A Symmetric Smoother for the Nonsymmetric Interior Penalty Discontinuous Galerkin Discretization K. Johannsen The Center for Subsurface Modeling (CSM) The Institute for Computational Engineering and Sciences (ICES) The University of Texas, Austin, TX 78712, USA
Abstract We consider the nonsymmetric interior penalty discontinuous Galerkin discretization (NIPG) for the Laplace equation on nondegenerate, quasi-uniform grids using polynomial degrees of order p ≥ 1. We show that the classical smoothing property holds for a variant of the symmetric Gauss-Seidel iteration in the Euclidian norm. Multigrid convergence can be concluded by standard arguments. Numerical results that confirm the theoretical findings are presented. AMS Subject Classification: 65F10, 65F50, 65N22, 65N30, 65N55. Key words: smoothing property, multigrid method, sparse matrices, discontinuous Galerkin methods 1 Introduction The application of discontinuous Galerkin methods to elliptic or parabolic problems dates to the early 1970’s, see e.g. [2, 3, 5, 13, 23, 24, 37, 38]. In the 1980’s and early 1990’s, development focused mainly on applications to hyperbolic equations. Recently their application to elliptic and parabolic problems has gained much interest due to new developments in the late 1990’s [1,4,6,8,11,12,25–29,31–36]. As a consequence, solution methods for discrete problems has become a field of active research. It is well-known that multigrid methods are among
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the fastest methods available for solving discrete systems arising from the discretization of elliptic partial differential equations. For conforming as well as for many nonconforming discretizations theory and application of multigrid methods is well-established. For an overview, see [16,39,40] and the citations therein. Theory and practice is less developed for the discontinuous Galerkin methods. Multigrid methods for the symmetric interior penalty discontinuous Galerkin (SIPG) [37] and the symmetric local discontinuous Galerkin (LDG) [11] have been investigated in [9, 10, 15, 21, 22]. Multigrid solution strategies for the nonsymmetric variants of discontinuous Galerkin discretizations have been investigated on a less rigorous level through Fourier analysis [18–20] and in the application practice [7]. A mathematically rigorous treatment is not know to the author. In this paper we investigate the smoothing property of a variant of the symmetric Gauss-Seidel as applied to the nonsymmetric discontinuous Galerkin method (NIPG). We consider arbitrary unstructured, quasi-uniform meshes and polynomial degrees of any order. The classical smoothing property, as described in for example [16], is shown to hold in the Euclidian norm. Employing standard arguments, h-uniform W-cycle multigrid convergence may be established. Numerical experiments are carried out that confirm the theoretical findings. The remainder of this paper is organized as follows. In the next section, we introduce the model problem and its discretization. In section 3 we define the smoothing iteration and investigate its smoothing property. Numerical examples are discussed in section 4. Finally, concluding remarks are given in section 5. 2 Model Problem and Discretization Let d = 2, 3, Ω ⊂ Rd , a convex polygonal domain, f ∈ L2 (Ω), and u ∈ H01 (Ω) the weak solution of −∆u = f, u = 0,
x ∈ Ω, x ∈ ∂Ω.
(1a) (1b)
The solution is fully regular, i.e. u ∈ H 2 (Ω) ∩ H01 (Ω). Let T be a nondegenerate, quasi-uniform triangulation with ∪ e∈T e = Ω, where e is a triangle or a quadrilateral if d = 2, and a tetrahedron, prism, pyramid or hexahedron if d = 3. The nondegeneracy condition requires that there exists a parameter ρ > 0 such that, if h e = diam(e) then e contains a ball with radius ρh e . We denote by h = maxe∈T he
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the mesh size of T. Let Γ = ∪e∈T ∂e denote the set of all edges. With T we associate the broken Sobolev space H1 (T) := {u ∈ L2 (Ω)|u|e ∈ H1 (e)∀e ∈ T}. Let ne denote the outer normal of e ∈ T and Γe,f := e¯∩ f¯ the common interface of e, f ∈ T. We define the average of a vector valued function u ∈ (H1 (T))d by ( u|e on ∂e ∩ ∂Ω hui := u|e + u|f )/2 on Γe,f ⊂ Γ \∂Ω and the jump of a scalar function u ∈ H 1 (T) by ( u|e ne on ∂e ∩ ∂Ω [u] := . u|e ne + u|f nf on Γe,f ⊂ Γ \∂Ω The variational formulation of (1) on grid T for the NIPG method reads as follows: Find u ∈ H1 (T) such that B(u, v)=L(v)
∀v ∈ H1 (T)
(2)
with B(u, v):=
XZ e∈T
e
∇u · ∇vdx −
Z
Γ
µ + h∇vi · [u]ds + h Γ Z
h∇ui · [v]ds
Z
Γ
[u] · [v]ds
and L(v)
:=
XZ e∈T
f vdx.
e
The parameter µ is called the penalty parameter. Let the symmetric and antisymmetric part of B(·, ·) be defined by Bs (u, v) := 1/2(B(u, v) + B(v, u)), Ba (u, v) := 1/2(B(u, v) − B(v, u)),
(3a) (3b)
for u, v ∈ H1 (T). We discretize the variational problem using a finite dimensional ansatz-space V := {u ∈ L2 (Ω)|u|e ∈ Pk (e)∀e ∈ T} ⊂ H1 (T),
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where Pk (e) denotes the space of polynomials on e with degree smaller than or equal to k. The discrete problem results from (2), using V for the test an trial functions, i.e. Find u ∈ V such that B(u, v)=L(v)
∀v ∈ V.
(4)
Note that Bs is positive definite for any µ > 0 due to the Dirichlet boundary conditions [27]. The following estimate holds: Lemma 1 Let Bs (·, ·) and Ba (·, ·) be defined by (3). Then Ck |Ba (u, v)| ≤ √ (Bs (u, u) + Bs (v, v)) µ
∀u, v ∈ V,
(5)
with a constant C ∈ R only dependent on the shape regularity parameter ρ of the mesh. Proof Let u, v ∈ V and γ > 0. Then Z Z X h∇ui · [v]ds ≤ Γ
e∈T
∂e
|∇u · [v]| ds
Z Z 1X 1 2 2 ≤ (∇u · ne ) ds + |[v]| ds γ 2 γ ∂e ∂e e∈T Z X ck 2 γ Z 1 2 |∇u| dx + |[v]|2 ds, ≤ 2h e 2γ Γ e∈T
where we have used an inverse inequality from [30]. The constant c √ depends on ρ only. Choosing γ = h/k cµ, we get the estimate # Z r " Z Z k c X µ 2 2 h∇ui · [v]ds ≤ |∇u| dx + |[v]| ds , 2 µ h Γ Γ e e∈T
leading to the assertion above.
Remark 1 (i) For d = 2, k = 1, and T a uniform square grid, the constant from Lemma 1 may be chosen as C = 2 1/4 . (ii) For d = 3, k = 1, and T a uniform cubic grid, the constant from Lemma 1 may be chosen as C = 6.751/4 .
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3 Smoother In this section we investigate the symmetric Gauss-Seidel iteration and its smoothing property as applied to the model problem defined in the previous section. We start with some basic definitions and lemmata [16, 17]. Let N > 0, x, b ∈ RN , and A, W ∈ RN ×N . We denote by k · k0 and (·, ·) the Euclidian norm and the Euclidian scalar product, respectively and by k · kA and (·, ·)A the energy norm and the energy scalar product, respectively with respect to a matrix A > 0. Note that kW kA = kA1/2 W A−1/2 k0 . With ρ(A) we denote the spectral radius of A, with m(A) := maxi |{j|Aij 6= 0}| the maximum number of non-zero entries per row. Definition 1 Let A ∈ RN ×N . We define |A| ∈ RN ×N by |A|2 = AT A,
|A| symmetric positive semi-definite.
Lemma 2 Let A ∈ RN ×N be regular, (i) k|A|α k0 = kAkα0 , ∀α ≥ 0 and (ii) A|A|−1 is unitary. Proof (i) k|A|α k0 = k(AT A)α/2 k0 = ρ((AT A)α/2 ) = ρ(AT A)α/2 = kAkα0 . (ii) Let Q := A|A|−1 . Then QQT = A|A|−2 AT = I. For the solution of the linear system of equations Ax = b we consider the linear, consistent iteration xm+1 = xm − W −1 (Axm − b) with the iteration matrix S = I − W −1 A. For the definition of the smoothing property we must consider a sequence of equations Al xl = bl , l ≥ 0 and a sequence of linear iterations Sl = Il − Wl−1 Al , l ≥ 0. The discrete problems may arise from discretizations of a boundary value problem and usually vary in dimension. In the context of multigrid methods, the so-called smoothing property plays an important role. We recall its definition here. Definition 2 The linear iterations S l , l ≥ 0 are said to fulfill the smoothing property if kAl Slν k0 ≤ kAl k0 η(ν)
ν>0
for a function η(ν) → 0, ν → ∞ independent of l. A sufficient condition for the smoothing property is given in Lemma 3.
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Lemma 3 Let Al , Wl be regular, θ > 1 and Sl,θ := I − θWl−1 Al . If kSl,θ k|Wl | ≤ 1 and kWl k0 ≤ CkAl k0 uniformly in l, then Sl := Sl,1 fulfills the smoothing property with an η(ν) ≤ Cν −1/2 , where C depends only on θ. Proof For ease of notation, we omit the index l as all estimates obtained are uniform in l. It holds that AS ν = W (I − S)S ν = W |W |−1/2 |W |1/2 (I − S)S ν |W |−1/2 |W |1/2 , and, therefore, kAS ν k0 = kW |W |−1/2 k0 k(I − S)S ν k|W | kW k1/2 . From Lemma 2 we derive 1/2
kW |W |−1/2 k0 = kW |W |−1 |W |1/2 k0 = k|W |1/2 k0 = kW k0 and, therefore,
kAS ν k0 = kW k0 k(I − S)S ν k|W | ≤ CkAk0 k(I − S)S ν k|W | . Due to [14], there is a function η(ν) ≤ Cν −1/2 , with C dependend only on θ > 1 such that k(I − S)S ν k|W | ≤ η(ν) ∀ν. This completes the proof. Now we can state the main result of this paper. Theorem 1 Let A = As + Aa , with As = 1/2(A + AT ) > 0, Aa = 1/2(A − AT ) and m(A) ≤ m0 . Let As = D − L − LT be the standard additive splitting with D diagonal and L a strictly lower triangular matrix. The approximation of the symmetric Gauss-Seidel iteration as applied to As is given by W := As + LD −1 LT . Let ξ > 0, 0 < θ < Θ = 2/(1 + ξ 2 ) and −ξAs ≤ iAa ≤ ξAs .
(6)
Then the smoothing property holds for S θ = I − θW −1 A. Proof We show that kSΘ kW ≤ 1,
kW k0 ≤ CkAk0 .
(7)
The assertion of the theorem follows then from Lemma 3. (i) Let T = ΘW A−1 . The first condition of (7) is equivalent to kSΘ vk2W = ((I − T )v, (I − T )v)W ≤ (v, v)W , (T v, T v)W ≤ 2(T v, v)W , ∀v 0 ≤ (v, (2T −1 − I)v)W , 0 ≤ (v, (2W A
−1
∀v
W − ΘW )v),
∀v
∀v
⇔ ⇔
⇔
⇔
A Symmetric Smoother for NIPG
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and, because W is positive definite, 0 ≤ (v, (2W 1/2 A−1 W 1/2 − ΘI)v) −1/2
With X = As
−1/2
Aa As
∀v.
⇔
,
0 ≤ (v, (2W 1/2 A−1/2 (I + X)−1 A−1/2 W 1/2 − ΘI)v),
0 ≤ (v, (2W
1/2
A
−1/2
2 −1
(I − X )
A
−1/2
W
−1 1/2 ΘA1/2 As ≤ 2(I − X 2 )−1 s W
1/2
− ΘI)v),
∀v
∀v
Θ(I − X ) ≤ 2As−1/2 W As−1/2 Θ(As − Aa A−1 s Aa ) ≤ W 2
⇔
⇔
⇔
⇔
and, because W ≥ As , the latter condition follows from Θ(As − Aa A−1 s Aa ) ≤ A s
⇔
(As−1/2 iAa As−1/2 )2 ≤ (2/Θ − 1)I p p − 2/Θ − 1As ≤ iAa ≤ 2/Θ − 1As ,
⇔
where i denotes the complex unit. From (6) the first condition of (7) follows. (ii) Let M := D −1/2 As D −1/2 . Then M = M T and Mii = 1 ∀i. Since M is positive definite it follows that |M ij | ≤ 1. Therefore, |(D −1/2 W D −1/2 )ij |=|(D −1/2 (D − L)D −1 (D − L)T D −1/2 )ij | ≤
N X k=1
|(M )ik ||(M )kj | ≤ m0 .
Denoting by k · k∞ the maximum norm, we derive kD −1/2 W D −1/2 k0 ≤ kD −1/2 W D −1/2 k∞ ≤ m20 . Therefore, W ≤ m20 D and kW k0 ≤ m20 kDk0 ≤ m20 kAs k0 = m20 max |(x, As x)| kxk0 =1
= m20
max |(x, Ax)|
kxk0 =1
≤ m20 |
max
kxk0 =1,kyk0 =1
|(y, Ax)|
= m20 kAk0 . This proves the second condition of (7) and completes the proof.
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The method defined in Theorem 1 is called the symmetric GaussSeidel method with damping parameter θ. If the parameter ξ in the spectral bound (6) is smaller than 1, the symmetric Gauss-Seidel iteration fulfills the smoothing property without damping. It may be applied to the model problem introduced in the previous section. To this end, we introduce an appropriate local basis {ϕ i }i=0,...,N ⊂ V, where N denotes the dimension of V. Let x, b ∈ R N and A ∈ RN ×N be defined by X u= (x)k ϕi , (b)j := L(ϕj ), (A)ij := B(ϕj , ϕi ) k
for 1 ≤ i, j ≤ N . Then the model problem (4) is equivalent to Ax = b.
(8)
The symmetric part of A corresponds to the symmetric part of B(·, ·) (As )ij = Bs (ϕj , ϕi ), and, analogously, Aa corresponds to Ba (·, ·). The smoothing property of the symmetric Gauss-Seidel iteration for this problem is stated as follows: Corollary 1 Let µ > 0, k ≥ 1. There exists a θ 0 > 0 such that the damped Gauss-Seidel method applied to (8) fulfills the smoothing property for all θ < θ0 . Proof Since Bs (·, ·) is positive definite, As is too. Due to the local basis functions, A is sparse. By Lemma 1, estimate (5) holds, which √ is equivalent to (6) with ξ = 2Ck/ µ. From Theorem 1 follows the assertion above. Remark 2 According to Remark 1 the symmetric Gauss-Seidel √ is a smoother if (i) d = 2, k = 1, T a uniform square grid, √ µ > 4 2, and (ii) d = 3, k = 1, T a uniform cubic grid and µ > 6 3. Corollary 2 Let k ≥ 2 and µ > 0. There exists a θ 0 > 0 and a ν ≥ 1 such that the W (ν, ν)-cycle multigrid iteration using the damped Gauss-Seidel smoother exhibits an h-independent convergence rate ρ < 1 for the model problem (1). Proof Due to the regularity of the model problem, the discretization is of second order in L2 (Ω) [28]. Standard arguments lead to h-independent multigrid convergence [16].
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4 Numerical experiments The theory as presented in the previous section guarantees uniform multigrid convergence only for the W -cycle iteration and polynomial degrees k ≥ 2. The restriction to k ≥ 2 is due to the approximation property of the NIPG discretization, which is not optimal in L 2 . Numerical experiments indicate that, for odd polynomial degrees, L2 -optimality is achieved in practice, i.e. second order accuracy is observed for k = 1. This is sufficient to prove uniform multigrid convergence. Therefore, h-uniform convergence should be observed for k ≥ 1. Furthermore, numerical experiments indicate that uniform multigrid convergence may be achieved for the V -cycle multigrid as well. To investigate the performance of the multigrid method using the the smoother introduced in the previous section we carry out some numerical experiments. Let Ω =]0, 1[ 2 and f = 0. We investigate the model problem (1) with the trivial solution u ≡ 0. Let L ≥ 1, T 0 be the uniform grid consisting of 5 × 5 square elements and T l be the uniform refinement of Tl−1 , 0 < l ≤ L. On each grid level we discretize the problem using the NIPG method with a polynomial basis Pk , k = 1, 2, 3. The basis on each element results from a GramSchmidt orthogonalization with respect to L 2 of the monomials given in P1 := span(1, x, y),
(9a) 2
2
P2 := span(1, x, y, x , xy, y ),
(9b)
P3 := span(1, x, y, x2 , xy, y 2 , x3 , x2 y, xy 2 , y 3 ).
(9c)
Lexicographical ordering is chosen for the elements of the grids. Within the elements the basis functions are ordered according to the sequence of monomials in (9). In this way the enumeration of all degrees of freedom is fixed. We investigate the convergence of a V -cycle multigrid iteration with ν pre- and post-smoothing steps of the symmetric Gauss-Seidel iteration, as described in the previous section, and an exact solve on grid T0 . For the prolongation we use the natural embedding; for the restriction we use its transpose. The iterations performed to evaluate the convergence are started with a random initial guess. The convergence is measured by ρ10,20 :=
kd20 k2 kd10 k2
1/10
,
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T2 T3 T4 T5 T6
0.4
ρ10,20
0.3
0.2
0.1
0 1
µ
2
3
Fig. 1. Convergence of the V (1, 1)-cycle multigrid method as applied to the model problem using the basis P1 in dependence on the penalty parameter µ. The graphs correspond to different grid levels.
where di denotes the defect after the i-th iteration. We investigate the convergence in dependence on the grid level L, the penalty parameter µ, the order k of the polynomial basis, damping factor θ of the smoother and the number of pre- and post smoothing steps ν. 4.1 Variations with respect to L, k and µ In this subsection we investigate the V (1, 1)-cycle iteration (ν = 1) and use the symmetric Gauss-Seidel without damping (θ = 1). First we consider a discretization using the linear basis P 1 and L ∈ [2, 6]N ,
µ ∈ [0, 3] ∩ 0.1Z.
(10)
The evaluation of ρ10,20 is displayed in Fig. 1. As may be clearly seen the multigrid iteration converges uniformly with respect to the mesh size, but it exhibits a strong dependence on the penalty parameter. A fast multigrid convergence may be observed only in a small range of penalty parameters. This behavior is in agreement with the theoretical findings of the previous section. An insufficiently large penalty parameter requires a damping factor θ < 1. An increasing penalty
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1
T2 T3 T4 T5 T6
0.8
ρ10,20
0.6
0.4
0.2
0 2
3
µ
4
5
Fig. 2. Convergence of the V (1, 1)-cycle multigrid method as applied to the model problem using the basis P2 in dependence on the penalty parameter µ. The graphs correspond to different grid levels.
parameter increases the Euclidian norm of the stiffness matrix, leading to a deterioration of the multigrid smoothing. This deterioration is not balanced by the approximation which is not improved. The same behavior may be observed using the polynomial basis P 2 and P3 , as can be seen in Fig. 2 and Fig. 3 respectively. In contrast to the case of linear basis functions P 1 , the band of fast multigrid convergence is shifted to higher values of µ, corresponding to a deteriorated estimate (5) corresponding to an increased constant ξ in (6). Correspondingly, the interval for µ in (10) is changed. In the case of P3 the investigations could be performed on grid levels 2 through 5 only, due to memory contraints.
4.2 Variations with respect to ν and θ To extend the band of acceptable convergence rates, the number ν of smoothing steps and possibly the damping factor θ of the smoother must be varied. Note that an increase of ν leads to increased costs. We investigate the behavior of the multigrid iteration with respect to the model problem using the P2 basis.
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T6 1
T2 T3 T4 T5
0.8
ρ10,20
0.6
0.4
0.2
0 5
6
µ
7
8
Fig. 3. Convergence of the V (1, 1)-cycle multigrid method as applied to the model problem using the basis P3 in dependence on the penalty parameter µ. The graphs correspond to different grid levels.
To extend the band of acceptable convergence rates to higher penalty parameters, an increasing number ν = 1, 2, 4, 8 of pre- and post-smoothing steps for the non-damped smoothing iteration are used. The results are shown in Fig. 4. Whereas the the lower bound of convergence for the penalty parameter remains essentially unchanged, the upper bound is increased. This result may be derived from the estimate kAS ν k0 ≤ CkAk0 ν −1/2 for the smoothing property and the fact that kAk0 ∝ C 0 µ, for large µ. In Fig. 4 we see an increase of the upper bound by roughly 1.7 for a doubling of the number of smoothing steps, indicating that the estimate obtained from the smoothing property is too pessimistic. To extend the band of acceptable convergence to smaller values of the penalty parameter, the number of smoothing steps must be increased, while decreasing the damping factor of the smoothing. We choose, quite arbitrarily, θ = ν −1 , ν = 1, 2, 4, 8. The results of this numerical experiment are shown in Fig. 5. An extension to smaller values for the penalty parameter may be observed for ν = 1, 2, 4.
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1
0.8
ρ10,20
0.6
0.4
ν ν ν ν
0.2
=1 =2 =4 =8
0 1
µ
10
100
Fig. 4. Multigrid convergence for the model problem using the basis P2 in dependence on the penalty parameter µ. An increasing number ν of pre- and postsmoothing steps is used.
5 Conclusions In this paper we investigated the smoothing property for a variant of the symmetric Gauss-Seidel smoother as applied to the NIPG discontinuous Galerkin discretization of the Poisson equation. The smoothing property was shown to hold for nondegenerate grids, polynomial degrees k ≥ 1, and positive penalty parameters. For quadratic and higher order polynomials, uniform multigrid convergence was shown by standard arguments. Depending on the penalty parameter used for the discretization, the smoothing iteration must be damped. The dependence has been given explicitly. Numerical experiments confirm the theoretical findings. Though the theoretical estimates presented here do not seem to be sharp, they provide an excellent understanding of the qualitative behavior of the scheme under consideration. Acknowledgements The support of this work by the J. Tinsley Oden Faculty Fellowship Research Fund is gratefully acknowledged. Furthermore, the support of P. Bastian, Heidelberg University, Germany, who made a software
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ν ν ν ν
= 1, θ = 2, θ = 4, θ = 8, θ
=1 = 1/2 = 1/4 = 1/8 n16
0.8
ρ10,20
0.6
0.4
0.2
0 0.1
µ
1
10
Fig. 5. Multigrid convergence for the model problem using the basis P2 in dependence on the penalty parameter µ. An increasing number ν of pre- and postsmoothing steps is used, while the damping parameter is varied with θ = ν −1 .
implementation of discontinuous Galerkin methods available is gratefully acknowledged. References 1. T. Arbogast, S. Bryant, C. Dawson, F. Saaf, C. Wang, and M.F. Wheeler. Computational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation. J. Comp. Appl. Math., 74(12):19–32, 1996. 2. D.N. Arnold. An iterior penalty finite element method with discontinuous elements. J. Numer. Anal., 19:742–760, 1982. 3. I. Babuska and M. Zlamal. Nonconforming elements in the finite element method with penalty. SIAM J Numer. Anal., 10:863–875, 1973. 4. I. Babusky, C.E. Baumann, and J.T. Oden. A discontinuous hp finite element method for diffusion problems. J. Comp. Phys., 146:491–516, 1998. 5. G. Baker. Finite elment methods for elliptic equations using nonconforming elements. Math. Comp., 31:45–59, 1977. 6. F. Bassi and S. Rebay. A higher-order accurate discontinuous finite element method for the numerical solution of the navier-stokes equations. J. Comp. Phys., 131:267–279, 1997. 7. P. Bastian and V. Reichenberger. Multigrid for higher order discontinuous Galerkin finite elements applied to groundwater flow. Technical Report 200037, SFB 359, Heidelberg University, 2000.
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8. C.E. Baumann and J.T. Oden. A discontinuous hp finite element method for convection-diffusion problems. Comp. Meth. Appl. Mech. Eng., 175(3-4):311– 341, 1999. 9. S. Brenner and Li-Yeng Sung. c0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comp., 2223:83–118, 2005. 10. S. Brenner and Jie Zhao. Convergence of multigrid algorithms for interior penalty methods. Appl. Num. Anal. Comp. Math., 2(1):3–18, 2005. 11. B. Cockburn and C.W. Shu. The local discontinuous Galerkin finite element method for convection diffusion systems. SIAM J. Num. Anal., 35:2440–2463, 1998. 12. C. Dawson, S. Sun, and M.F. Wheeler. Compatible algorithms for coupled flow and transport. Comp. Meth. Appl. Mech. Eng., 193:2565–2580, 2004. 13. J. Douglas and T. Dupont. Interior penalty procedures for elliptic and parabolic Galerkin methods. Lecture notes in Physics, 58:207–216, 1976. 14. A. Ecker and W. Zulehner. On the smoothing property of multigrid methods in the non-symmetric case. Numer. Linear Algebra Appl., 3:161 – 172, 1996. 15. J. Gopalakrishnan and G. Kanschat. A multilevel discontinuous Galerkin method. Numer. Math., 3:527–550, 2003. 16. W. Hackbusch. Multi-Grid Methods and Applications. Springer-Verlag, Berlin, Heidelberg, 1985. 17. W. Hackbusch. Iterative Solution of Large Sparse Systems of Equations. Applied Mathematical Sciences, Vol 95. Springer-Verlag, New York, U.S.A., 1993. 18. P.W. Hemker, W. Hoffmann, and M.H. van Raalte. Fourier two-level analysis for discontinuous Galerkin discretization with linear elements. Technical Report MAS-R0217, ISSN 1386-3703, CWI, Amsterdam, 2002. 19. P.W. Hemker, W. Hoffmann, and M.H. van Raalte. Two-level fourier analysis of a multigrid approach for discontinuous Galerkin discretisation. Technical Report MAS-R0206, ISSN 1386-3703, CWI, Amsterdam, 2002. 20. P.W. Hemker and M.H. van Raalte. Fourier two-level analysis for higher dimensional discontinuous Galerkin. Technical Report MAS-R0227, ISSN 1386-3703, CWI, Amsterdam, 2002. 21. G. Kanschat. Preconditioning methods for local discontinuous Galerkin discretizations. SIAM J. Sci. Comp., 25(3):815–831, 2003. 22. G. Kanschat. Block preconditioner for ldg discretizations of linear incompressible flow problems. J. Sci. Comp., 22-23:371–384, 2005. 23. J.T. Oden and L.C. Wellford Jr. Discontinuous finite element approximations for the analysis of shock waves in nonlinear elastic materials. J. Comp. Phys., 19(2):179–210, 1975. 24. P. Percell and M.F. Wheeler. A local residual finite element procedure for elliptic equations. SIAM J. Num. Anal., 15(4):705–714, 1978. 25. B. Riviere. Discontinous Galerkin finite element methods for solving the miscible displacement problem in porous media. PhD thesis, The University of Texas at Austin, 2000. 26. B. Riviere and M.F. Wheeler. Non conforming methods for transport with nonlinear reaction. Contemporary Mathematics, 295:421–432, 2002. 27. B. Riviere, M.F. Wheeler, and V. Girault. Improved energy estimates for inerior penalty, constraint and discontinuous Galerkin methods for elliptic problems. Part i. Computational Geosciences, 3:337–360, 1999.
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