Discontinuous Galerkin method for a class of elliptic multi-scale problems

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 000; 00: 6 [Version: 00/09/8 v.0] Discontinuous Galerkin method for a class of elliptic multi-scale problems Ling Yuan and Chi-Wang Shu,, Department of Mechanical Engineering and Applied Mechanics, Universit of Pennslvania, Philadelphia, PA 904 Division of Applied Mathematics, Brown Universit, Providence, RI 09 SUMMARY In this paper we develop a discontinuous Galerkin (DG method for solving a class of second order elliptic multi-scale problems. The main ingredient of this method is to use a non-polnomial multiscale approximation space in the DG method to capture the multi-scale solutions using coarse meshes without resolving the fine-scale structure of the solution. We perform analsis on the approximation, stabilit and error estimates, and provide numerical results to demonstrate the proposed method. Copright c 000 John Wile & Sons, Ltd. ke words: Discontinuous Galerkin method; multi-scale approximation. Introduction In this paper, we design a discontinuous Galerkin (DG method based on non-polnomial basis functions [] for solving a class of multi-scale second order elliptic partial differential equations (a ε (x u = f(x in Ω ( with the boundar condition u = 0 on Ω, where Ω is a rectangular domain and the coefficient a ε (x is an oscillator function involving a small scale ε, for example it could be a ε (x = a(x, x ε, however the force function f(x does not involve this small scale. A standard discontinuous Galerkin method [, 0] using piecewise polnomial approximation spaces can be used to solve (, however it would require a ver fine mesh to resolve the small scale ε in the solution, which might be too costl or impossible on toda s computers. Correspondence to: Division of Applied Mathematics, Brown Universit, Providence, RI shu@dam.brown.edu Research of this author is partiall supported b ARO grant W9NF and NSF grant DMS Received 0 March 007 Copright c 000 John Wile & Sons, Ltd. Revised xx March 007

2 L. YUAN AND C.-W. SHU We will explore the usage of non-polnomial multi-scale approximation spaces in the DG method, consisting of special basis functions constructed from the PDEs to capture the microscale structure information of the solutions so that the solutions of multi-scale PDEs can be well approximated even under coarse meshes. Similar approaches have been used b Babuška et al. in [3, 4, 5] and b Hou and Wu in [] for continuous finite element methods, in which both theoretical proofs and numerical experiments were provided to show that more accurate results can be obtained b using the multi-scale approximation spaces instead of piecewise polnomial spaces. We extend the methodolog to the DG methods, providing analsis on the approximation, stabilit and error estimates, and numerical results to demonstrate the proposed method. The DG method, comparing with the continuous finite element methods, has the advantage of more flexibilit on non-conforming meshes with hanging nodes, easier h-p adaptivit, and more importantl easier patching of the local multi-scale approximation spaces in different cells, especiall for multi-dimensional problems, since no continuit across cell interfaces is required. This paper is organized as follows. In Section, we give a brief review of the DG method for elliptic PDEs. In Section 3, the new multi-scale approximation spaces are introduced and some approximation results are given. In Section 4, stabilit and approximation properties are proven for the DG method based on the multi-scale approximation spaces. Then, error estimates are obtained in Section 5. In Section 6, a numerical example in one space dimension is provided to demonstrate the performance of the DG method. Concluding remarks and plans for future work are given in Section 7.. The discontinuous Galerkin method In this section we give a brief review of some discontinuous Galerkin finite element methods for elliptic problems [,, 7, 8]. For the sake of simplicit, we onl consider the second order elliptic problem in -D (a(xu x x = f(x, 0 x ( where a(x 0. We first rewrite the problem as a first order sstem (aw x = f, w u x = 0. (3 Assuming = (, x j+, j =,, N, is a partition of [0, ], we multipl both equations b test functions v and τ, respectivel, and integrate over the cell, followed b an integration b parts to obtain the weak formulation awv x dx a j+ w j+ v j+ + a j w j v j = fvdx, (4 wτdx + uτ x dx u j+ j+ + u j τ j = 0. (5 The general formulation of the DG method for the elliptic problem ( is: Find U, W V h such that aw v x dx a j+ Ŵ j+ v + a j+ j Ŵ j v+ = fvdx, (6 j Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

3 DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC MULTI-SCALE PROBLEMS 3 W τdx + Uτ x dx Ûj+ τ + j+ j = 0 (7 for all test functions v, τ V h. Here V h is a finite dimensional finite element space containing functions which are discontinuous across cell interfaces. For traditional DG methods, these functions are piecewise polnomials. In this paper, we consider multi-scale approximation spaces. The equations (6-(7 are called the flux formulation. We can also eliminate the auxiliar variable W from the flux formulation (6-(7 to obtain a tpical finite element formulation, which is called the primal formulation. There are man choices for the fluxes Û and Ŵ. Different choices will give us different DG methods. We will just give two examples below. First, the flux formulation of the LDG method [9, 7, 8] is: (6-(7 with the following choice of fluxes Û j+ = U, Ŵ j+ j+ = W + +[U] j+ j+ ; or Û j+ = U +, Ŵ j+ j+ = W +[U] j+ j+ where [U] = U + V. Secondl, the primal formulation of the Babuška-Zlámal method [6] is: Find U V h such that ( η j+ U x v x dx + a j+ x [U] j+ [v] j+ = fvdx, j= for all test functions v V h, where η j+ requirement for the lower bound η 0 = inf j η j+ j= is a positive constant. There is a theoretical for maintaining stabilit and rates of convergence of this method: η 0 x k if dim V h Ij = k +. For the multi-scale problem (, a direct numerical solution of this problem b the DG method based on standard piecewise polnomial spaces is difficult when ε is ver small, even with modern supercomputers. The reason is that piecewise polnomial spaces do not approximate well the solution of ( when the mesh is not refined enough. We need to use a ver fine mesh in the computation to get good numerical results, hence the computational cost will be huge. The core idea of the DG method in this paper is to construct the finite element basis functions which can capture the small scale information. In this DG method, the basis functions are constructed from the elliptic differential operator, hence the are adapted to the local properties of the differential operator in each cell. This idea of the construction of basis functions was first used b Babuška et al. to -D problems [4, 5] and to a special class of -D problems [3], and also b Hou and Wu to some -D elliptic problems with rapidl oscillating coefficients [], in the context of continuous finite elements. For continuous finite element methods, there exists difficult in enforcing continuit of basis functions at the -D cell interfaces. However, for the DG methods, continuit at the element interfaces is not needed, thus eliminating this difficult. 3. Multi-scale approximation spaces The multi-scale approximation spaces are constructed as below: S = {φ : (a ε (x φ K = 0} (8 Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

4 4 L. YUAN AND C.-W. SHU and S k = {φ : (a ε (x φ K P k (K} for k (9 where K denotes the cell in the space discretization. If we define P (K = {0}, (8 can also be denoted b (9 for k =. First consider the -D elliptic problem with the boundar condition where (a ε (xu x x = f(x, 0 x (0 u(0 = u( = 0, ( 0 < α a ε (x β < +. ( The multi-scale approximation space (9 for this problem is explicitl given { { S k x }} = v : v Ij span, a ε (ξ dξ, ξ x x j a ε (ξ dξ,, (ξ k a ε dξ (3 (ξ ( where = + x j+. This multi-scale approximation space (3 has a ver good approximation propert for the solution of the problem (0. This can be shown in the following lemmas. Lemma 3.: Let u(x be the exact solution of (0. There exists some v(x S k (k such that for all j: u(x v(x C(α, β f H k (( k+/, x (4 where C(α, β is a constant depending on α, β and independent of and ε. Proof: The proof is inspired b the fundamental theorem of calculus u(x = u( + u (d = u( + a ε (u ( a ε ( d ( = u( + a ε (u (d x a ε (ξ dξ = u( + a ε ( u x ( ( a ε (ξ dξ x a ε (ξ dξ d(a ε (u ( = u( + a ε ( u x ( ( a ε (ξ dξ a ε (ξ dξ f(d where in the last equalit we have used the differential equation (0. When k =, let v = u( + a ε ( u ( a ε (ξ dξ. We can see v S. And we have u(x v(x = ( a ε (ξ dξ f(d ( / d a ε (ξ dξ f (d α ( 3/ f L ( (5 / Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

5 DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC MULTI-SCALE PROBLEMS 5 for all x. When k, we can find some p P k ( such that f p L ( C f H k (( k. Here and below C is a generic constant independent of the functions and meshes sizes. Let v = u( + a ε ( u ( a ε (ξ dξ ( a ε (ξ dξp(d. We can easil verif that v Sk since ( ( x ξ a ε (ξ dξ p(d = p(d a ε (ξ dξ and ξ p(d P k (. Then we can obtain u(x v(x = ( x a ε (ξ dξ (f( p(d ( / d x a ε (ξ dξ (f( p( d α ( 3/ f p L ( C(α, β( k+/ f H k ( (6 for all x. Combining (5 and (6 will finish the proof. Next, we estimate the approximation rate in the L norm. Lemma 3.: Let u(x be the exact solution of (0 and P h be the L projection operator into the space S k. There exists a constant C(α, β such that: u P h u L (0, C(α, β f H k (0,( x k+. (7 Proof: We choose the same v as that in Lemma 3.. Squaring both sides of (4 and then integrating in the cell, we obtain, for an j, Therefore, u v L ( C(α, β f H k ( ( k+. / u P h u L (0, u v L (0, = j u v L ( C(α, β j f H k ( ( k+ C(α, β f H k (0, ( xk+. Taking square roots on both sides finishes the proof. We now show a numerical example in Table I for the approximation to the solution of the elliptic problem (0. We choose f = x, a = + x + sin( πx ε. (8 Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

6 6 L. YUAN AND C.-W. SHU Table I. L and L -errors of the approximation to the solution of problem (0 with the choice of (8. Uniform mesh with N cells. N S space S space ε = 0. ε = 0. L -error order L -error order L -error order L -error order E-04.4E E-06.59E E E E E E E E E E E E E E E E E E E E E-0.99 ε = 0.0 ε = 0.0 L -error order L -error order L -error order L -error order 0 5.9E-04.38E E-06.60E E E E E E E E E E E E E E E E E E E E E-0.73 with different choices of ε. We can see that we obtain the optimal order of the approximation rate (equal to the dimension of the local approximation space when using the approximation spaces S k (k =,. We also show that the usual inverse inequalit holds for the multi-scale approximation spaces. Lemma 3.3: (Inverse Inequalit For an φ S k, we have the following inverse inequalit φ L ( C( / φ L ( (9 for some constant C. { Proof: Onl the proof for k = will be given here., } x a ε (ξ dξ is a set of basis functions of the local space S Ij. We can transfer it to an orthogonal basis {b 0, b } with b 0 =, b = ( a ε (ξ dξ d. First we prove that for each i, b i satisf the inverse inequalit (9. The proof is trivial for i = 0. For i =, we have b = ( a ε (ξ dξ d a ε (ξ dξ d x d α α, that is b L ( α. (0 Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

7 DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC MULTI-SCALE PROBLEMS 7 From smmetr, we have ( ( a ε (ξ dξ d dx = 0. Hence there exists some x 0 such that We claim that for x > x 0, ( ( ( x a ε (ξ dξ d = (0 a ε (ξ dξ d = 0. a ε (ξ dξ d > 0, since x 0 a ε (ξ dξ + 0 Similarl, for x < x 0, we have ( x a ε (ξ dξ d < 0. Without loss of generalit, we ma assume x 0. We then have ( ( ( b L ( = d a ε (ξ dξ dx j+ / x 0 ( + xj x 0 x 0 ( ( xj+ / x 0 ( x a ε (ξ dξ d = x 0 a ε (ξ dξ d > 0. ( d a ε (ξ dξ dx ( ( xj+ / ( ( a ε (ξ dξ d dx d dx x 0 a ε (ξ dξ ( j+ / x 0 β (x x 0dx ( 5 64β. That is, we have b L ( ( 3/. ( 8β Combining the two inequalities (0 and (, we obtain b L ( C( / b L (. For an φ S, we can write φ = c 0 b 0 + c b, and we have φ L ( ( c 0 b 0 L (I + j c b L (I C( j / c 0 b 0 L ( + c b L ( = C( / φ L (. Now the proof is complete for k =. Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

8 8 L. YUAN AND C.-W. SHU We now consider a special -D elliptic multi-scale problem with the boundar condition (a ε (xu x x (b ε (u = f(x,, (x, Ω ( u(x, = 0, (x, Ω where 0 < α a ε (x, b ε ( β < +. (3 We propose a -D multi-scale approximation space for this -D elliptic problem to be { { S k = v : v K span, x K x K a ε (ξ dξ K x K b ε (η dη, a ε (ξ dξ, K K b ε (η dη, η K b ε (η dη,, x K ξ x K a ε (ξ dξ, (ξ x K k (ξ x K k a ε dξ, (ξ x K a ε dξ (ξ K b ε (η dη,, x K a ε (ξ dξ (η K k (η K k }} b ε dη, (η b ε dη. (η K Here K is a two dimensional cell with the barcenter at the point (x K, K. In the following lemma we will prove the approximation propert for S. Lemma 3.4: Let u(x, be the exact solution of (. There exists some v(x, S such that for all cell K: u v L (K C(α, β f L (K( K, (x, K (4 where K = diam(k and C(α, β is a constant depending on α, β and independent of K and ε. Proof: This lemma is a generalization of the theorems in [3] and so is the procedure of its proof. We define x = x K a ε (ξ dξ, ỹ = K b ε (η dη, ũ( x, ỹ = u(x,, ãε ( x = a ε (x, b ε (ỹ = b ε (, and f( x, ỹ = f(x,. Change the variable x and in ( to x and ỹ, we obtain a new elliptic PDE b ε ũ x x ã ε ũỹỹ = ã ε bε f, ( x, ỹ K (5 From the Bernstein Theorem in [3], we have ũ H ( K C(α, β ãε bε f L ( K C(α, β f L (K From the approximation theor of polnomial spaces, we can find a linear polnomial ṽ = c 0 + c x + c ỹ such that K ũ ṽ L ( K C ũ H ( K ( K. Now we denote v(x, = ṽ( x, ỹ = c 0 + c x K a ε (ξ dξ + c K b ε (η dη. We also have K C K. The approximation propert can then be obtained u v L (K = ũ ṽ L ( K C ũ H ( K ( K C(α, β f L (K( K. Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

9 DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC MULTI-SCALE PROBLEMS 9 This complete the proof. Next, we will prove the L approximation propert for S. Lemma 3.5: Let u(x, be the exact solution of ( and P h be the L projection operator into the space S. There exists a constant C(α, β such that: u P h u L (Ω C(α, β f L (Ω (6 where = max K K. Proof: We choose the same v as that in Lemma 3.4. Squaring both sides of (4 and then integrating over the cell K, we obtain u v L (K C(α, β f L (K ( K 4, K. Therefore, u P h u L (Ω u v L (Ω C(α, β K f L (K ( K 4 C(α, β f L (Ω 4. Taking square roots on both sides finishes the proof. 4. Boundedness and stabilit of the multi-scale DG operator, and approximation properties Here we denote the primal form B h (, as the left-hand-side of the primal formulation of the DG method. In this section, we discuss the boundedness and stabilit of B h and the approximation properties of the space S k with respect to some appropriate norm. For simplicit, we will onl provide the proof for the Babuška-Zlámal DG method in -D following the proof in []. The proof can also be easil generalized to other DG methods and to -D along the lines of []. 4.. Boundedness Lemma 4.: (Boundedness There exists some constant C b such that where the norm is defined as with the norm defined as B h (w, v C b w v, w, v S k (7 v = v H (0, + v N v = ( x k [v] j+. Proof: For the Babuška-Zlámal method, we have ( a ε (x B h (w, v = a ε j+ w x v x dx + η j+ [w] x j+ [v] j+. j= Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

10 0 L. YUAN AND C.-W. SHU From (, we can easil obtain j= for some constant C. We also have ( a ε (+ η j+ [w] x j+ [v] j+ a ε w x v x dx C w H (0, v H (0, (8 for some constant C. Combining (8 and (9 completes the proof. 4.. Stabilit Lemma 4.: (Stabilit There exists some constant C s such that Proof: From (, we can easil obtain C ( x k (sup η j+ w v (9 j B h (v, v C s v, v S k. (30 j= for some constant C. We also have ( a ε (+ η j+ [v] x j+ [v] j+ a ε v x v x dx C v H (0, (3 C ( x k (inf j η j+ v (3 for some constant C. The combination of (3 and (3 gives us (30. To make C s independent of x, the lower bound of η j+ must be sufficientl large, i.e. inf j η j+ = O(( x k Approximation Lemma 4.3: (Approximation Let u(x be the exact solution of (0. There exists some interpolant u I (x S k (k such that u u I C a ( x k f H k (0, (33 Proof: We choose u I as the usual continuous interpolant, then the jumps of u u I will vanish on the boundaries. We also have u u I H (0, C( x k f H k (0,, proven in [4]. Then The proof is completed. u u I = u u I H (0, C( x k f H k (0,. 5. Error estimates We now prove an error estimate for the DG method based on the multi-scale approximation spaces b using the properties of consistenc, boundedness, stabilit and approximation discussed previousl. Again, onl the results for the Babuška-Zlámal DG method is provided. Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

11 DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC MULTI-SCALE PROBLEMS Lemma 5.: (Error estimates Let u(x be the exact solution of (0 and u h be the numerical solution computed b the multi-scale Babuška-Zlámal DG method. There exists some constant C independent of ε such that u u h L (0, C( x k+ f H k (0,. (34 Proof: The proof here mainl follows the lines in []. The Babuška-Zlámal DG method is not consistent. Instead of satisfing the consistenc condition, it satisfies B h (u, v = 0 fvdx + where u is the exact solution of (0 and {w} = (w + w +. This method is neither adjoint consistent. We also have B h (v, ψ = 0 vgdx + a ε (+ {u x} j+ [v] j+, v H (, j. (35 a ε (+ [v] j+ {ψ x} j+, v H (, j, where ψ is the exact solution of the adjoint problem of (0, which is with the boundar condition (a ε (xψ x x = g(x, 0 x (36 From the stabilit of the Babuška-Zlámal method, we obtain ψ(0 = ψ( = 0. (37 C s u I u h B h (u I u, u I u h + B h (u u h, u I u h =: T + T. (38 Since u I is the continuous interpolant of u in S k, we can use the continuit of u u I to estimate T : We also have T = T C u I u u I u h C( x k u I u h f H k (0,. (39 a ε (+ {u x} j+ [u I u h ] j+ C( x k u I u h u H (0, C( x k u I u h f L (0, (40 where the second inequalit comes from the regularit of the elliptic PDE (0 and the first inequalit is estimated b using the auxiliar inequalit a ε (+ {u x} j+ [v] j+ = N C v ( x k+ {u x } j+ a ε (+ ( xk+/ {u x } j+ [v] j+ ( x k / / C( x k v u H (0, (4 Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

12 L. YUAN AND C.-W. SHU with the last step following from the trace inequalit which is proven in []. Substituting (39 and (40 back into (38, we get and b the triangle inequalit. For the L -error estimate, we have u u h L (0, B h(u u h, ψ u I u h C( x k f H k (0,, u u h C( x k f H k (0,. a ε (+ {ψ x} j+ [u u h] j+ =: T 3 + T 4. (4 Let ψ I be the continuous interpolant of ψ in S k, then B h (u, ψ I = 0 fψ Idx. Therefore, T 3 = B h (u u h, ψ ψ I C u u h ψ ψ I C x u u h u u h L (0,. (43 The term T 4 can be estimated b using the auxiliar inequalit (4 T 4 C( x k u u h ψ H (0, C( x k u u h u u h L (0,. (44 Substituting (43 and (44 back into (4, we get which finishes the proof. u u h L (0, C( x k+ f H k (0, 6. A numerical example In this section, we present a numerical example of using the Babuška-Zlámal DG method based on the proposed multi-scale approximation spaces for solving the elliptic multi-scale problem (0 with f = x, a = + x + sin( πx ε (45 with ε = 0. and ε = 0.0 respectivel. The numerical results are shown in Table II. We can clearl observe that the solutions are numericall well resolved and the expected (k + -th order of accurac is achieved without an resonance effects when the mesh size x changes from larger than ε to smaller than ε. Our numerical implementation does show an amplification of the round-off errors to around the level at 0 7 in double precision, probabl due to the large condition number of the mass matrix. We will explore more accurate implementation of the method in the future. 7. Concluding remarks DG methods based on multi-scale approximation spaces for solving a class of second order elliptic PDEs with multi-scale solutions are studied in this paper. The basis functions of the Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

13 DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC MULTI-SCALE PROBLEMS 3 Table II. L and L -errors for the Babuška-Zlámal DG method based on multi-scale approximation spaces. N uniform cells. S space N ε = 0. ε = 0.0 L -error order L -error order L -error order L -error order E-04.45E E-04.7E 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 S space N ε = 0. ε = 0.0 L -error order L -error order L -error order L -error order 5 3.0E E E E E E E E E E E E E E E E E E E E-07. multi-scale approximation spaces are constructed from the differential operators to capture the micro-scale structure information of solutions so that the solutions of multi-scale PDEs can be well approximated. Both theoretical proofs and numerical experiments show that, compared to the usual piecewise polnomial spaces, more accurate results are obtained for coarse meshes. No resonance effects appear, that is, the rate of convergence is independent of ε. From a practical point of view, we have onl considered the one dimensional problem and a special class of two dimensional problems for which the multi-scale basis functions can be explicitl constructed. This renders the DG scheme ver efficient, with a cost comparable to that for the regular DG scheme with polnomial basis functions. For general multi-dimensional problems, the multi-scale basis functions ma not be available explicitl and must also be computed numericall [], which would increase the computational cost of the multi-scale DG method tremendousl. In future work we will generalize both the multi-scale DG method and its analsis to broader classes of multi-dimensional problems with higher order of accurac. We will also explore applications of the multi-scale DG method in fluid dnamics and semi-conductor device simulations. REFERENCES Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6

14 4 L. YUAN AND C.-W. SHU. Arnold DN. An interior penalt finite element method with discontinuous elements. SIAM Journal on Numerical Analsis 98; 9: Arnold DN, Brezzi F, Cockburn B, Marini LD. Unified analsis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analsis 00; 3: Babuška I, Caloz G, Osborn J. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM Journal on Numerical Analsis 994; 3: Babuška I, Osborn J. Generalized finite element methods: Their performance and their relation to mixed methods. SIAM Journal on Numerical Analsis 983; 0: Babuška I., Osborn J. Finite element methods for the solution of problems with rough input data. In Singularities and Constructive Methods for Their Treatment, Springer-Verlag Lecture Notes in Mathematics, Grisvard P, Wendland W, Whiteman JR (eds, Springer-Verlag: Berlin-New York, 985; Babuška I, Zlámal M. Nonconforming elements in the finite element method with penalt. SIAM Journal on Numerical Analsis 973; 0: Castillo P, Cockburn B, Perugia I, Schötzau D. An a priori error analsis of the local discontinuous Galerkin method for elliptic problems. SIAM Journal on Numerical Analsis 000; 38: Cockburn B, Kanschat G, Perugia I, Schötzau D. Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM Journal on Numerical Analsis 00; 39: Cockburn B, Shu CW. The local discontinuous Galerkin method for time-dependent convection-diffusion sstems SIAM Journal on Numerical Analsis 998; 35: Cockburn B, Shu CW. Runge-Kutta Discontinuous Galerkin methods for convection-dominated problems Journal of Scientific Computing 00; 6: Hou TY, Wu XH. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Phsics 997; 34: Yuan L, Shu CW. Discontinuous Galerkin method based on non-polnomial approximation spaces. Journal of Computational Phsics 006; 8: Copright c 000 John Wile & Sons, Ltd. Int. J. Numer. Meth. Fluids 000; 00: 6