Superconvergence of the Direct Discontinuous Galerkin Method for Convection-Diffusion Equations

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1 Superconvergence of the Direct Discontinuous Galerkin Method for Convection-Diffusion Equations Waixiang Cao, Hailiang Liu, Zhimin Zhang,3 Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 0093, China Department of Mathematics, Iowa State University, Ames, Iowa Department of Mathematics, Wayne State University, Detroit, Michigan 480 Received October 05; accepted 6 July 06 Published online in Wiley Online Library wileyonlinelibrary.com). DOI 0.00/num.087 This paper is concerned with superconvergence properties of the direct discontinuous Galerkin DDG) method for one-dimensional linear convection-diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a k-th and k + ) -th order superconvergence rate of the DDG approximation at nodes and Lobatto points, respectively, and a k + ) -th order of the derivative approximation at Gauss points, where k is the polynomial degree. Moreover, we also prove that the DDG solution is superconvergent with an order k + to a particular projection of the exact solution. Numerical experiments are presented to validate the theoretical results. 06 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: , 06 Keywords: convection-diffusion equations; direct discontinuous Galerkin methods; superconvergence I. INTRODUCTION In this paper, we study the superconvergence of the direct discontinuous Galerkin DDG) method for the one-dimensional linear convection-diffusion equation t u + x f u) = x u, x, t) [a, b] [0, T ], ux,0) = u 0 x), x [a, b],.) Correspondence to: Waixiang Cao, Beijing Computational Science Research Center, Beijing 00094, China wxcao@csrc.ac.cn) Contract grant sponsor: China Postdoctoral Science Foundation; contract grant number: 05M57006 Contract grant sponsor: National Natural Science Foundation of China NSFC); contract grant numbers: 5006, 4703, and Contract grant sponsor: US National Science Foundation NSF); contract grant number: DMS Contract grant sponsor: US National Science Foundation NSF); contract grant number: DMS-3636 and RNMS KI-Net) 079 H. Liu) 06 Wiley Periodicals, Inc.

2 CAO, LIU, AND ZHANG where the linear flux f = αu. For simplicity, we only consider the case α 0 and the periodic boundary condition. For equations containing higher-order spatial derivatives, such as the convection-diffusion equation.), the discontinuous Galerkin DG) method [ 6] cannot be directly applied, due to the discontinuous solution space at the element interfaces, which is not regular enough to handle higher derivatives. Several DG methods have been suggested in the literature to solve this problem, including the local discontinuous Galerkin LDG) method [7], the interior penalty IP) method [8 0], and the method introduced by Baumann-Oden [, ], and the direct DG discretization methods see, e.g., [3 5]). In 009, Liu and Yan [6] introduced a new direct DG method for convection-diffusion problems. The basic idea of this method is to directly force the weak formulation of the PDE into the DG method, without introducing any auxiliary variables or rewriting the original equation into a first-order system. It was also discussed in this paper how a careful choice of numerical fluxes can be made to ensure stability and accuracy of the DDG method. Later, Liu et al. studied a set of more effective choices of parameters in the DDG numerical fluxes [7, 8], and proved an optimal L error estimates for the DDG methods solving convection-diffusion problems in [9]. The key feature of DDG methods lies in its special choice of the numerical flux for the solution derivative at cell interfaces. This gives DDG methods extra flexibility and advantage over the IPDG method [0] and the LDG method [7]. The DDG method is the only known DG method which can satisfy the maximum principle for Fokker-Planck equations up to third order [], yet both IPDG method and LDG method can be proved to satisfy maximum principle with only up to second order of accuracy even for diffusion without drift see e.g. []). The additional parameter in the DDG numerical flux also makes it possible for the DG method for Fokker Planck type equations to satisfy some entropy dissipation law at the discrete level, see, e.g. [3, 4]. The superconvergence result obtained for a nontrivial β presented in this paper provides another example of advantages of the DDG method. The main purpose of this paper is to study and reveal the superconvergence phenomenon of the DDG method for the convection-diffusion equation.). To the best of our knowledge, there was no any superconvergence result of the DDG method for these problems in the literature. Our results include the superconvergence properties of the DDG approximation at nodal and Lobatto points, and the derivative approximation at Gauss points, as well as the supercloseness between the DDG solution and a particular projection of the exact solution. To be more precise, we prove that, under a careful choice of the parameters in the DDG numerical fluxes, the DDG solution is superconvergent at all Gauss derivative approximation) and Lobatto points, with an order k + and k +, respectively; and the error between the DDG solution and the Gauss Lobatto projection of the exact solution achieves k + ) -th order. We further prove a k-th order superconvergence rate of the DDG approximation at nodes. As we may recall, all these superconvergence results are the same as the counterpart finite element methods see. e.g., [5 7]). Moreover, the influence of the choice of the parameters in the numerical fluxes on the superconvergence is also discussed. The analysis is based on the correction function idea, which is motivated from its successful application to the FEM and finite volume methods FVM) for Poisson equations [8, 9], and the DG or LDG methods for hyperbolic and parabolic equations [30 3]. The essence of the correction idea is to construct a specially designed function which vanishes or is of high order at some special points. With the correction function, we first construct a special interpolation function of the exact solution, and then prove the interpolation function is superconvergent to the numerical solution, with the highest order k. Finally the supercloseness between the interpolation function and the numerical solution gives the desired superconvergence results at some special points: Gauss, Lobatto, and nodal points. Numerical Methods for Partial Differential Equations DOI 0.00/num

3 SUPERCONVERGENCE OF THE DDG METHOD 3 The main difficulty in the superconvergence analysis lies in the construction of the correction function or the special interpolation function) and the choice of the parameters in the numerical fluxes. The correction function here is greatly different from those in [30 3], due to the different model problems and numerical schemes. The existence of the convection-diffusion term f u) and the special choice of numerical fluxes make the construction of the correction function more sophisticated. It is also different from the correction functions of FEM or FVM for elliptic equations in [8, 9] because of the time-dependent feature. Furthermore, as the choice of the parameters in the numerical fluxes influences the superconvergence directly a feature not shared by those methods in [8 3]), how a careful choice of the parameters in the numerical fluxes to ensure some desired superconvergence results including superconvergence at Gauss and Lobatto points) is also a technical question needed to handle. The rest of the paper is organized as follows. In section, we recall the DDG schemes and the basic stability results for linear convection-diffusion equations, following [9]. Sections 3 and 4 are the main body of the paper, where superconvergence results for zero flux and linear flux are proved separately, with suitable initial discretization and careful choice of numerical fluxes. Finally, some numerical examples are presented in section 5 to support our theoretical findings. Throughout this paper, we adopt standard notations for Sobolev spaces such as W m,p D) on subdomain D equipped with the norm m,p,d and seminorm m,p,d. When D =, we omit the index D; and if p =,wesetw m,p D) = H m D), m,p,d = m,d, and m,p,d = m,d. We use the notation A B to indicate that A can be bounded by B multiplied by a constant independent of the mesh size h. A B stands for A B and B A. II. DDG SCHEMES To discretize the weak formulation, we divide the interval =[a, b] with a mesh: a = x <x3 < <x N+ = b, and the mesh size h = h j = x j+ x j, and the interval I j =[x j, x j+ ]. We denote by v + and v the right and left limits of any function v, and define [v] =v + v, {v} = v+ + v. Define the k-degree discontinuous finite element space V h = {v h : v h Ij P k I j ), j Z N }, where P k I j ) denotes the set of all polynomials of degree no more than k on I j, and Z r = {,,..., r} for any positive integer r. The DDG method for.) is to find u h V h such that for any v h V h t u h, v h ) j = f u h ) x u h, x v h ) j + ˆ fu h ) + x u h )v h + u h {u h }) x v h ) Ij..) Here ˆ fu h ) and x u h are numerical fluxes, and ϕ, v) j = ϕvdx, I j ) ) v Ij = v x v x +, ϕ, v L. j+ j Numerical Methods for Partial Differential Equations DOI 0.00/num

4 4 CAO, LIU, AND ZHANG Following [9], we take the numerical fluxes ˆ fu h ) = αu h and x u h = β 0 h [u h]+{ x u h } + β h[ x u h].) with β i, i = 0, satisfying the following stability condition where Ɣβ ) = sup ν P k [,] β 0 Ɣβ ), ν) β s ν)). ν s)ds Summing up all j and using the periodic boundary condition, we have from.) t u h, v h ) j + fu h ) + x u h, x v h ) j + fu ˆ h ) + x u h )[v h ]+[u h ] { x v h }) j+ = 0, j= or equivalently, j= where u h, v h ) = N j= u h, v h ) j, and Au h, v h ) = Fu h, v h ) = j= t u h, v h ) + Au h, v h ) Fu h, v h ) = 0, v h V h,.3) x u h, x v h ) j + j= f u h ), x v h ) j + j= x u h [v h ]+{ x v h } [u h ]) j+,.4) j= fu ˆ h )[v h ] j+..5) For any function v h V h satisfying the periodic boundary condition, we have see [9]) where γ 0, ) is some positive constant and j= Fv h, v h ) 0, Av h, v h ) γ v h E, N v h E = j= I j x v h dx + j= β 0 h [v h] j+..6) Then d dt v h 0 + γ v h E tv h, v h ) + Av h, v h ) Fv h, v h ), v h V h..7) III. SUPERCONVERGENCE FOR ZERO FLUX f = 0 Before the study of superconvergence, we begin by introducing some notations, two important projections P h and I h, and the supercloseness between I h v and P h v for any smooth function v, which will be frequently used in our later superconvergence analysis. Numerical Methods for Partial Differential Equations DOI 0.00/num

5 A. Preliminaries SUPERCONVERGENCE OF THE DDG METHOD 5 First, for a smooth function v, we define a global projection P h v V h of v by P h v, ϕ h ) j = v, ϕ h ) j, ϕ h P k I j ), j Z N, 3.) x P h v) := β 0 h [P h v]+{ x P h v)} + β h[ x P hv)] j+ = x vx j+ ), 3.) {P h v} j+ = vx j+ ), j Z N. 3.3) Note that we use the periodic boundary condition at x N+, and P h only needs to satisfy the conditions 3.) 3.3) when k =. It is shown in [9] that the global projection P h is uniquely defined, and v P h v 0 h k+ v k+. 3.4) Second, denote by L m the Legendre polynomial of degree m, and {φ m } 0 polynomials, on the interval [, ]. That is, φ 0 = s, φ = s+, and φ m+ s) = s L m s )ds, m, s [, ]. the series of Lobatto The orthogonal property of Legendre polynomials gives Moreover, there holds the identity φ m ) = φ m ) = 0, m. 3.5) φ m = m L m L m ), m. Therefore, we have for all m, r, φ m φ r ds = m )r ) L m L m )L r L r )ds = 0, if m r = 0, ±. Let L j,m and φ j,m be the Legendre and Lobatto polynomials of degree m on the interval I j, respectively. That is, 3.6) x x j x L j,m x) = L m s), φ j,m x) = φ m s), s = j+, m 0. h For any smooth function v, we have the following expansion in each element I j, vx) = v j,m φ j,m x), x I j, m=0 where v j,0 = v ) x + j, vj, = v ) x j+, vj,m = m Numerical Methods for Partial Differential Equations DOI 0.00/num I j x vl j,m dx, m. 3.7)

6 6 CAO, LIU, AND ZHANG It has been proved in [9] that Now we define the Gauss Lobatto projection of v by v j,m+ h i+ p v i+,p,ij, 0 i m. 3.8) I h v Ij := v j,m φ j,m x). 3.9) By 3.5) 3.6) and a scaling from [, ] to I j,wehave ) ) φ j,m x + j = φj,m x j+ = 0, φj,m+ P m I j ), m, which yields Moreover, since m=0 v I h v) ) x j+ = v Ih v) ) x + j = 0, v Ih v) P k. x v I h v)x) = h m=k+ we obtain from the orthogonality of the Legendre polynomials v j,m L j,m x), x I j, 3.0) x v I h v) P k. 3.) To estimate the error I h v P h v for any smooth function v, we need the following important result. Lemma 3.. Suppose M is an N N block circulant matrix with the first row [CBO...O] and the last row [BO...OC], where O is an zero matrix, and both C and B are nonsingular matrix satisfying detc ± B) = 0. Then ρm ) c, where ρm) denotes the spectral radius of M, and c is some bounded constant independent of N. Proof. For any vector or matrix X, we first denote by X the L norm of X. Let ω j = e i πj N, j = 0,..., N ben roots of with i ωj ) 0 =, and j = 0 ω j. It is easy to verify from the properties of the circulant matrix Mζ j = C + B j )ζ j, ζ j = I, j, j,..., N j ) T, where I is the identity matrix. By denoting ϒ = ) ζ0 ζ 0, ζ ζ,..., ζ N, = diagc + B 0, C + B,..., C + B N ), ζ N Numerical Methods for Partial Differential Equations DOI 0.00/num

7 we have M = ϒ ϒ. By a direct calculation, ζ l ) T ζ j = SUPERCONVERGENCE OF THE DDG METHOD 7 N j l)πm m=0 ei N 0 0 N j l)πm m=0 ei N Then ϒ is an orthogonal matrix, which yields ϒ = ϒ =. ) ) 0, ifj = l, 0 = ) 0 0, ifj = l. 0 0 On the other hand, since detc ± B) = 0, both M and are nonsingular matrices. Then each C + B j is invertible and thus, ρm ) = M ϒ ϒ maxdetc + B j ). j Then the desired result follows. Now we are ready to show the supercloseness between I h v and P h v. Lemma 3.. For any function v H k+,ifβ = kk+), then Proof. Suppose P h v I h v has the following formula in I j, P h v I h v 0 h k+ v k+. 3.) P h v I h v)x) = v j,m L j,m x), x I j. m=0 Recalling the definition of P h and the properties of I h,wehave P h v I h v, ϕ h ) j = 0, ϕ h P k I j ), {P h v I h v} j+ = 0, x P h v) x I h v)) j+ = x v x I h v)) j+, which yields v j,m = 0, m k, and where L m ) v j,m + L m ) v j+,m ) = 0, m=k m=k g 0 m) v j,m + g m) v j+,m ) = h x v x I h v)) j+ = b j, g 0 m) = β 0 L m ) + L m ) 4β L m ), g m) = β 0 L m ) + L m ) + 4β L m ). 3.3) Numerical Methods for Partial Differential Equations DOI 0.00/num

8 8 CAO, LIU, AND ZHANG Let M be an N N block circulant matrix with the first row [CBO...O] and the last row [BO...OC], where O is an zero matrix, and ) ) Lk ) L C = k ) Lk ) L, B = k ). g 0 k ) g 0 k) g k ) g k) Then { v j} N j= with vj = v j,k, v j,k ) satisfies the linear system MX = b, X = v,..., v N ) T, b = 0, b,0,b,...,0,b N ) T. 3.4) It is proved in [9] that detc ± B) = 0 when β 0 >Ɣβ ). Then the conclusion in Lemma 3. gives j= m=k v j,m X M b b j. Now we estimate the term b j, j Z N. Noticing that v I h v is continuous across the point x j+, we have ) b j = h x v x I h v)) j+ = h { x v I h v)} j+ + β h[ x v I hv)] j+. Plugging 3.8) into the above identity and using the scaling from [, ] to I j, j Z N yields b j = v j,m L m ) 4β L m )) + v j+,ml m ) + 4β L m ))) = m=k+ v j,m + ) m v j+,m ) β mm )), m=k+ where the coefficient v j,m is the same as in 3.5), and in the last step, we have used the following identity j= L m ±) = ±) m, L m ±) = ±)m mm ). Consequently, if β = kk+),wehave b j = v j,m + ) m v j+,m ) β mm )), m=k+ which yields, together with 3.6), Here we use the notation I N+ = I. Note that b j h k+ v k+,ij + v k+,ij+ ). 3.6) P h v I h v 0 h N Then the desired result 3.0) follows. j= m=k N v j,m h b j. Numerical Methods for Partial Differential Equations DOI 0.00/num j=

9 SUPERCONVERGENCE OF THE DDG METHOD 9 We end this subsection with an integral projection Dx, which is defined as, for any function v, x D x v I j := vdx. x j Apparently, Moreover, noticing that D x v I hv)x) = m=k+ v j,m x D x v 0 h v ) x j φ j,m dx = m=k+ v x j,m L j,m L j,m )dx, x I j, m x j we have all for k, D x v I hv) ) x j+ = D x v I hv) ) x + j = 0, j ZN. 3.6) Similarly, since we have Then v P h v) Ij = m= v P h v) P 0, k, ṽ j,m L j,m x), ṽ j,m = m + v P h v, L j,m ) h j. D x v P hv)x ) = D j+ x v P hv)x + ) = 0, j Z j N, k. 3.7) The identities 3.6) and 3.7) indicate that both Dx v I hv) and Dx v P hv) are continuous on the whole domain when k. B. Superconvergence at Gauss and Lobatto Points To study the superconvergence properties of the DDG solution u h at Gauss and Lobatto points, we first analyze the supercloseness of u h toward the Gauss Lobatto projection I h u of the exact solution u, and then use the supercloseness result between u h and I h u to obtain the superconvergence at Gauss and Lobatto points. Our analysis is along this line. Define { } H h = v : v Ij H I j ), j Z N, and for all ϕ, v H h, let aϕ, v) = Aϕ, v) + ϕ, v) Numerical Methods for Partial Differential Equations DOI 0.00/num

10 0 CAO, LIU, AND ZHANG and v E = v E + v 0, where E is defined in.6). For any v h V h, the inverse inequality holds. Then { x v h } j+ x v h 0,,I j + x v h 0,,I j+ h x v h 0,I j + x v h 0,I j+ ), j Z N. Similarly, we get h[ x v h] j+ h x v h 0,,I j + h x v h 0,,I j+ h x v h 0,I j + x v h 0,I j+ ). Consequently, h { x v h } + β h[ x v h]) j+ j= j= I j x v h dx. Recalling the definition of A, ) in.4), we have for all ϕ h, v h V h, Then N Aϕ h, v h ) ϕ h E j= ϕ h E v h E. I j x v h dx + h N ) { x v h } j+ + h [v h ] j+ j= av h, v h ) γ v h E, aϕ h, v h ) ϕ h E v h E, ϕ h, v h V h. 3.8) By the Lax Milgram lemma, there exists a unique function w u V h such that aw u, v h ) = t u P h u), v h ), v h V h. 3.9) Choosing v h = w u in the above equation and using the coercivity of a, ), we obtain for any k, γ w u E tu P h u), w u ) = D x tu P h u), x w u ) h k+ t u k+ w u E, where we have used the integration by parts and 3.7) in the second step, and 3.4), 3.5) in the last step. Noticing u t = u xx, then w u E h k+ u t k+ h k+ u k ) Similarly, taking time derivative on both sides of 3.9) and choosing v = t w u yields t w u E h k+ u tt k+ h k+ u k+5. 3.) We have the following superconvergence results for u h I h u and u h P h u. Numerical Methods for Partial Differential Equations DOI 0.00/num j=

11 SUPERCONVERGENCE OF THE DDG METHOD Theorem 3.3. Let u H k+5 and u h be the solution of.) and.), respectively. Suppose the initial value u h x,0) is chosen such that Then there holds for all k and t > 0, Denote u h,0) I h u 0 0 h k+ u 0 k+3. 3.) u h P h u), t) 0 + u h I h u), t) 0 h k+ sup u, τ) k ) e = u u h, ξ = u h P h u + w u, η = u P h u + w u, where w u is the solution of 3.9). Noticing that the exact solution u also satisfies.3), we obtain, by choosing v h = ξ in.7) and using the orthogonality t e, v h ) + Ae, v h ) = 0, v h V h, d dt ξ 0 tη, ξ)+ Aη, ξ) = t u P h u), ξ)+ t w u, ξ)+ Aw u, ξ)+ Au P h u, ξ) = t w u, ξ) w u, ξ)+ Au P h u, ξ). Here in the last step, we have used the identity 3.9). By integration by parts, there holds for all ϕ H h and v h V h Aϕ, v h ) = ϕ, x v h) + = ϕ, x v h) + By the properties of P h, we obtain Consequently, [ϕ x v h ]+ x ϕ[v h ]+{ x v h } [ϕ]) j+ j= x ϕ[v h ] {ϕ} [ x v h ]) j+. 3.4) j= Au P h u, v h ) = 0, v h V h. 3.5) d dt ξ 0 tw u, ξ) w u, ξ) t w u 0 + w u 0 ) ξ 0. In light of 3.0) 3.), we have which yields ξ, t) 0 ξ,0) 0 + h k+ t 0 d dt ξ 0 h k+ u k+5, u, τ) k+5 dτ ξ,0) 0 + Th k+ sup u, τ) k+5. Numerical Methods for Partial Differential Equations DOI 0.00/num

12 CAO, LIU, AND ZHANG Due to the special choice of the initial solution, 3.) and 3.0), we get and thus, ξ,0) 0 h k+ u 0 k+ + w u,0) 0 h k+ u 0 k+3, Then the desired result 3.3) follows from 3.) and 3.0). ξ, t) 0 h k+ sup u, τ) k ) Remark 3.4. The result of Theorem 3.3 indicates that the DDG solution u h is superclose with an order k + to the global projection P h u and the Gauss Lobatto projection I h u of the exact solution. We denote by g j,m, j, m) Z N Z k and l j,m, j, m) Z N Z k+ the Gauss points of degree k and Lobatto points of degree k + on the interval I j, respectively. That is, g j,m and l j,m are separately zeros of L j,k and φ j,k+. As a direct consequence of Theorem 3.3, we have the following superconvergence results of the derivative approximation at Gauss points and the function value approximation at Lobatto points. Corollary 3.5. Suppose all the conditions of Theorem 3.3 hold. Then for any fixed t 0, T ], e g = Nk j= i= x u u h ) g j,i, t)) h k+ sup u, τ) k+5, 3.7) k+ e l = u u h ) l j,i, t)) h k+ sup u, τ) k ) Nk + ) j= i= Proof. In each element I j, j Z N, note that v I h v)l j,i ) = m=k+ v I h v) x g j,i ) = h v j,m φ j,m l j,i ), i Z k+, m=k+ v j,m L j,m g j,i ), i Z k, where v j,m is defined by 3.7). Therefore, for any v H k+, we have from 3.8) which yields v I h v)l i,j ) h k+ v k+,ij, x v I h v)g i,j ) h k+ v k+,ij, k+ v I h v) l j,i )) h k+ v k+, Nk + ) Numerical Methods for Partial Differential Equations DOI 0.00/num j= i=

13 Nk j= SUPERCONVERGENCE OF THE DDG METHOD 3 x v I h v) g j,i )) h k+ v k+. i= On the other hand, since I h u u h V h, we have from the inverse inequality k+ I h u u h ) l j,i )) Nk + ) j= i= I h u u h 0, x I h u u h ) g j,i )) Nk h I h u u h 0. j= i= Then the desired results 3.7) 3.8) follow from 3.3) and the triangle inequality. C. Superconvergence at Nodes As a direct consequence of 3.3), we have the following superconvergence result at nodes. ) {u u h } N j+ = N j= j= {I h u u h } j+ ) I h u u h 0 h k+ sup u, τ) k+5. However, our numerical results show that the superconvergence rate at nodes can reach k when k 3, which is better than the order k + we provide above. Therefore, new analysis tools are needed to prove the k-th superconvergence order for k 3. We will adopt the idea of correction function in our superconvergence analysis. That is, we construct a specially designed function w V h such that w vanishes or is of high order at nodes to guarantee that the error is small enough to be compatible with the superconvergence error estimate at nodes) and u h u I 0 = u h P h u + w 0 h k cu), where cu) is a constant dependent on the exact solution u our later superconvergence analysis shows that cu) can be bounded by sup τ [0,T ] u, τ) k+ ). In light of.7) and 3.5), d dt u h u I 0 + γ u h u I E tu h u I ), u h u I ) + Au h u I, u h u I ) = t u u I ), u h u I ) + Au u I, u h u I ) = t u u I ), u h u I ) + Aw, u h u I ). 3.9) Therefore, to achieve our superconvergence goal, we need to construct a function w V h such that the right hand side in 3.9) is of high order. To be more precise, we shall construct a function w V h satisfying t u u I ), u h u I ) + Aw, u h u I ) h k c u) u h u I E + u h u I 0 ). Numerical Methods for Partial Differential Equations DOI 0.00/num

14 4 CAO, LIU, AND ZHANG Here again, c u) is dependent on some norm of the exact solution u. To design w, we first denote w 0 = u P h u and define a serial of functions w i, i k )/ as follows. w i, x v h) = t w i, v h ), v h P k I j )\P I j ), 3.30) x w i := β 0 h [w i ]+{ x w i } + β h[ x w i] j+ = 0, 3.3) {w i } j+ = ) Here k denotes the maximal integer no more than k. Due to the existence of the global projection P h, the function w i, i k )/ defined in 3.30) 3.3) is uniquely determined. Moreover, we have the following estimate for w i, i k )/. Lemma 3.6. For any k 3, suppose w i, i k )/ is defined by 3.30) 3.3). Then w i Ij := m=k i c i j,m L j,m, 3.33) where L j, = 0 and c i j,m are some bounded constants. Furthermore, if r t u H k++i, r = 0,, i k )/, there holds r t w i 0 h k++i u k++i+r). 3.34) Proof. We will show 3.33) by induction. First, we suppose in each element I j w i Ij := c i j,m L j,m. m=0 For any v h P k \P, since x v h P k, by choosing x v h = L j,m, m Z k in 3.30), we immediately obtain c i j,m = m + t w i D x h D x L j,m)dx, m k. 3.35) I j Noticing that D x D x c j,m = m + h L j,m P m+ I j ) and u P h u) P k, then t u P h u)d x D x L j,m)dx = 0, m k 4, I j which indicates that 3.33) is valid for i =. Suppose 3.33) holds for all i, i k )/. Then In light of 3.35), we have c i+ j,m = m + h w i P k i. t w i D x D x L j,m)dx = 0, m <k i 3. I j Consequently, 3.33) is also valid for i +. Then the proof of 3.33) is complete. Numerical Methods for Partial Differential Equations DOI 0.00/num

15 SUPERCONVERGENCE OF THE DDG METHOD 5 We next estimate the coefficients cj,m i. By 3.35) and 3.5), we have c i j,m h t w i 0,Ij D x D x L j,m 0,Ij h 3 t w i 0,Ij, m k. To estimate cj,m i, m = k, k, we obtain from 3.3) 3.3) m=k m=k k L m )c i j,m + L m )c i j+,m ) = k g 0 m)c i j,m + g m)c i j+,m ) = m=k i m=k i L m )c i j,m + L m )c i j+,m ), g 0 m)c i j,m + g m)c i j+,m ), where g 0, g are the same as in 3.3). By the same argument as in Lemma 3., we have Consequently, w i 0 j= m=k N j= m=k i c i j,m ) k j= m=k i c i j,m ) h 3 t w i 0. hc i j,m ) ) h t w i 0, i k )/. Taking time derivative on both sides of 3.30) 3.3), the three identities still hold. In other words, we can replace w i by t w i in 3.30) 3.3). Similarly, 3.30) 3.3) are still valid if we take m t w i instead of w i, where m. Then following the same arguments as what we did for w i,weget m t w i 0 h m+ t w i 0, m, i k )/. By recursion formula, there holds for all i k )/ and r =0, In light of 3.4), we have r t w i 0 h i r+i t w 0 0. m t w 0 0 = m t u P h u) 0 h k+ m t u k+ h k+ u k++m, m. Then 3.34) follows. This finishes our proof. Now we define the special correction function w as w = k )/ i= w i. 3.36) Theorem 3.7. Let u and u h be the solution of.) and.), respectively, and u I = P h u w be the special interpolation function with w defined by 3.36), 3.30) 3.3). For all k 3, if u H k+, then u h u I ), t) 0 u h u I ),0) 0 + h k sup u, τ) k+, t > ) Numerical Methods for Partial Differential Equations DOI 0.00/num

16 6 CAO, LIU, AND ZHANG Proof. By 3.33) and the orthogonality of P hu,wehave Then for any function v h P, which gives, together with 3.30) w i P, 0 i k )/. t w i, v h ) = 0 = w i+, x v h), 0 i k )/, w i+, x v h) = t w i, v h ), v h V h, i k )/. Consequently, for any v h V h, we have from 3.4) and 3.3) 3.3), t u P h u), v h ) + Aw, v h ) + t w, v h ) = t w r, v h ), ifk = r +, t u P h u), v h ) + Aw, v h ) + t w, v h ) = t w r, v h ) = D x tw r, x v h ), ifk = r. Here in the last step for the even k = r, we have used integration by parts, and the fact that w r P, which yields D x w r x ) = D j+ x w r x + ) = 0. j In light of 3.5) and 3.34), we have for all k 3 t u P h u), v h ) + Aw, v h ) + t w, v h ) C h k u k+ v h 0 + v h E ) C h 4k u k+ + C 3 v h 0 + γ v h E, where C i, i 3 are some bounded constants independent of the mesh size h. Plugging the above inequality into 3.9) yields and thus, d dt u I u h 0 h4k u k+ + u I u h 0, t u I u h ), t) 0 u I u h ),0) 0 + h4k sup u, τ) k+ + u I u h ), τ) 0 dτ. 0 By the Gronwall inequality see, e.g., [33], p.9), we have ) u I u h ), t) 0 + et ) u I u h ),0) 0 + h4k sup u, τ) k+. Then the desired result 3.37) follows. Numerical Methods for Partial Differential Equations DOI 0.00/num

17 SUPERCONVERGENCE OF THE DDG METHOD 7 Theorem 3.8. Let u H k+ be the solution of.), and u h be the solution of.). Suppose u I = P h u w with w defined by 3.36), 3.30) 3.3), and the initial value u h x,0) is taken such that Then there holds for all k 3 and t 0, T ] j= u h u I ),0) 0 h k u 0 k ) e n = u {u h }) x N j+, t)) h k sup u, τ) k ) Proof. have First, for any fixed t 0, T ], by the special choice of the initial value and 3.37), we u I u h ), t) 0 h k sup u, τ) k+. On the other hand, it is easy to obtain from 3.3) that {w, t)} j+ = 0. Then ux j+, t) = {u, t)} j+ = {P h u w), t)} j+ = {u I, t)} j+. Since u h u I V h, the inverse inequality holds, and thus ) {u I } {u h }) x N j+, t)) u I u h ), t) 0,,I N j u I u h ), t) 0. Then j= This finishes our proof. j= e n u I u h ), t) 0 h k sup u, τ) k+. Remark 3.9. We would like to point out that, based on the superconvergence results in Theorem 3.3, Corollary 3.5 and Theorem 3.8, the choice of β in.) has influence on the superconvergence of the DDG solution u h at Gauss and Lobatto points, as well as the supercloseness between u h and I h u. However, it does not affect the superconvergence rate of u h at nodes. We will demonstrate these points in our numerical experiments. Remark 3.0. In our superconvergence analysis in Corollary 3.5 and Theorem 3.8, the initial discretizations 3.) and 3.38) are of great significance. To obtain the k + -th order superconvergence rate at Gauss and Lobatto points, the initial value u h x,0) = I h u 0 or P h u 0 is a valid choice. However, to achieve the k goal at nodal points, it is indicated from Theorem 3.8 that the initial error should also reach the same superconvergence rate, which imposes a stronger condition on the initial discretization. A natural way of initial discretization to obtain 3.46) is to choose u h x,0) = u I x,0),oru h x,0) =ũ I x,0), where { u I if k = r ũ I = 3.40) u I w r if k = r + Numerical Methods for Partial Differential Equations DOI 0.00/num

18 8 CAO, LIU, AND ZHANG The latter is a valid choice due to the fact that w r,0) 0 h k u 0 k, when k = r +. D. Initial Discretization In this subsection, we would like to demonstrate how to implement the initial discretization as it plays an important role in our superconvergence analysis. If we take the initial data u h x,0) = I h u 0 or u h x,0) = P h u 0, then we can obtain u h x,0) by only using the information of the initial value u 0 and 3.)-3.3) and 3.9). While if the initial data is chosen as u h x,0) = u I x,0) or u h x,0) = ũ I x,0), the initial discretization is somewhat complicated. We next show how to calculate u I x,0) only using the information of u 0. The same technique can be applied to ũ I. Noticing that then r t u t=0 = r x u 0, 0 r k )/, r t w 0 t=0 = r t u P hu) t=0 = r x u 0 P h r x u 0). On the other hand, taking time derivative on both sides of 3.30) 3.3), we can calculate the term t rw i+ t=0 from the information of t r+ w i t=0. Now we divide the initial discretization into the following steps:. Compute P h x ru 0),0 r k )/ by 3.) 3.3) and set t rw 0 = x ru 0 P h x ru 0).. Calculate t rw i+, i k )/,0 r k i by 3.30) 3.3) with w i+ replaced by t rw i+ and t w i replaced by t r+ w i. 3. Figure out u h x,0) = P h u 0 k )/ i= w i. IV. SUPERCONVERGENCE FOR LINEAR FLUX f = αu The superconvergence analysis for the convection-diffusion equation with f = αu is similar to that for the zero flux f = 0. To simplify our proof, we only consider the case in which α is a constant in the rest of this section. We first introduce another global projection Q h. For a given smooth function v, the projection Q h v V h is defined by Q h v, ϕ h ) j = v, ϕ h ) j, ϕ h P k I j ), j Z N, 4.) x Q h v) fq ˆ h v)) j+ = x vx j+ ) αv)x j+ ), 4.) {Q h v} j+ = vx j+ ). 4.3) The uniqueness and existence of the projection Q h have been proved in [9]. Moreover, there holds u Q h u 0 h n u n, 0 n k ) We have the following superconvergence result of Q h v I h v. Numerical Methods for Partial Differential Equations DOI 0.00/num

19 Lemma 4.. For any function v H k+,ifβ = kk+), then SUPERCONVERGENCE OF THE DDG METHOD 9 Q h v I h v 0 h k+ v k+. 4.5) Here we omit the proof since it is similar to that for zero flux, the only difference lies in a modification of g 0 where g 0 is given in 3.3). By the definition of Q h,wehave g 0 m) = g 0 m) αh, Au Q h u, v h ) Fu Q h u, v h ) = αu Q h u), x v h ), v h V h. 4.6) We modify the correction function w i V h as follows. For all k, w i, i k isthe function satisfying w i, x v h) j = t w i, v h ) j αw i, x v h ) j, v h P k I j )\P I j ), 4.7) x w i fw ˆ i )) j+ = 0, 4.8) {w i } j+ = 0, j Z N. 4.9) Here w 0 = u Q h u. Note that the definition of w i is the same as that of the global projection Q h, except the right hand side. Therefore, w i, i k defined in 4.7) 4.9) is uniquely determined. Lemma 4.. Suppose w i V h, i Z k is defined by 4.7) 4.9), and in each element I j, j Z N, w i Ij := Then there holds for any positive integer r j= c i j,m L j,m. m=0 ) h r t ci j,m ) h l r t u l, 0 m k, 4.0) where 0 l maxk m, k + + i). Consequently, r t w i 0 h k++i r t u k++i. 4.) Proof. Since 4.) is a direct consequence of 4.0), we only prove 4.0) in the following. We will show 4.0) by induction. First, noticing that 4.7) 4.9) still hold by taking time derivative of r-th order, then we choose x v h = L j,m, m k in 4.7) to derive h m + r t ci+ j,m = +r t w i, D x D x Numerical Methods for Partial Differential Equations DOI 0.00/num L j,m) j α r t w i), D x L j,m) j, m k. 4.)

20 0 CAO, LIU, AND ZHANG Since w 0 = u Q h u) P k and Dx letting i = 0 in 4.) On the other hand, note that r t c j,m D x L j,m P m+ I j ), D L j,m P m+ I j ), we have, by = 0, m = 0,..., k 4. D x D x L j,m 0,Ij h 5, D x L j,m 0,Ij h 3, m 0. By choosing i = 0, m = k in 4.) and using the estimate 4.4) and the fact u t = u xx αu x, we obtain for all 0 n k + ) h r t c j,k ) h +r t w h r t w 0 0 j= h n+ r t u n + +r t u n ) h n+ r t u n+. Moreover, if k 3, we have t rw 0, Dx L j,k 3) j = 0, and thus, which yields r t c j,k 3 =m + h j= +r t I j w 0 D x D x L j,k 3)dx h 3 +r t u Q h u) 0,Ij, ) h r t c j,k 3 ) h +r t u Q h u) 0 h n+ +r t u n h n+ r t u n+. To estimate t rc j,m, m = k, k, we follow the same argument as in Lemma 3. to obtain h j= m=k ) r t c j,m ) h k j= m=k 3 x r t c j,m ) ) h n+ r t u n+. Consequently, 4.0) is valid for all r 0 and m,0 m k in case i =. Now we suppose 4.0) is valid for i, i k and prove that the same result holds true for i +. In fact, by 4.) and the orthogonality of the Legendre polynomials, m+ ) j,m h r+ t c i j,p + h r t ci+ p=0 m+ ) r t ci j,p, m k. For any positive integer p, p m +, by choosing n maxk m, k + i), we obtain from the induction hypothesis, j= p=0 ) h +r t c i j,p ) h n +r t u n = h n r t u n +, Numerical Methods for Partial Differential Equations DOI 0.00/num

21 SUPERCONVERGENCE OF THE DDG METHOD which yields h ) max h +r t c i j,p ) h n + r t u n + = h n r t u n, 0 p m+ j= where n = n + maxk m, k + + i + )). Similarly, we can obtain ) h max h r t ci j,p ) h n r t u n. 0 p m+ j= Then a direct calculation indicates that 4.0) is valid for i + when 0 m k. By 4.8) 4.9) and the same argument as in Lemma 3., we obtain h j= m=k ) r t ci+ j,m ) h k j= m=0 ) r t ci+ j,m ) h k++i r t u k++i. Therefore, 4.0) also holds true for i + when m = k, k. Thus 4.0) is valid for i +. This finishes our proof. To study the superconvergence properties of u h at Gauss and Lobatto points, we define w = w, u I = Q h u w = Q h u w. Then a direct calculation from.5), 3.4), 4.8) 4.9) yields Aw, v h ) Fw, v h ) = w, x v h) αw, x v h ). 4.3) On the other hand, choosing v h = u h u I = ξ in.7) and using the orthogonal property, we get d dt ξ 0 + γ ξ E tu u I ), ξ)+ Au u I, ξ) Fu u I, ξ). 4.4) In light of 4.6) 4.9), we have for all ṽ P k \P, t u u I ), ṽ) + Au u I, ṽ) Fu u I, ṽ) = t w, ṽ) αw, ṽ x ). While for any v P, noticing that x v P 0 and u Q h u) P 0, we have from 4.6) and thus, Au Q h u, v) Fu Q h u, v) = 0, u u I ) t, v) + Au u I, v) Fu u I, v) = t w, v) αw, x v) + t u Q h u), v) = t w, v) D x tu Q h u) + αw, x v). Here in the last step, we have used the integration by parts and the fact that D x tu Q h u)x ) = D j+ x tu Q h u)x ) = 0. j Numerical Methods for Partial Differential Equations DOI 0.00/num

22 CAO, LIU, AND ZHANG Since any v h V h can be decomposed into v h = v +ṽ with v P, ṽ P k \P, we obtain t u u I ), v h ) + Au u I, v h ) Fu u I, v h ) = t w, v h ) αw, x v h ) D x tu Q h u), x v), which yields, together with 4.4) and the fact that v 0 + ṽ 0 v h 0, d dt u h u I 0 tw 0 + w 0 + D x tu Q h u) 0 + u h u I. By 4.) and the Gronwall inequality, u h u I ), t) 0 u h u I ),0) 0 + h k+ sup u, τ) k+5, t >0. Consequently, if we choose the initial value u h x,0) = I h u 0 x) or u h x,0) = Q h u 0 x), then Thus, u h u I ),0) 0 I h u Q h u),0) 0 + w,0) 0 u 0 k+. u h u I ), t) 0 h k+ sup u, τ) k+5, t >0. By following the same argument as in the case f = 0, we obtain the same superconvergence results at Gauss and Lobatto points for the convection-diffusion flux f = αu. That is, the results in Corollary 3.5 are valid for the convection-diffusion equation f = αu. Now we focus on our attentions to the k-th superconvergence rate of the DDG approximation at nodes for k 3. We modify the correction function w as k w = w i, 4.5) where w i is defined by 4.7) 4.9). In this case, 4.3) still holds. Then for all ṽ P k \P, i= u u I ) t, ṽ) + Au u I, ṽ) Fu u I, ṽ) = t w k, ṽ) αw k, ṽ x ) h k t u k ṽ 0 + ṽ ). For any v P I j ), we suppose v = v j,0 L j,0 + v j, L j,. Then there holds for all i, i k, t w i, v) j αw i, v x ) j = In light of 4.0), we have N j= j= h t c i j,0 v j,0 + h ) 3 tc i j, v j, αc i j,0 v j,. h t c i j,0 ) + hc i j,0 ) + h 3 t c i j, ) ) h k t u k. Numerical Methods for Partial Differential Equations DOI 0.00/num

23 Then we use the Cauchy Schwartz inequality to obtain SUPERCONVERGENCE OF THE DDG METHOD 3 t w i, v) w i, v x ) h k t u k v 0 + v ), v P 0. Noticing that u Q h u) P when k 3, we get t u u I ), v) + Au u I, v) Fu u I, v) = t w, v) αw, v x ) Then for all v h = v +ṽ V h, h k t u k v 0 + v ). t u u I ), v h ) + Au u I, v h ) Fu u I, v h ) h k t u k v h 0 + v h ). Consequently, we have from 4.4), the Cauchy Schwartz inequality, and the Gronwall inequality, u h u I ), t) 0 u h u I ),0) 0 + h k sup t u, τ) k, t >0. If the initial value is taken as u h x,0) = u I x,0), we immediately obtain u h u I ), t) 0 h k sup t u, τ) k h k sup u, τ) k+, t >0. Then by the same arguments as in Theorem 3.8, the superconvergence result 3.39) follows for the convection-diffusion flux f = αu. Remark 4.3. Following the argument as in the constant coefficient case, we can obtain the same superconvergence results for the variable coefficient α, where α is sufficiently smooth. V. NUMERICAL EXAMPLES In this section, we present numerical examples to verify our theoretical findings. To test the superconvergence phenomenon of u h at Gauss, Lobatto and nodal points, as well as the supercloseness of u h toward I h u, we will measure in our numerical experiments the errors u h I h u 0 and e l, e g, e n defined in Theorem 3.8 and Corollary 3.5). In our experiments, we will use DDG scheme.) for spatial discretization, and the fourthorder Runge Kutta method for time discretization with the time step t = 0.00h. We obtain our meshes by equally dividing 0, π) into N4,...,64) subintervals. Example. We consider the following equation with the periodic boundary condition: The exact solution is u t = u xx, x, t) 0, π) 0, T), ux,0) = sinx). u = e t sinx). We take different ways of initial discretizations and different values of β 0, β ) to test the influence of the initial errors and the choice of parameters in the DDG numerical fluxes.) on the superconvergence rate. Numerical Methods for Partial Differential Equations DOI 0.00/num

24 4 CAO, LIU, AND ZHANG TABLE I. Errors and corresponding convergence rates for zero flux with u h,0) =ũ I and β = kk+). k N e l Order e n Order e g Order u h I h u 0 Order 4 3.8e-03.46e e e e e e e e e e e e e e e e e e e e-05.76e e e e e e e e e e e e e e e e e e e e e e-05 4.e e e e e e e e e e e e e e e e e TABLE II. Errors and corresponding convergence rates for zero flux with u h,0) =ũ I and β = kk+). k N e l Order e n Order e g Order u h I h u 0 Order 4.76e e-03.04e e e e e e e e e e e e e e e e e e e e e-04.03e e e e e e e e e e e e e e e e e We first choose u h,0) = ũ I,0) with ũ I,0) defined in 3.40). Note that this special initial discretization satisfies both 3.) and 3.38). We list in Table I the various errors and the corresponding convergence rates for k =,3,4,andT =, with the parameter β 0, β ) taken as 0, ),, ), 6, ), respectively. Note that the choice of β , β ) assures β = for all kk+) k =, 3, 4. From Table I, we observe a convergence rate k + for e l and a rate k + for e g. These results are consistent with the superconvergence results given in Corollary 3.5, which indicates that the superconvergence of the derivative approximation at Gauss points and the function value approximation at Lobatto points exist. Moreover, the convergence rates provided in 3.7) 3.8) are optimal. We also observe a superconvergence rate k for e n when k =, 4, which confirms the superconvergence results in 3.39). While, it seems that the convergence rate of e n can reach k+ when k = 3, two-order higher than that in 3.39). As predicted in 3.3), the error u h I h u 0 is convergent with an order k + for k =, 3. While, it seems that the convergence rate of u h I h u 0 is one order better than that in 3.3) when k =4. To test the influence of β on the superconvergence rate, we also consider the case in which β = kk+). Table II demonstrates the error data and the corresponding convergence rate for k =, 4, with β 0, β ) =, 4 ), 0, ) respectively. Clearly, we observe a rate k + for e l and u h I h u 0, and a rate k for e g, which means that the superconvergence properties of u h at Gauss and Lobatto points, as well as the supercloseness between u h and I hu, disappear. Therefore, Numerical Methods for Partial Differential Equations DOI 0.00/num

25 SUPERCONVERGENCE OF THE DDG METHOD 5 TABLE III. Errors and corresponding convergence rates for zero flux with u h,0) = I h u 0 and β = kk+). k N e l Order e n Order e g Order u h I h u 0 Order 4.e-0.75e-03.34e-0.63e-0 8.8e e e e e e e e e e e e e e e e e e e-04.3e e e e e e e e e e e e e e e e e e-05.6e-08.5e-04.e e e e e e e e e e e e e e e e e TABLE IV. Errors and corresponding convergence rates for zero flux with u h,0) = I h u 0 and β = kk+). k N e l Order e n Order e g Order u h I h u 0 Order 4 4.4e-03.36e-03.9e e e e e e e e e e e e e e e e e e e e-07.4e e e e e e e e e e e e e e e e e e as we indicate in Remark 3.9, the value of β affects the superconvergence of u h at Gauss and Lobatto points. However, we can still observe the superconvergence phenomenon at nodes, which indicates that the superconvergence properties of u h at nodes are independent of the choice of β. As a comparison group, we also test another way of initial discretization. That is, u h,0) = I h u 0. Note that this choice of initial discretization satisfies the condition 3.30) while does not satisfy 3.38). Two cases, i.e. β = and β kk+) =, are considered. Listed in kk+) Table III are the corresponding errors and convergence rates in case β =. That is, we kk+) choose β 0, β ) =, ), 6, ), 8, ) for k =, 3, 4, respectively. Table IV demonstrates the 4 40 corresponding numerical results in case β = for k =, 4 with β kk+) 0, β ) = 6, ),, ). 4 Again, we observe similar superconvergence phenomena as in the correction initial disrcretization case. To be more precise, when β =, the DDG solution is superconvergent with an kk+) order k + to the Gauss Lobatto projection of the exact solution, as well as the function value approximation at Lobatto points; and for the error of derivative approximation at Gauss points, the convergence rate is k +. While the superconvergence phenomena disappears when β =. kk+) It should be pointed out that, we still observe an order k for e n in both cases β = and kk+) β =, although the initial condition 3.38) is not satisfied. In addition, the convergence kk+) rate of e n can reach k + for k = 3, 4 when β =, two-order better than that in 3.39). kk+) Numerical Methods for Partial Differential Equations DOI 0.00/num

26 6 CAO, LIU, AND ZHANG TABLE V. Errors and corresponding convergence rates for linear flux with u h,0) =ũ I and β = kk+). k N e l Order e n Order e g Order u h I h u 0 Order e e-03.79e e e e e e e e e e e e e e e e e e e-04.88e e e e e e e e e e e e e e e e e e e e-04.6e e e e e e e e e e e e e e e e e e e- 6.0 TABLE VI. Errors and corresponding convergence rates for linear flux with u h,0) =ũ I and β = kk+). k N e l Order e n Order e g Order u h I h u 0 Order e e-03.08e e e e e e e e e e e e e e e e e e e-04.87e-05.55e e e e e e e e e e e e e e e e e e Example. We consider the linear convection-diffusion equation u t + αu) x = u xx, x, t) 0, π) 0, T), ux,0) = e sinx)/ with the periodic boundary condition and α = + cosx t). The exact solution is u = e sinx t)/. Listed in Tables V and VI are error data and the corresponding convergence rate for the final time T = in cases β = and β kk+) =. In Table V, the parameter is chosen as β 0, β ) = 0, kk+) ),, ), 6, ) for k =, 3, 4, respectively, and in Table VI, 4 40 β 0, β ) =, ) for both k =, 4. Similar as the zero flux case, we observe a k + ) -th order 4 superconvergence rate for e l and u h I h u 0, and a k + ) -th order for e g when β =. kk+) However, those superconvergence phenomena disappears when β =. Different from the kk+) errors e l, e g, we observe a k-th superconvergent order for the error e n in both β = and kk+) β =. As we may recall, all these numerical results are consistent with our theoretical kk+) findings. Moreover, all the superconvergence rates in 3.7) 3.8) and 3.39) are optimal, i.e., the error bounds are sharp. Numerical Methods for Partial Differential Equations DOI 0.00/num

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