. TEMIMI subject to one of the following three kinds of boundary conditions (mixed Dirichlet- eumann, purely Dirichlet, and periodic) which are commonly encountered in practice: (1b) (1c) (1d) u(a) = u l, u (b) = u r, u(a) = u l, u(b) = u r, u(a) = u(b), u (a) = u (b), where f(x) is a smooth function on [a,b]. For the sake of simplicity, we shall consider here only the mixed boundary conditions (1b). This assumption is not essential. If (1c) or (1d) are chosen, the DG method can be easily designed and our results remain true. In this paper, the diffusion constant ǫ is a positive parameter and the velocity c a nonnegative constant. The choice of c > 0 guarantees that the location of the boundary layer is at the outflow boundary x = 1. In our error analysis, we assume that ǫ = O(1). However, our numerical examples indicate that the analysis techniques in this paper is still valid for singularly perturbed problems when Shishkin meshes are used, see section 4. In order to obtain the weak DG formulation, we divide the computational domain Ω = [a,b] into subintervals = [x k 1,x k ], k = 1,...,, where a = x 0 < x 1 < < x = b. We denote the length of by h k = x k x k 1. We also denote h = max h k and h min = min h k as the length of the largest and smallest 1 k 1 k subinterval, respectively. Here, we consider regular meshes, that is h λh min, where λ 1 is a constant (independent of h) during mesh refinement. If λ = 1, then the mesh is uniformly distributed. In this case, the nodes and mesh size are defined by x k = a+kh, k = 0,1,2,...,, h = b a. Throughout this paper, we define v(x k ) = lim v(x k +s) and v(x + s 0 k ) = lim v(x k + s 0 + s) to be the left limit and the right limit of the function v at the discontinuity point x k. We also use [v](x k ) = v(x + k ) v(x k ) to denote the jump of v at x k. The weak DG formulation is obtained by multiplying (1a) on each element by a smooth test function v and integrating over. After integrating by parts we obtain the following weak formulation: (2) (ǫu cu)(x k 1 )v(x k 1 ) (ǫu cu)(x k )v(x k ) ǫu(x k 1 )v (x k 1 ) +ǫu(x k )v (x k ) (ǫv +cv )udx = fvdx. We define the piecewise-polynomial space V p h as the space of polynomials of degree at most p in each subinterval, i.e., V p h = {v : v P p ( ), k = 1,...,}, where P p ( ) is the space of polynomials of degree at most p on. ote that polynomials in the space V p h are allowed to have discontinuities across element boundaries. ext, we approximate the exact solution u(x) by a piecewise polynomial u h (x) V p h. We note that u h is not necessarily continuous at the endpoints of. The discrete formulation consists of finding u h V p h such that: v V p h and k =

6 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 407 1,,, (3) (ǫû h cû h )(x k 1 )v(x + k 1 ) (ǫû h cû h )(x k )v(x k ) ǫû h(x k 1 )v (x + k 1 ) +ǫû h (x k )v (x k I ) (ǫv +cv )u h dx = fvdx, k wherethe numericalfluxes û h and û h are, respectively, the discrete approximations to the traces of u and u at the nodes. In order to complete the definition of the discrete DG method we need to select û h and û h on the boundaries of. For the mixed boundary conditions (1b), we take the following alternating numerical fluxes; see Cheng and Shu [28]: { ul, k = 0, û h (x k ) = u h (x k ), k = 1,...,, { u û h(x k ) = h (x + k ), k = 0,..., 1, (4a) u r, k =. The numerical fluxes associated with the Dirichlet boundary conditions (1c) can be taken as u l, k = 0, û h (x k ) = u h (x k ), k = 1,..., 1, u r, k =, { u û h(x k ) = h (x + k ), k = 0,..., 1, (4b) u h (b )+ p h (u h (b ) u r ), k =. We note that, if the periodic boundary conditions (1d) are used then the numerical fluxes can be taken as (4c) û h (x k ) = u h (x k ), û h(x k ) = u h (x+ k ), k = 0,...,. otation, definitions, and preliminary results. In our analysis we need the pth-degree Legendre polynomial defined by Rodrigues formula [1] d p L p (ξ) = 1 2 p p! dξ p[(ξ2 1) p ], 1 ξ 1, which satisfies the following properties: Lp (1) = 1, Lp ( 1) = ( 1) p, L p ( 1) = p(p+1) 2 ( 1) p+1, and (5) 1 1 L k (ξ) L p (ξ)dξ = 2 2k +1 δ kp, where δ kp is the Kronecker symbol. Mapping the physical element = [x k 1,x k ] into a reference element [ 1,1] by the standard affine mapping (6) x(ξ,h k ) = x k +x k h k 2 ξ, we obtain the p-degree shifted Legendre polynomial L p,k (x) = L ( 2x xk x k 1 p. h k ) In this paper, we define the L 2 inner product of two integrable functions u and v on the interval as u,v) Ik = u(x)v(x)dx. Denote u 0,Ik = ((u,u) Ik ) 1/2 to be the standard L 2 -norm of u on. Moreover, the standard L -norm of u on is defined by u,ik = sup x u(x). on

7 408 M. BACCOUCH AD H. TEMIMI Let H s ( ), where s = 0,1,..., denote the standard Sobolev space of square integrable functions on with all derivatives u (j), j = 0,1,...,s being square integrable on i.e., { } H s ( ) = u : u (j) 2 dx <, 0 j s, andequippedwiththenorm u s,ik = ( s j=0 of a function u on is given by u s,ik = u (s) 0,Ik. ) 1/2.TheH u (j) 2 s (I 0, )-seminorm k We also define the norms on the whole computational domain Ω as follows: ( ) 1/2 ( ) 1/2 u 0,Ω = u 2 0,, u s,ω = u 2 s,, u,ω = max u 1 k,. The seminorm on the whole computational domain Ω is defined as u s,ω = ( ) 1/2.We u 2 s, notethat ifu H s (Ω), s = 1,2,..., the norms u s,ω onthe ( s whole computational domain is the standard Sobolev norm 1/2. u (j) 0,Ω) 2 For convenience, we use u Ik and u to denote u 0,Ik and u 0,Ω, respectively. For p 1, we consider two special projection operators, P ± h, which are defined as follows: For any smooth function u, the restriction of P h u to is the unique polynomial in P p ( ) satisfying (7a) (P h u u)vdx = 0, v Pp 1 ( ), and (P h u u)(x k ) = 0. Similarly, the restriction of P + h u to is the unique polynomial in P p ( ) satisfying (7b) (P + h u u)vdx = 0, v Pp 1 ( ), and (P + h u u)(x+ k 1 ) = 0. These special projections are used in the error estimates of the DG methods to derive optimal L 2 error bounds in the literature, e.g., in [30]. They are mainly used to eliminate the jump terms at the element boundaries in the error estimates in order to prove the optimal L 2 error estimates. In our analysis, we need the following well-known projection results. The proofs can be found in [31]. Lemma 2.1. The projections P ± h u exist and are unique. Moreover, for any u H p+1 ( ) with k = 1,,, there exists a constant C independent of the mesh size h such that (u P ± Ik h u)(s) Ch p+1 s k u p+1,ik, (u P ± (8) h u)(s) Ch p+1 s u p+1,ω,s = 0,1,2. j=0 In addition to the projections P ± h, we also need another projection P h which is defined as follows: For any smooth function u, the restriction of P h u to is the unique polynomial in P p ( ) satisfying the following p +1 conditions: For p = 1, we require the two conditions (9a) ( P h u u) (x + k 1 ) = 0, ( P h u u)(x + k 1 ) = 0.

8 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 409 For p = 2, we require the two conditions in (9a) and ( P h u u)(x k ) = 0 i.e., (9b) ( P h u u) (x + k 1 ) = 0, ( P h u u)(x + k 1 ) = 0, ( P h u u)(x k ) = 0, and, for p 3, we require the three conditions in (9b) and the following p 2 conditions (9c) ( P h u u)vdx = 0, v P p 3 ( ). The existence and uniqueness of P h is provided in the following lemma. Lemma 2.2. The operator P h exists and unique. Moreover, we have the following a priori error estimates: For p = 1, (10) (u P h u)(x k ) h 3/2 k u 2,Ik. Furthermore, for p 1, (u Phu) (s) Ch p+1 s k u p+1,ik, Ik (u P h u) (s) (11) Ch p+1 s u p+1,ω, s = 0,1,2. Proof. For p = 1, Ph u is the first-degree Taylor polynomial for u about x k 1. It can be seen from (9a) that P h u is uniquely given by P h u(x) = u(x + k 1 )+u (x + k 1 )(x x k 1), x. Similarly, for p = 2, the three conditions in (9b) give the unique polynomial P h u(x) =u(x + k 1 )+u (x + k 1 )(x x k 1) + u(x k ) h ku (x + k 1 ) u(x+ k 1 ) h 2 (x x k 1 ) 2, x. k ow, we assume p 3. We are only going to proof the uniqueness, and since we are working with a linear system of equations, the existence is equivalent to the uniqueness. Assume that w 1 and w 2 are two polynomials in P p ( ) which satisfy (9b) and (9c). Then the difference w = w 1 w 2 satisfies the following p+1 conditions (12) w (x + k 1 ) = 0, w(x+ k 1 ) = 0, w(x k ) = 0, wvdx = 0, v P p 3 ( ). We note the w can be expressed in terms of the Legendre polynomials w(x) = p i=0 c il i,k (x), x. Usingthe orthogonalityrelation(5) andthe p 2conditions wvdx = 0, v P p 3 ( ), we get w(x) = c p 2 L p 2,k (x)+c p 1 L p 1,k (x)+c p L p,k (x), x. ow using the three conditions w (x + k 1 ) = 0, w(x+ k 1 ) = 0, w(x k ) = 0 and the properties of Legendre polynomials L i,k (x k 1) = ( 1) i+1 i(i+1) h k, L i,k (x k 1 ) = ( 1) i, and L i,k (x k ) = 1, we get the following linear system of equations ( 1) p 1(p 2)(p 1) c p 2 +( 1) pp(p 1) c p 1 +( 1) p+1(p+1)p c p = 0, h k h k h k ( 1) p 2 c p 2 +( 1) p 1 c p 1 +( 1) p c p = 0, c p 2 +c p 1 +c p = 0.

9 410 M. BACCOUCH AD H. TEMIMI A direct computation reveals that the determinant of the coefficient matrix of this linear system is 4(1 2p) h k 0. Thus, the system has the trivial solution c p 2 = c p 1 = c p = 0. Thus w(x) = 0 which completes the proof of the existence and uniqueness. ext, we will prove (10). We note that, for p = 1, Ph u is the first-degree Taylor polynomial for u about x k 1. Using Taylor s formula with integral remainder, we have (13) u(x) = P h u(x)+ x x k 1 (x k 1 t)u (t)dt. Thus, (u P xk h u)(x k ) = (x k 1 t)u (t)dt x k 1 xk x k 1 x k 1 t u (t) dt. Using x k 1 t h k, x and applying the Cauchy-Schwarz inequality, we obtain (14) ( xk ) 1/2 xk (u Ph u)(x k ) hk u (t) dt h 3/2 k u (t) 2 dt = h 3/2 k u 2,Ik. x k 1 x k 1 The proof of (11) is similar to that of (8) and is omitted. We would like to remark that the operator P h u is introduced only for the purpose of technical proof of error estimates and superconvergence. Finally, we recall some inverse properties of the finite element space V p h which will be used in our error analysis: For any v h V p h, there exists a positive constant C independent of v h and h, such that v (s) (15a) h Ch s k v h Ik, s 1, k = 1,...,, Ik (15b) v h (x + k 1 ) + v h (x k ) Ch 1/2 k v h Ik, k = 1,...,. From now on, the notation C, C 1, C 2, etc. will be used to denote positive constants that are independent of the discretization parameters h, but which may depend upon (i) the exact smooth solution of the differential equation (1a) and its derivatives and (ii) the diffusion constant ǫ. Furthermore, all the constants will be generic, i.e., they may represent different constant quantities in different occurrences. 3. Global convergence and superconvergence error analysis In this section, we investigate the optimal convergence and superconvergence properties of the DG method. We prove that the DG solution and its derivative are O(h 2p ) superconvergent at the downwind and upwind mesh points, respectively. In order to prove these results, we need to derive some error equations. Throughout this paper, e = u u h denotes the error between the exact solution of (1) and the numerical solution defined in (3).

10 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 411 We subtract (3) from (2) with v V p h and we use the numerical flux (4a) to obtain the DG orthogonality condition for the error e on : (16a) A k (e;v) = 0, v V p h, where the bilinear form A k (e;v) is given by A k (e;v) = (ǫe (x + k 1 ) ce(x k 1 ))v(x+ k 1 ) (ǫe (x + k ) ce(x k ))v(x k ) (16b) +ǫe(x k )v (x k ) ǫe(x k 1 )v (x + k 1 ) (ǫv +cv )edx. Performing a simple integration by parts on the last term of A k (e;v) yields A k (e;v) = ǫe (x + k )v(x k )+ǫe (x + k 1 )v(x+ k 1 ) (17) +[e](x k 1 )(ǫv +cv)(x + k 1 )+ (ǫv +cv)e dx. Using another integration by parts, we get (18) A k (e;v) = ǫ[e ](x k )v(x k )+[e](x k 1)(ǫv +cv)(x + k 1 )+ ( ǫe +ce )vdx. We note that, with the numerical fluxes (4a), the jumps of e and e at an interior point x k aredefined as[e](x k ) = e(x + k ) e(x k ) and[e ](x k ) = e (x + k ) e (x k ). Since e(x 0 ) = e (x + ) = 0, the jumps at the endpoints of the computational domain are given by [e](x 0 ) = e(x + 0 ) e(x 0 ) = e(x+ 0 ), [e ](x ) = e (x + ) e (x ) = e (x ), ext, we state and prove the following results needed for our analysis. Theorem 3.1. Let u be the exact solution of (1). Let p 1 and u h be the DG solution of (3) with the numerical fluxes (4a), then there exists a positive constant C which depends on u and ǫ but independent of h such that (19) max e (x + j 1 ) Ch p e. j=1,..., Proof. Weconstructthefollowingauxiliaryproblem: findafunctionv H 1 ([x j 1,b]) such that (20) ǫv +cv = 0, x (x j 1,b], subject to the boundary condition V(x j 1 ) = 1 ǫ, where j = 1,..., is a fixed integer. This problem has the following exact solution V(x) = 1 ( ǫ exp c ) (21) ǫ (x x j 1), x Ω 1 = [x j 1,b]. Using (21), we have the regular estimate V p+1,ω1 C. On the one hand, taking v = V in (17) and using (20), we have for all k = j,..., A k (e;v) = ǫe (x + k )V(x k )+ǫe (x + k 1 )V(x+ k 1 ) +[e](x k 1 )(ǫv +cv)(x + k 1 I )+ (ǫv +cv)e dx k = ǫe (x + k )V(x k)+ǫe (x + k 1 )V(x k 1),

11 412 M. BACCOUCH AD H. TEMIMI which, after summing over, k = j,..., and using the fact that V(x j 1 ) = 1/ǫ and e (x + ) = 0, gives (22) A k (e;v) = ǫe (x + )V(x )+ǫe (x + j 1 )V(x j 1) = e (x + j 1 ). k=j On the other hand, adding and subtracting P h V to V and using (16a) with v = P h V P p ( ), we write A k (e;v) = A k (e;v P h V)+A k (e; P h V) = A k (e;v P h V). Using (17) with v = V P h V, we obtain A k (e;v) = ǫe (x + k )(V P h V)(x k )+ǫe (x + k 1 )(V P h V)(x + k 1 ) +[e](x k 1 )(ǫ(v P h V) +c(v P h V))(x + k 1 ) + (ǫ(v P h V) +c(v P h V))e dx. Applying the properties of the projection P h (9a), we get A k (e;v) = ǫe (x + k )(V P h V)(x k )+ (ǫ(v P h V) +c(v P h V))e dx. Summing over the elements, k = j,..., and using (22), we obtain (23) e (x + j 1 ) = k=j (ǫ(v P h V) +c(v P h V))e dx ǫe (x + k )(V P h V)(x k ). k=j We consider the cases p = 1 and p 2 separately. We first consider the case p 2. Since (V P h V)(x k ) = 0 by (9b), (23) reduces to e (x + j 1 ) = k=j (ǫ(v P h V) +c(v P h V))e dx. Applying the Cauchy-Schwarz inequality and the estimate (11) yields e (x + j 1 ) ( ǫ (V Ph V) +c V Ph V ) e dx k=j (ǫ (V Ph V) 0,Ω1 +c V Ph V ) 0,Ω1 e 0,Ω1 ) (ǫc 1 h p V p+1,ω1 +cc 2 h p+1 V p+1,ω1 e Ch p e. Taking the maximum of both sides, we obtain the estimate (19).

12 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 413 ext, we consider the case p = 1. Using the same steps as above and the estimate (10), (23) gives e (x + j 1 ) k=j + ( ǫ (V Ph V) +c V Ph V ) e dx ǫ e (x + k ) V Ph V)(x k ) k=j Ch e + ǫ e (x + k ) h 3/2 V 2,Ik Ch e +Ch 3/2 ǫ e (x + j ) k=j Ch e +Ch 3/2 max j=1,..., e (x + j 1 ), since e (x + b a ) = 0. Using the fact that h min b a h, we get e (x + j 1 ) Ch e +Ch 1/2 max e (x + j 1 ). j=1,..., Consequently, (1 Ch 1/2 )max j=1,..., e (x + j 1 ) Ch e. Hence, for all h 1/2, we have 1 2C 1 2 max j=1,..., e (x + j 1 ) (1 Ch 1/2 ) max j=1,..., j=1 e (x + j 1 ) Ch e. We conclude that for p 1, max j=1,..., e (x + j 1 ) Ch p e, which completes the proof of (19). ext, we state and prove optimal L 2 error estimate for e. Theorem 3.2. Under the same conditions as in Theorem 3.1, there exists a constant C such that (24) e C h p. Proof. We consider the following auxiliary problem: find a function U H 1 (Ω) such that (25) ǫu +cu = e, x (a,b] subject to U(a) = 0. The above initial-value problem has a unique solution U H 1 (Ω) U(x) = 1 x ( c ) (26) exp ǫ ǫ (s x) e (s)ds, that verify the following regular estimates a (27) U C e, U 1,Ω C e, U 2,Ω C( e + e ), U(b) C e. Taking v = U in (17) and using (25), we obtain A k (e;u) = ǫe (x + k )U(x k)+ǫe (x + k 1 )U(x k 1)+[e](x k 1 )e (x + k 1 )+ (e ) 2 dx,

13 414 M. BACCOUCH AD H. TEMIMI which, after summing over all elements and using the fact that U(a) = e (x + ) = 0, gives A k (e;u) = ǫe (x + )U(x )+ǫe (x + 0 )U(x 0) (28) = + [e](x k 1 )e (x + k 1 )+ [e](x k 1 )e (x + k 1 )+ e 2. (e ) 2 dx Adding and subtracting P h U to U and using (16a) with v = P h U P p ( ), we get (29) A k (e;u) = A k (e;u P h U)+A k (e; P h U) = A k (e;u P h U). Applying (17) with v = U P h U and using the properties of the projection P h (9a), i.e., ( P h u u) (x + k 1 ) = ( P h u u)(x + k 1 ) = 0, we obtain A k (e;u) = ǫe (x + k )(U P h U)(x k ) (30) + (ǫ(u P h U) +c(u P h U))e dx. Summing over all the elements, k = 1,...,, we arrive at A k (e;u) = ǫ e (x + k )(U P h U)(x k ) (31) + Combining (28) and (31), we get (32) where b a (ǫ(u P h U) +c(u P h U))e dx. e 2 = T 1 +T 2 +T 3, T 1 = b a (ǫ(u P h U) +c(u P h U))e dx, T 2 = ǫ e (x + k )(U P h U)(x k ), T 3 = ext, we will estimate T k, k = 1,2,3 one by one. [e](x k 1 )e (x + k 1 ). Estimate of T 1. Applying the Cauchy-Schwarz inequality and using the estimate (11) yields (33) T 1 ǫ ( (U Ph U) +c U Ph U ) e ) ǫ (Ch p u p+1,ω +cch p+1 u p+1,ω e C 1 h p e. Estimate of T 2. We consider the cases p = 1 and p 2 separately. We first consider the case p 2. Using the properties of the projection P h (9b), we have

14 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 415 ( P h U U)(x k ) = 0. Thus, T 2 = 0 for p 2. ext, we consider the case p = 1. Using (19) with p = 1, (10), and the regularity estimate (27), we obtain T 2 ǫ e (x + k ) (U P h U)(x k ) (Ch e )(h 3/2 U 2,Ik ) = Ch 5/2 e U 2,Ik Ch 5/2 e 1/2 U 2,Ω C 3 h 5/2 1/2 e ( e + e ) Since b a h, we get T 2 C 3 h 2 e ( e + e ) Using the smoothness of u, we have e = u u h u + u h C since, for p = 1, u h is piecewise constant. Furthermore, e = u C since u h = 0 for p = 1. We conclude that (34) T 2 C 2 h 2, for p = 1 and T 2 = 0, for p 2. Estimate of T 3. Using the estimate (19), we have T 3 [e](xk 1 ) e (x + k 1 ) Ch p e [e](xk 1 ). ext, we will estimate [e](xk 1 ). Taking v = 1 in (18) and using (16a), we get [e](x k 1 ) = ǫ[e ](x k )+ (ǫe ce )dx = ǫe (x + k ) ǫe (x + k 1 ) c e dx. Using the estimate (19) and applying the Cauchy-Schwarz inequality yields [e](x k 1 ) ǫ e (x + k ) +ǫ e (x + k 1 ) +ch 1/2 e 0,Ik ǫch p e +ǫch p e +ch 1/2 e 0,Ik C 1 h p e +ch 1/2 e 0,Ik. Summing over all elements, applying the Cauchy-Schwarz inequality, and using the fact that b a, we get h min b a h [e](xk 1 ) C 1 h p e +ch 1/2 e 0,Ik Therefore, we conclude that (35) C 1 h p e +ch 1/2 1/2 e C 1 (b a)h p 1 e +c(b a) 1/2 e C 2 h p 1 e +C 3 e C 4 e, p 1. T 3 C 3 h p e 2. ow, combining (32) with (33), (34), (35) and applying the inequality ab 1 2 a b2, we get, for p 2, e 2 C 1 h p e +C 3 h p e C2 1h 2p e 2 +C 3 h p e 2,

15 416 M. BACCOUCH AD H. TEMIMI which gives e 2 C 2 1h 2p +2C 3 h p e 2. Similarly, if p = 1, we have an extra term T 2 C 2 h 2. Thus, for all p 1, we have e 2 Ch 2p +Ch p e 2 Ch 2p +Ch e 2, Hence, for all h 1 2C, we have 1 2 e 2 (1 Ch) e 2 Ch 2p, which completes the proof of (24). In the following corollary, we state and prove 2p-order superconvergence of the solution s derivative at the upwind points. Corollary 3.1. Under the same conditions as in Theorem 3.1, there exists a constant C such that (36) max e (x + j 1 ) Ch 2p. j=1,..., Proof. Combining the estimates (19) and (24), we immediately obtain (36). ext, we state and prove optimal L 2 error estimates for e. Theorem 3.3. Under the same conditions as in Theorem 3.1, there exists a constant C such that (37) e C h p+1. Proof. The main idea behind the proof of (37) is to construct the following adjoint problem: find a function W such that (38) ǫw cw = e, x (a,b), subject to W(a) = 0, ǫw (b)+cw(b) = 0. This boundary-value problem has the following exact solution W(x) = 1 x ( y ( c ) ) e(s) exp ǫ a b ǫ (s y) ds dy + 1 e c ǫ (x a) b ( y ( c ) ) e(s) exp ǫ ǫ (s y) ds dy, that satisfy the regular estimate a b (39) W 2,Ω C e. On the one hand, taking v = W in (16b) and using (38), we obtain A k (e;w) = (ǫe (x + k 1 ) ce(x k 1 ))W(x k 1) (ǫe (x + k ) ce(x k ))W(x k) +ǫe(x k )W (x k ) ǫe(x k 1 )W (x + k 1 ) (ǫw +cw )edx (40) = (ǫe (x + k 1 ) ce(x k 1 ))W(x k 1) (ǫe (x + k ) ce(x k ))W(x k) +ǫe(x k )W (x k ) ǫe(x k 1 )W (x k 1 )+ e 2 dx.

16 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 417 Summing over all elements and using W(x 0 ) = e(x 0 ) = e (x + ) = 0 and ǫw (x )+ cw(x ) = 0, gives (41) A k (e;w) = (ǫe (x + 0 ) ce(x 0 ))W(x 0) (ǫe (x + ) ce(x ))W(x ) +ǫe(x )W (x ) ǫe(x 0 )W (x 0 )+ e 2 = e(x )(ǫw (x )+cw(x ))+ e 2 = e 2. On the other hand, adding and subtracting P h W to W and applying (16a) with v = P h W P p ( ) yields (42) A k (e;w) = A k (e;w P h W)+A k (e; P h W) = A k (e;w P h W). We consider the cases p = 1 and p 2 separately. We first consider the case p 2. Using (17) and the property of the projection P h, (42) gives (43) ( A k (e;w) = ǫ(w Ph W) +c(w P h W) ) e dx, which, after summing over all elements and applying the Cauchy-Schwarz inequality yields A k (e;w) = (ǫ(w P h W) +c(w P h W))e dx ( ǫ (W Ph W) +c W Ph W ) e Applying the standard interpolation error estimate (11), the regularity estimate (39), and the estimate (24), we get (44) A k (e;w) (ǫc 1 h W 2,Ω +cc 2 h 2 W 2,Ω )C 3 h p (ǫc 1 hc 4 e +cc 2 h 2 C 4 e )C 3 h p C(h p+1 +h p+2 ) e Ch p+1 e. Combining the two formulas (41) and (44) yields e 2 Ch p+1 e, which completes the proof of (37) in the case p 2. ext, we consider the case p = 1. We note that (41) is still valid. However (43) is not since P h W is defined by the two conditions (9a). Thus, we proceed differently. Adding and subtracting P h W to W, using (16a), (17), and the properties of the operator P h (9a), we obtain A k (e;w) = A k (e;w P h W) (45) = ǫe (x + k )(W P h W)(x k )+ (ǫ(w P h W) +c(w P h W))e dx.

17 418 M. BACCOUCH AD H. TEMIMI Summing over all the elements, k = 1,..., and using (41), we get e 2 = + ǫe (x + k )(W P h W)(x k ) (ǫ(w P h W) +c(w P h W))e dx ǫ e (x + k ) (W Ph W)(x k ) + ( ǫ (W P h W) +c W P h W ) e dx. Applying the Cauchy-Schwarz inequality, using the estimate (36) with p = 1, and using the standard interpolation error estimate (10) i.e., (W Ph W)(x k ) h 3/2 k W 2,Ik, k = 1,...,, we obtain e 2 ǫ(c 1 h 2 )(h 3/2 k W 2,Ik )+ ( ǫ (W Ph W) +c W Ph W ) e C 2 ǫh 7/2 W 2,Ik + ( ǫ (W Ph W) +c W Ph W ) e. ApplyingtheCauchy-Schwarzinequality ( a ) 1/2 ( 1/2, kb k a2 k k) b2 the error estimates (11), (24), and the regular estimate (39), we get (46) ( ) ( 1/2 ) 1/2 e 2 b a C 2 ǫh 7/2 W 2 2,I h k ) + (ǫc 3 h W 2,Ω +cc 4 h 2 W 2,Ω C 5 h = ( C 2 (b a)ǫh 3 +C 5 ( ǫc3 h 2 +cc 4 h 3)) W 2,Ω ( C 2 (b a)ǫh 3 +C 5 ( ǫc3 h 2 +cc 4 h 3)) C 6 e C(1+h)h 2 e, Thus, we get e C(1+h)h 2 = O(h 2 ), which completes the proof of (37) in the case p = 1. Finally, we state and prove 2p-order superconvergence of the solution at the downwind points. Theorem 3.4. Under the same conditions as in Theorem 3.1, there exists a constant C such that e(x (47) j ) Ch 2p, j = 1,...,. Proof. Again, the main idea behind the proof of (47) is to construct the following auxiliary problem: find a function ϕ H 2 ([a,x j ]) such that (48) ǫϕ cϕ = 0, x [a,x j ), subject to ϕ(x j ) = 0, ϕ (x j ) = 1 ǫ,

18 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 419 where j = 1,..., is a fixed integer. This problem has the following exact solution ϕ(x) = 1 ( c )) (49) 1 exp( c ǫ (x j x), x Ω 2 = [a,x j ]. Using (49), we can easily show the following estimates ϕ(a) = 1 ( c ) ) (50) exp( c ǫ (x j a) 1 = C > 0, ϕ p+1,ω2 C. We follow the reasoning of Adjerid et al. [11, Theorem 2.2]. First we prove (47) for the case p 3. For p 3, let us consider an interpolation operator I h which is definedasfollows: Foranysmoothfunction ϕ, I h ϕ V p h H2 (Ω) andtherestriction of I h ϕ to is the unique polynomial in P p ( ) satisfying: for each k = 1,...,j, I h ϕ(x + k 1 ) = ϕ(x+ k 1 ), (I hϕ) (x + k 1 ) = ϕ (x + k 1 ), I h ϕ(x k ) = ϕ(x k ), (I hϕ) (x k ) = ϕ (x k ), and I h ϕ interpolates ϕ at p 3 additional distinct points x k, k = 1,...,p 3 in (x k 1,x k ). In our analysis, we need the following a priori error estimate [31]: For any ϕ H p+1 (Ω 2 ), there exists a constant C independent of the mesh size h such that (51) ϕ I h ϕ s,ω2 Ch p+1 s ϕ p+1,ω2, s = 0,1,2. On the one hand, taking v = ϕ in (16b) and using (48), we obtain (52) A k (e;ϕ) = (ǫe (x + k 1 ) ce(x k 1 ))ϕ(x+ k 1 ) (ǫe (x + k ) ce(x k ))ϕ(x k ) +ǫe(x k )ϕ (x k ) ǫe(x k 1 )ϕ (x + k 1 ) (ǫϕ +cϕ )edx = (ǫe (x + k 1 ) ce(x k 1 ))ϕ(x k 1) (ǫe (x + k ) ce(x k ))ϕ(x k) +ǫe(x k )ϕ (x k ) ǫe(x k 1 )ϕ (x k 1 ). Summing over all the elements, k = 1,...,j and using ϕ(x j ) = e(x 0 ) = 0, and ϕ (x j ) = 1/ǫ, gives (53) A k (e;ϕ) = (ǫe (x + 0 ) ce(x 0 ))ϕ(x 0) (ǫe (x + j ) ce(x j ))ϕ(x j) +ǫe(x j )ϕ (x j ) ǫe(x 0 )ϕ (x 0 ) = ǫe (x + 0 )ϕ(x 0)+e(x j ). On the other hand, adding and subtracting I h ϕ to ϕ and using (16a), we write A k (e;ϕ) as (54) A k (e;ϕ) = A k (e;ϕ I h ϕ)+a k (e;ϕ) = A k (e;ϕ I h ϕ). Using (16b) and the properties of the operator I h, we obtain (55) A k (e;ϕ) = (ǫe (x + k 1 ) ce(x k 1 ))(ϕ I hϕ)(x + k 1 ) (ǫe (x + k ) ce(x k ))(ϕ I hϕ)(x k ) +ǫe(x k )(ϕ I hϕ) (x k ) ǫe(x k 1 )(ϕ I hϕ) (x + k 1 ) (ǫ(ϕ I h ϕ) +c(ϕ I h ϕ) )edx I k = (ǫ(ϕ I h ϕ) +c(ϕ I h ϕ) )edx.

19 420 M. BACCOUCH AD H. TEMIMI Summing over all the elements, k = 1,...,j, we get (56) A k (e;ϕ) = (ǫ(ϕ I h ϕ) +c(ϕ I h ϕ) )edx. Combining the two formulas (53) and (56) yields (57) ǫe (x + 0 )ϕ(x 0)+e(x j ) = (ǫ(ϕ I h ϕ) +c(ϕ I h ϕ) )edx. Using the estimates (50) and (36) and applying the Cauchy-Schwarz inequality yields e(x j ) ǫ e (x + 0 ) ϕ(x0 ) + (ǫ (ϕ I h ϕ) +c (ϕ I h ϕ) ) e dx (58) ǫ(c 0 h 2p )(C 1 )+(ǫ (ϕ I h ϕ) 0,Ω2 +c (ϕ I h ϕ) 0,Ω2 ) e 0,Ω2 ǫc 0 C 1 h 2p +(ǫ (ϕ I h ϕ) 0,Ω2 +c (ϕ I h ϕ) 0,Ω2 ) e. Applying the standard interpolation error estimate (51) and the estimate (37), we get e(x j ) ǫc 0 C 1 h 2p +(ǫc 2 h p 1 ϕ p+1,ω (59) which completes the proof of (47) when p 3. +cc 3 h p ϕ p+1,ω )C 4 h p+1 = O(h 2p ), For p = 2, let us consider another interpolation operator T h which is defined as follows: For any smooth function ϕ, T h ϕ V 2 h and the restriction of T hϕ to is the unique polynomial in P 2 ( ) satisfying: for each k = 1,,j, T h ϕ(x + k 1 ) = ϕ(x+ k 1 ), T hϕ(x k ) = ϕ(x k ), (T hϕ) (x k ) = ϕ (x k ). In our analysis, we need the following a priori error estimate [31]: For any ϕ H 3 (Ω 2 ), there exists a constant C independent of the mesh size h such that (60) ϕ T h ϕ s,ω2 Ch 3 s ϕ 3,Ω2, s = 0,1,2. Adding and subtracting T h ϕ to ϕ, using (16a), and the properties of the operator T h, we obtain A k (e;ϕ) = A k (e;ϕ T h ϕ) (61) = (ǫe (x + k 1 ) ce(x k 1 ))(ϕ T hϕ)(x + k 1 ) (ǫe (x + k ) ce(x k ))(ϕ T hϕ)(x k ) +ǫe(x k )(ϕ T hϕ) (x k ) ǫe(x k 1 )(ϕ T hϕ) (x + k 1 ) (ǫ(ϕ T h ϕ) +c(ϕ T h ϕ) )edx = ǫe(x k 1 )(ϕ T hϕ) (x + k 1 ) (ǫ(ϕ T h ϕ) +c(ϕ T h ϕ) )edx.

20 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 421 Summing over all the elements, k = 1,...,j and using (53), we arrive at ǫe (x + 0 )ϕ(x 0)+e(x j ) = ǫe(x k 1 )(ϕ T hϕ) (x + k 1 ) (ǫ(ϕ T h ϕ) +c(ϕ T h ϕ) )edx. Applying the Cauchy-Schwarz inequality and using the estimates(50), (36), and the standardinterpolationerrorestimates (ϕ Th ϕ) (x + k 1 ) Ch 2 k ϕ (3),Ik, k = 1,...,j, we obtain (62) e(x j ) ǫ e (x + 0 ) ϕ(x 0 ) + ǫ e(x k 1 ) (ϕ T h ϕ) (x + k 1 ) + (ǫ (ϕ T h ϕ) +c (ϕ T h ϕ) ) e dx ǫ(c 0 h 4 )(C 1 )+C 2 ǫh 2 ϕ (3),Ω2 e(x k 1 ) + (ǫ (ϕ T h ϕ) 0,Ω2 +c (ϕ T h ϕ) 0,Ω2 ) e 0,Ω2. Applying the standard interpolation error estimate (60) and the estimate (37), we get (63) e(x j ) ǫc 0 C 1 h 4 +C 2 ǫh 2 ϕ (3),Ω2 +(ǫc 2 h ϕ 3,Ω +cc 3 h 2 ϕ 3,Ω )C 4 h 3 ( ) C h 4 +h 2 e(x k 1 ). e(x k 1 ) ext, we use induction and (63) to prove e(x j ) Ch 4. Taking v = 1 in (16b) with k = 1, using (16a) and e(x 0 ) = 0, we get (64) 0 = A 1 (e;1) = ǫe (x + 0 ) ce(x 0 ) ǫe (x + 1 )+ce(x 1 ) = ǫe (x + 0 ) ǫe (x + 1 )+ce(x 1 ). Applying (36) with p = 2, we obtain e(x 1 ) = ǫ c (e (x + 1 ) e (x + 0 )) ǫ c ( e (x + 1 ) + e (x + 0 ) ) (65) ǫ c (C 0h 4 +C 1 h 4 ) = Ch 4. ext, if we assume that e(x k ) C 1 h 4, k < j, then (63) yields e(x j ) ( ) ( ) C h 4 +h 2 e(x k 1 ) C h 4 +h 2 C 1 h 4 C ( h 4 +C 1 h 5) = O(h 4 ), which completes the proof of (47) for the case p = 2.

21 422 M. BACCOUCH AD H. TEMIMI Finally, we consider the case p = 1. We introduce an interpolation operator L h which is defined as follows: For any smooth function ϕ, L h ϕ Vh 1 and the restrictionofl h ϕto istheuniquepolynomialinp 1 ( )satisfying: foreachk = 1,...,j, L h ϕ(x + k 1 ) = ϕ(x+ k 1 ), L hϕ(x k ) = ϕ(x k ). The following a priori error estimate [31] holds: For any ϕ H 2 (Ω 2 ), there exists a constant C independent of the mesh size h such that (66) ϕ L h ϕ s,ω2 Ch 2 s ϕ 2,Ω2, s = 0,1. Adding and subtracting L h ϕ to ϕ, using (16a), (16b), and the properties of the operator L h, we obtain A k (e;ϕ) = A k (e;ϕ L h ϕ) = (ǫe (x + k 1 ) ce(x k 1 ))(ϕ L hϕ)(x + k 1 ) (ǫe (x + k ) ce(x k ))(ϕ L hϕ)(x k ) +ǫe(x k )(ϕ L hϕ) (x k ) ǫe(x k 1 )(ϕ L hϕ) (x + k 1 ) (ǫ(ϕ L h ϕ) +c(ϕ L h ϕ) )edx = ǫe(x k )(ϕ L hϕ) (x k ) (67) ǫe(x k 1 )(ϕ L hϕ) (x + k 1 ) (ǫ(ϕ L h ϕ) +c(ϕ L h ϕ) )edx. Since L h ϕ is a linear function on, we have (L h ϕ) (x) = 0. Thus, (67) simplifies to A k (e;ϕ) = ǫe(x k )(ϕ L hϕ) (x k ) ǫe(x k 1 )(ϕ L hϕ) (x + k 1 ) (ǫϕ +c(ϕ L h ϕ) )edx. Summing over all the elements, k = 1,...,j, using e(x 0 ) = 0 and (53) we arrive at ǫe (x + 0 )ϕ(x 0)+e(x j ) = ǫe(x j )(ϕ L hϕ) (x j ) ǫe(x 0 )(ϕ L hϕ) (x + 0 ) (ǫϕ +c(ϕ L h ϕ) )edx = ǫe(x j )(ϕ L hϕ) (x j ) (ǫϕ +c(ϕ L h ϕ) )edx. Applying the Cauchy-Schwarz inequality and using the estimates(50), (36), and the standard interpolation error estimates (ϕ L h ϕ) (x + k 1 ) Ch k ϕ,ik, k = 1,...,j, we obtain e(x j ) ǫ e (x + 0 ) ϕ(x 0 ) +ǫ e(x j ) (ϕ L h ϕ) (x j ) + (ǫ ϕ +c (ϕ L h ϕ) ) e dx ǫ(c 0 h 2 )(C 1 )+C 2 ǫh ϕ,ij e(x j ) +(ǫ ϕ 0,Ω2 +c (ϕ L h ϕ) 0,Ω2 ) e 0,Ω2.

22 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 423 Applying the standard interpolation error estimate (66) and the estimate (37), we get e(x j ) ǫc 0 C 1 h 2 +C 2 ǫh ϕ,ω2 e(x j ) +(ǫ ϕ 0,Ω +cc 3 h ϕ 2,Ω )C 4 h 2 (68) C ( h 2 +h e(x j ) ), which gives (1 Ch) e(x j ) Ch 2. Therefore, for small h, e(x j ) = O(h 2 ). Thus, we have completed the proof of (47) for the case p = 1. WE conclude that the superconvergence result (47) is valid for all p e 10-4 e' p=1, Slp=-2.13 p=2, Slp=-3.01 p=3, Slp=-3.99 p=4, Slp= p=1, Slp=-1.09 p=2, Slp=-1.99 p=3, Slp=-2.99 p=4, Slp= Figure 1: The L 2 -norm of the error e (left) and the derivative of error e (right) for Example 4.1 versus using p = 1,2,3,4. Table 1. Maximum errors e at the downwind points for Example 4.1 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order 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 umerical Experiments The purpose of this section is to validate the superconvergence results of this paper. We use the DG method and carry out several experiments by numerically solving the model problem (1a) subject to either mixed Dirichlet-eumann or purely Dirichlet boundary conditions. In addition, we use the DG method to solve a nonlinear boundary-value problem to show numerically that the achieved results are still valid for the nonlinear case. In all computations, we have used uniform and non-uniform meshes and observed similar results. We compute the maximum DG errors e at the downwind point of each element and then take the maximum over all

23 424 M. BACCOUCH AD H. TEMIMI Table 2. Maximum errors e at the upwind points for Example 4.1 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order e Order e Order e Order 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 elements, k = 1,...,. Similarly, the maximum DG errors e is computed at upwind point of each element and by taking the maximum over all elements, i.e., e = max 1 k e(x k ), e = max 1 k e (x + k 1 ). Example 4.1. We consider the following convection-diffusion problem subject to the mixed boundary conditions 0.5u +u = (x 1)sinh(x)+(1 0.5x)cosh(x), x [0,4], u(0) = 0, u (4) = 4sinh(4)+cosh(4). The exact solution is given by u(x) = xcosh(x). We solve this problem using the DG method on uniform meshes having = 10, 20, 30, 40, 50, 60 elements and using the spaces P p with p = 1,2,3, and 4. In Figure 1, we plot e and e versus in a log-log graph in order to obtain the convergence rates for e and e. We conclude that e = O(h p+1 ) and e = O(h p ). In Tables 1 and 2, we present the maximum errors e and e as well as their order of convergence. These tables show that the DG errors e and e are O(h 2p ) superconvergent, respectively, at the downwind and upwind endpoints of each subinterval. These results are in full agreement with the theory e 10-4 e' p=1, Slp=-2.54 p=2, Slp=-3.14 p=3, Slp=-4.02 p=4, Slp= p=1, Slp=-1.08 p=2, Slp=-2.05 p=3, Slp=-3.00 p=4, Slp= Figure 2: The L 2 -norm of the error e (left) and the derivative of error e (right) for Example 4.2 versus using p = 1,2,3,4.

24 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS 425 Table 3. Maximum errors e at the downwind points for Example 4.2 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order 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 Table 4. Maximum errors e at the upwind points for Example 4.2 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order e Order e Order e Order 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 Example 4.2. In this example, we consider the following convection-diffusion problem subject to the Dirichlet boundary conditions u +u = e x (sin(x) cos(x)), x [0,π], u(0) = u(π) = 0. The exact solution is given by u(x) = e x sinx. We solve this problem using the DG method on uniform meshes having = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 elements and using the spaces P p with p = 1,2,3 and 4. In order to obtain the convergence rates for e and e, we plot e and e versus in Figure 2 using a log-log graph. We observe that e = O(h p+1 ) and e = O(h p ). The maximum errors e and e as well as their order of convergence shown in Tables 3 and 4 indicate that the DG errors e and e are O(h 2p ) superconvergent, respectively, at the downwind and upwind points of each element. These results confirm the theoretical findings of this paper. Example 4.3. We consider the following singularly perturbed problem subject to the Dirichlet boundary conditions ǫu +u = exp(x), x [0,1], u(0) = 0, u(1) = 0,

25 426 M. BACCOUCH AD H. TEMIMI u u h u(x), u h (x) u'(x), u' h (x) x u' u' h x Figure 3: The solutions u and u h (left) and u and u h (right) for Example 4.3 with ǫ = 10 4 using =20 with p = u u h u(x), u h (x) u'(x), u' h (x) x u' u' h x Figure 4: The solutions u and u h (left) and u and u h (right) for Example 4.3 with ǫ = 10 6 using =20 with p = u u h u(x), u h (x) u'(x), u' h (x) x u' u' h x Figure 5: The solutions u and u h (left) and u and u h (right) for Example 4.3 with ǫ = 10 8 using =20 with p = 2. where ǫ is a small positive parameter. The exact solution is given by (1 exp(1))exp( x 1 ǫ )+(1 exp( 1 ǫ ))exp(x)+exp(1 1 ǫ ) 1 u(x) = (1 ǫ)(1 exp( 1 ǫ )), ǫ 1, exp(1) exp(1) 1 (exp(x) 1) xexp(x), ǫ = 1. We note that the true solution has a boundary layer with the width O(ǫ lnǫ ) at the boundary x = 1. We solve this problem with ǫ = 10 4,10 6,10 8 using the polynomial spaces V p h, p = 1,2,3,4 on Shishkin meshes [55] having elements, where in an even positive integer, and using a mesh transition parameter τ = (2p + 1)ǫln( + 1) which denotes the approximate width of the boundary layer.

26 DG METHOD FOR COVECTIO-DIFFUSIO PROBLEMS p=1, Slp=-2.35 p=2, Slp= p=3, Slp=-4.09 p=4, Slp= e e' p=1, Slp=-1.02 p=2, Slp=-2.04 p=3, Slp=-3.04 p=4, Slp= Figure 6: The L 2 -norm of the error e (left) and the derivative of error e (right) for Example 4.3 with ǫ = 10 4 versus using p = 1,2,3, p=1, Slp=-2.29 p=2, Slp= p=3, Slp=-4.09 p=4, Slp= e 10-8 e' p=1, Slp=-1.02 p=2, Slp=-2.04 p=3, Slp=-3.04 p=4, Slp= Figure 7: The L 2 -norm of the error e (left) and the derivative of error e (right) for Example 4.3 with ǫ = 10 6 versus using p = 1,2,3, e 10-8 e' p=1, Slp= p=2, Slp=-3.16 p=3, Slp=-4.10 p=4, Slp= p=1, Slp=-1.12 p=2, Slp=-2.06 p=3, Slp=-3.04 p=4, Slp= Figure 8: The L 2 -norm of the error e (left) and the derivative of error e (right) for Example 4.3 with ǫ = 10 8 versus using p = 1,2,3,4. The computational domain [0,1] is divided into two subintervals [0,1 τ] and [τ,1]. Each interval [0,1 τ] and [1 τ,1] is uniformly subdivided into 2 subintervals which yields a Shishkin mesh.

27 428 M. BACCOUCH AD H. TEMIMI Table 5. Maximum errors e at the downwind points for Example 4.3 with ǫ = 10 4 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order e e e e e e e e e e e e e e e e e e e e e e e e Table 6. Maximum errors e at the downwind points for Example 4.3 with ǫ = 10 6 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order e e e e e e e e e e e e e e e e e e e e e e e e Table 7. Maximum errors e at the downwind points for Example 4.3 with ǫ = 10 8 using p =1, 2, 3 and 4. p = 1 p = 2 p = 3 p = 4 e Order e e e e e e e e e e e e e e e e e e e e e e e e We plot the DG solution and its derivative in Figures 3-5 using = 20, p = 2, and ǫ = 10 4,10 6,10 8. We observe that the DG solutions do not have any oscillatory behavior near the boundary layer at the outflow boundary x = 1. The L 2 error norms shown in Figures 6-8, respectively, exhibit an O(h p+1 ) and O(h p ) convergence rates. In Tables 5-7 and 8-10, respectively, we present the maximum errors e and e aswell as their orderofconvergence. These tables showthat the DG errors e and e are O(h 2p ) superconvergent, respectively, at the downwind and upwind endpoints of each subinterval. These results indicate that the analysis techniques in this paper is still valid for singularly perturbed problems. The analysis remains an open problem for the DG method and will be investigated in the future.