Interpolation, Projection and Hierarchical Bases in Discontinuous Galerkin Methods

Interpolation, projection and hierarchical bases in discontinuous Galerkin methods

Lutz Angermann∗ and Christian Henke†

February 20, 2013

Abstract: The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev- Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities. The main focus is directed to applications to discrete conservation laws. Keywords: Interpolation, quadrature, hierarchical bases, inverse inequalities. 2010 Mathematics Subject Classiﬁcation: 65 M 60, 65 D 32.

1 Introduction

The topics covered in this paper belong to the fundamentals of the analysis of discontinuous Galerkin methods for partial differential equations. On the one hand, we have collected and reproved results from areas that are important for any study of finite element methods, such as the piecewise polynomial approximation in Sobolev spaces, quadrature formulas and inverse inequalities (Sections 2, 3, 4, 6). On the other hand, we have directed our attention to facts that are specifically related to particular techniques such as certain relations between lumping and quadrature effects or the investigation of fluctuation operators and shock-capturing terms by means of a hierarchical basis approach (Sections 4, 5). A major concern of our study was to trace the dependence of the constants on both the mesh width and the local polynomial degree. The paper is organized as follows. After a brief introduction, which introduces the basic notation, we investigate polynomial approximations by means of tensor product elements on affine partitions in Section 3. This includes estimates of the reference transformations, which we prove by the help of a general chain rule. In this way we get a certain insight into the structure of the occurring constants. After that we prove error estimates for the Lagrange interpolation and the L2-projection both with respect to the elements and with respect to the element edges in the scale of Sobolev- Slobodecki spaces. arXiv:1102.3100v2 [math.NA] 19 Feb 2013 In Section 4, we prepare some important notions such as quadrature formulas, lumping operators and discrete L2-projections for later purposes. In particular, we point out the importance of an suitable choice of the quadrature points for optimal (w.r.t. the local polynomial degree) error estimates of the Lagrange interpolation.

∗Institut fürMathematik, Technische UniversitätClausthal, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany. Email: [email protected] †Institut fürMathematik, Technische UniversitätClausthal, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany. Email: [email protected]

1 In the following section we investigate the projection and interpolation errors for Gauss-Lobatto nodes. For this purpose we extend the concept of the hierarchical modal basis to the so-called embedded hierarchical nodal basis and we prove error estimates for the Lagrange interpolation and for the discrete L2-projection which are optimal on the elements and almost optimal on the element edges. As a natural complement to the (direct) estimates from the previous sections, we present in Section 6 inverse inequalities that are based on generalizations of the Nikolski and Markov inequalities.

2 Basic notation and deﬁnitions

d Let Ω ⊂ R , d ∈ N, be a bounded polyhedral domain with a Lipschitzian boundary (see, e.g., [AF03, Def. 4.9]). Ω is subdivided by partitions T (in the sense of [EG04, Def. 1.49]) consisting d of tensor product elements T (closed as subsets of R ) with diameter hT := maxx,y∈T kx − yk`2 . p Here and in what follows the symbol k · k`p , p ∈ [1, ∞], denotes the usual ` -norm of (finite) real sequences. Furthermore the maximal width of a partition T is defined by h := max{hT : T ∈ T }. To indicate that a particular partition has the maximal width h, we will write Th. In this paper, the standard definition of finite elements {T,PT , ΣT } is used, see e.g. [EG04, Def. 1.23]. The finite element space is defined by

2 Wh := w ∈ L (Ω) : w|T ∈ PT ∀T ∈ Th , where l,∞ PT ⊂ W (T ) ∀T ∈ Th for some l ≥ 0.

Because of the last requirement, Wh is a subspace of

l,p n 2 l,p o W (Th) := w ∈ L (Ω) : w|T ∈ W (T ) ∀T ∈ Th , p ∈ [1, ∞].

A particular finite element {T,PT , ΣT } is generated by means of a reference element {T,ˆ P,ˆ Σˆ}, where the geometric reference element Tˆ is mapped onto the geometric element T by a C1- diffeomorphism FT : Tˆ → T. In the case of a Lagrange finite element {T,ˆ P,ˆ Σˆ} (in the sense of [EG04, Def. 1.27]) with the node ˆ k ˆ set N := {xˆ1,..., xˆ k }, n := #Σ, and the linear forms ndof dof k ˆ σî(ˆv) :=v ˆ(ˆxi), 1 ≤ i ≤ ndof , ∀vˆ ∈ P, the definitions −1 ˆ PT := {vˆ ◦ FT :v ˆ ∈ P } (1) and

−1 k σi(v) =σ î(v ◦ FT ) =σ î(ˆv), v(x) = (ˆv ◦ FT )(x) =v ˆ(ˆx), x = FT (ˆx), 1 ≤ i ≤ ndof , are used. The nodal basis of PT is obtained by an analogous transformation of the reference shape functions k {ϕˆ1,..., ϕˆ k } withϕ î(ˆxj) = δij, 1 ≤ i, j ≤ n . ndof dof

DEFINITION 1 A partition Th is called aﬃne if the mapping FT is aﬃne for all T ∈ Th, i.e. if d,d d FT (ˆx) = JT xˆ + bT with JT ∈ R , det JT 6= 0, bT ∈ R .

In addition, the following properties of Th are important.

2 DEFINITION 2 (locally quasiuniform, shape regular) A family of aﬃne partitions {Th}h>0 is called locally quasiuniform if there exists a constant σ0 > 0 such that

hT ∀h > 0 ∀T ∈ Th : σT := ≤ σ0, (2) %T where %T denotes the diameter of the largest ball contained in T.

DEFINITION 3 (quasiuniform) A family of partitions {Th}h>0 is called quasiuniform if it is locally quasiuniform and if a constant Cqu > 0 exists with

∀h > 0 ∀T ∈ Th : hT ≥ Cquh.

3 Polynomial approximation using tensor product elements

In what follows we will investigate thoroughly the approximation of functions on aﬃne partitions. ˆ d ˆ ˆ We set T := I ,I := [−1, 1] and P := Qk(T ) with ˆ α ˆ Qk(T ) := span {xˆ }, xˆ ∈ T , k ∈ N0 := N ∪ {0}. d α∈N0, kαk`∞ ≤k

d A functionϕ ˆα ∈ {ϕˆ1,..., ϕˆ k }, α ∈ , can be written as a product of univariate Lagrange ndof N0 k k k polynomials. Namely, denote by {ϕˆ0, ϕˆ1,..., ϕˆk} a basis of the space of univariate polynomials of d maximum degree k. Then, for any multiindex α ∈ N0 with kαk`∞ ≤ k, ϕˆ (ˆx) =ϕ ˆk (ˆx )ϕ ˆk (ˆx ) ··· ϕˆk (ˆx ). α α1 1 α2 2 αd d

The space Qk(T ) is defined according to (1). Similarly, based on the definition ˆ α ˆ Pk(T ) := span {xˆ }, xˆ ∈ T , k ∈ N0, d α∈N0, kαk`1 ≤k the space Pk(T ) of polynomials of maximum degree k can be introduced. As a consequence of the affine structure of the transformation between the reference element Tˆ and the element T we can prove the following estimates.

LEMMA 1 For l ≥ 0 and 1 ≤ p ≤ ∞, 1/∞ := 0, there exists a constant Cl,d ≥ 1 such that, for l,p T ∈ Th, Th aﬃne, and w ∈ W (T ), wˆ = w ◦ FT , the following estimates hold: l −1/p |wˆ|l,p,Tˆ ≤ Cl,dkJT k`2 | det JT | |w|l,p,T , (3) −1 l 1/p |w|l,p,T ≤ Cl,dkJT k`2 | det JT | |wˆ|l,p,Tˆ, (4) where Cl,d depends only on l and d. In particular, C0,d = 1. d Proof: By Fa`adi Bruno’s formula (see, e.g. [Joh02]), forw ˆ = w ◦ FT , |α| = l, P := N0 \{0} and    d X X  A := a : P → N0 : a(γ) = β and |a(γ)|γ = α  γ∈P γ∈P  we have the representation

h ˆγ ia(γ) ˆα P β P Q 1 (∂ FT )(ˆx) ∂ (w ◦ FT )(ˆx) = α! |β|≤|α|(∂ w)(FT (ˆx)) a∈A γ∈P a(γ)! γ! a(e ) P β P Qd 1 h ˆ i j = α! (∂ w)(FT (ˆx)) (∂jFT ) |β|≤|α| a∈A j=1 a(ej )! a(e ) P β P Qd 1 h ˆ i j = α! (∂ w)(FT (ˆx)) (∂jFT ) , |β|=l a∈A j=1 a(ej )!

3 P Pd Pd because the conditions l = |α| = | m |a(γ)|γ| = | |a(e )|e | = |a(e )| and |β| = γ∈N j=1 j j j=1 j Pd Pd | j=1 a(ej)| = j=1 |a(ej)| imply that |α| = |β| = l if “empty” sums are neglected. The absolute value of the left-hand side can be estimated as

P d+l−1 For p < ∞, using H¨older’sinequality for sums and observing that |β|=l 1 = l , on the reference element we have that

(p−1) d + l − 1 X k∂ˆαwˆkp ≤ Cp kJ klp k∂βw ◦ F kp , 0,p,Tˆ l α,d T `2 T 0,p,Tˆ |β|=l

Applying the substitution rule, we get

d + l − 1(p−1) k∂ˆαwˆkp ≤ Cp kJ klp | det J |−1|w|p . 0,p,Tˆ l α,d T `2 T l,p,T

Summing up w.r.t. α, the estimate

|wˆ|p = P k∂ˆαwˆkp l,p,Tˆ |α|=l 0,p,Tˆ P d+l−1(p−1) p lp −1 p ≤ |α|=l l Cα,dkJT k`2 | det JT | |w|l,p,T p d+l−1p lp −1 p ≤ max (Cα,d) l kJT k 2 | det JT | |w|l,p,T |α|=l `

d+l−1 proves the statement with Cl,d := l max (Cα,d) . For p = ∞ we see that |α|=l

ˆα |wˆ|l,∞,Tˆ = max k∂ wˆk0,∞,Tˆ |α|=l l P β ≤ max (Cα,d) kJT k 2 k |β|=l ∂ w (FT (ˆx)) k0,∞,Tˆ |α|=l ` l P β = max (Cα,d) kJT k 2 k |β|=l ∂ w (x) k0,∞,T |α|=l ` . l d+l−1 β ≤ max (Cα,d) kJT k 2 l k max ∂ w k0,∞,T |α|=l ` |β|=l l β = Cl,dkJT k 2 max k∂ wk0,∞,T ` |β|=l

ˆ Since FT : T → T is bijective, the second estimate follows obviously. J LEMMA 2 The following estimates are valid:

4 COROLLARY 1 Given a locally quasiuniform family {Th}h>0 of aﬃne partitions with the reference element Tˆ = Id. Then: √ hT −1 2 dσ0 kJT k`2 ≤ , kJT k`2 ≤ (5) 2 hT and d Y 1/2 d/2 d −d d ˆ d det JT = λi ≤ λmax = kJT k`2 ≤ 2 hT , |T | = |T || det JT | ≤ hT , (6) i=1 > where λi, 1 ≤ i ≤ d, are the eigenvalues of the matrix JT JT . Now we are ready to deﬁne the Lagrange interpolation operator as follows:

k ndof k 0 k X l,∞ ˆ Ih : C (T ) 3 v 7→ Ih v := v(xi)ϕi ∈ Qk(T ) ⊂ W (T ), xi ∈ NT := FT (N ). i=1 We mention that the results from this section on the Lagrange interpolation operator do not impose any conditions w.r.t. the location of the nodes. Later we will formulate statements about the interpolation operator which use the special properties of Gauss-Lobatto quadrature points. Simple calculations show that

k Ih v = v ∀v ∈ Qk(T ), (7) k kIh vkl,p,T ≤ Ckvk0,∞,T , l ≥ 0, p ∈ [1, ∞], (8)

k ndof k where C = C(T, {ϕi}i=1 , ndof , l, p) > 0. Furthermore we have the following error estimates.

LEMMA 3 Let T be an element of an aﬃne partition Th such that the corresponding family of partitions {Th}h>0 is locally quasiuniform. Assume that 1 ≤ l ≤ k +1, l ∈ N, p ∈ [1, ∞] and lp > d. k Then, for the Lagrange interpolation operator Ih , there exist constants C > 0 independent of hT such that

LEMMA 4 (interpolation inequality) Let l0, l1 ∈ N0, l0 6= l1, 1 ≤ p ≤ ∞ and G be a bounded domain with a Lipschitzian boundary. Deﬁne, for 0 < θ < 1, lθ := (1 − θ)l0 + θl1. Then

∀u ∈ W l0,p(G) ∩ W l1,p(G): kuk ≤ Ckuk1−θ kukθ . (9) lθ,p,G l0,p,G l1,p,G

d Proof of Lemma 4: We ﬁrst mention that the statement for the case G = R is a consequence of [Tri78, 1.3.3 (g)] and [BL76, Def. 6.2.2, Thm. 6.2.3, Thm. 6.2.4 and Thm. 6.4.5 (3),(4)]. If G is a bounded domain with a Lipschitzian boundary, then there exists a total extension operator EG (see [AF03, Thm. 5.24] or [Ste70, Ch. 6, Thm. 5]) such that

kuklθ,p,G = kEGuklθ,p,G (by [AF03, (i) in (5.17)]) ≤ kE uk d G lθ,p,R 1−θ θ ≤ CkEGuk d kEGuk d l0,p,R l1,p,R ≤ Ckuk1−θ kukθ (by [AF03, (ii) in (5.17)]). l0,p,G l1,p,G J

5 REMARK 1 Using further results on Stein’s extension operator [Kal85, p. 185 and Thm. 1], the interpolation inequality (9) can be extended to the parameter set l0, l1 ∈ R+, l0 6= l1, 1 ≤ p0, p1 ≤ ∞, as follows, where 1 := 1−θ + θ : pθ p0 p1

∀u ∈ W l0,p0 (G) ∩ W l1,p1 (G): kuk ≤ Ckuk1−θ kukθ . lθ,pθ,G l0,p0,G l1,p1,G Proof of Lemma 3: By the triangle inequality, (8), and the embedding theorem [AF03, Thm. 4.12, Part II], for 1 ≤ l ≤ k + 1, r ∈ N0, r ≤ l, lp > d we have that ˆk ˆk kvˆ − Ih vˆkr,p,Tˆ ≤ kvˆkr,p,Tˆ + kIh vˆkr,p,Tˆ ≤ kvˆkl,p,Tˆ + Ckvˆk0,∞,Tˆ l,p ˆ (10) ≤ Ckvˆkl,p,Tˆ ∀vˆ ∈ W (T ). ˆ ˆ Furthermore, by (7) and Pk(T ) ⊂ Qk(T ), ˆk ˆk ˆk |vˆ − Ih vˆ| ˆ ≤ kvˆ − Ih vˆk ˆ = inf k(Id − Ih )(ˆv +p ˆ)k ˆ r,p,T r,p,T ˆ r,p,T pˆ∈Pk(T ) ˆk ≤ inf k(Id − Ih )(ˆv +p ˆ)k ˆ, ˆ r,p,T pˆ∈Pl−1(T ) where Id denotes the identity operator. From (10) we see that ˆk |vˆ − Ih vˆ| ˆ ≤ C inf kvˆ +p ˆk ˆ ≤ C|vˆ| ˆ, r,p,T ˆ l,p,T l,p,T pˆ∈Pl−1(T ) where the last estimate is a consequence of the Deny-Lions lemma ([Cia91, Thm. 14.1]). Note that the constant C depends on the parameters of the reference element. The application of Lemma 1 results in an estimate on the element T : k −1 r 1/p ˆk |v − Ih v|r,p,T ≤ CkJT k`2 | det JT | |vˆ − Ih vˆ|r,p,Tˆ −1 r 1/p ≤ CkJT k`2 | det JT | |vˆ|l,p,Tˆ −1 r l−r ≤ C kJT k`2 kJT k`2 kJT k 2 |v|l,p,T r ` ≤ C hT hl−r|v| ≤ Chl−r|v| . ρT T l,p,T T l,p,T where we have used the condition (2) in the last step. In the case r ∈ R+ \ N0, we apply the interpolation inequality (9) with l0 := brc := max z, z∈Z, {z}≤r l1 := dre := min {z} and θ := r − brc: z∈Z, z≥r

k k 1−θ k θ |v − Ih v|r,p,T ≤ C|v − Ih v|brc,p,T |v − Ih v|dre,p,T l−r ≤ ChT |v|l,p,T , dre ≤ l. The proof of the second estimate runs similarly. For 1 ≤ l ≤ k + 1, p ∈ (1, ∞), and any face Eˆ of the reference element Tˆ with E = FT Eˆ we have, by the special trace theorem for the faces of Tˆ (see [Neˇc67, Thm. 2.5.4]) and (10), that ˆk ˆk l,p ˆ kvˆ − Ih vˆkl−1/p,p,Eˆ ≤ Ckvˆ − Ih vˆkl,p,Tˆ ≤ Ckvˆkl,p,Tˆ ∀vˆ ∈ W (T ). (11) Furthermore, [Neˇc67,Lemma 2.5.4] implies that, for any r ∈ [0, l − 1/p), ˆk ˆk l,p ˆ kvˆ − Ih vˆkr,p,Eˆ ≤ Ckvˆ − Ih vˆkl−1/p,p,Eˆ ∀vˆ ∈ W (T ). Combining this estimate with (11) we arrive, for p ∈ (1, ∞) and r ∈ [0, l − 1/p], at ˆk l,p ˆ kvˆ − Ih vˆkr,p,Eˆ ≤ Ckvˆkl,p,Tˆ ∀vˆ ∈ W (T ). (12)

6 In the case p = 1, we conclude from [Gag57, Thm. 1.II] by a similar argument as in [Neˇc67,Lemma 2.5.4] that ˆk ˆk l,1 ˆ kvˆ − Ih vˆkl−1,1,Eˆ ≤ Ckvˆ − Ih vˆkl,1,Tˆ ∀vˆ ∈ W (T ), hence, ˆk l,1 ˆ kvˆ − Ih vˆkr,1,Eˆ ≤ Ckvˆkl,1,Tˆ ∀vˆ ∈ W (T ) for all r ∈ [0, l − 1]. The case p = ∞ is a simple consequence of the fact that the trace operator is the classical restriction due to the embedding theorem [Neˇc67,Thm. 2.3.8]:

ˆk l,∞ ˆ kvˆ − Ih vˆkr,∞,Eˆ ≤ Ckvˆkl,∞,Tˆ ∀vˆ ∈ W (T ) for all r ∈ [0, l]. Thus (12) is proved for all p ∈ [1, ∞] with lp > d and all r ∈ [0, l − 1/p]. Now, if additionally r ∈ N0, we have that ˆk ˆk ˆk |vˆ − Ih vˆ| ˆ ≤ kvˆ − Ih vˆk ˆ = inf k(Id − Ih )(ˆv +p ˆ)k ˆ r,p,E r,p,E ˆ r,p,E pˆ∈Pk(T ) ˆk ≤ inf k(Id − Ih )(ˆv +p ˆ)k ˆ ˆ r,p,E pˆ∈Pl−1(T ) ≤ C inf kvˆ +p ˆk ˆ ≤ C|vˆ| ˆ. ˆ l,p,T l,p,T pˆ∈Pl−1(T ) Using the interpolation inequality (9) and performing the back-transformation, we get

k k 1−θ k θ |v − Ih v|r,p,E ≤ C|v − Ih v|brc,p,E|v − Ih v|dre,p,E −1 r 1/p ≤ CkJE k`2 | det JE| |vˆ|l,p,Tˆ 1/p −1 r l−r |E||Tˆ| ≤ C kJ k 2 kJ k 2 kJ k |v| T ` E ` T `2 |Eˆ||T | l,p,T r ≤ C hT hl−rh−1/p|v| ρE T T l,p,T l−1/p−r ≤ ChT |v|l,p,T , where we have used the simple estimate ρT ≤ ρE together with (2). J

COROLLARY 2 Let T be an element of an aﬃne partition Th such that the corresponding family of partitions {Th}h>0 is locally quasiuniform. Assume that l ∈ N, p ∈ [1, ∞] and lp > d. Then, for k the Lagrange interpolation operator Ih , there exist constants C > 0 independent of hT such that

k min{k+1,l}−r |v − Ih v|r,p,T ≤ ChT kvkl,p,T , 0 ≤ r ≤ l, k min{k+1,l}−1/p−r |v − Ih v|r,p,E ≤ ChT kvkl,p,T , 0 ≤ r ≤ l − 1/p, for all v ∈ W l,p(T ).

Proof: For 1 ≤ l ≤ k + 1, the statement coincides with Lemma 3. In the case k + 1 < l, r ∈ R+, Lemma 3 implies that k k+1−r |v − Ih v|r,p,T ≤ ChT |v|k+1,p,T . k+1−r min{k+1,l}−r From hT |v|k+1,p,T ≤ ChT kvkl,p,T the ﬁrst estimate follows. The proof of the second estimate runs analogously. J A further possibility of approximating functions in Sobolev spaces is given by the projection w.r.t. the L2 inner product.

2 k 2 2 DEFINITION 4 The orthogonal L -projection Ph : L (T ) → Qk(T ) is deﬁned, for v ∈ L (T ), by k (v − Ph v, w)0,T = 0 ∀w ∈ Qk(T ). (13)

7 In general, the relation (13) is equivalent to a system of linear algebraic equations. It can be solved easily provided an L2-orthogonal basis is used. For elements T of an affine partition and d any multiindex α ∈ N0, kαk`∞ ≤ k, we have that Z Z d ˆ ˆ Y 1 J ψαψβ dx = | det JT | ψαψβ dxˆ = | det JT | δαβ = ραδα,β, ˆ 2α + 1 T T i=0 i where ψˆ (ˆx) := ψˆα1 (ˆx )ψˆα2 (ˆx ) ··· ψˆαd (ˆx ), α α1 1 α2 2 αd d and 1 dαi αi αi αi ψˆ (ˆxi) := (ˆxi + 1) (ˆxi − 1) αi αi αi 2 αi! dxî denotes the αi-th one-dimensional Legendre polynomial of degree αi w.r.t.x î ∈ I, 1 ≤ i ≤ d. In k k Pndof J −1 this case and with an appropriate indexing, we see that Ph v = i=1 (ρi ) (v, ψi)0,T ψi. Summarized representations about Legendre polynomials can be found in [KS05, App. A] or [QV94, Ch. 4]. From the definition of the L2-projection, the following properties easily follow:

k Ph v = v ∀v ∈ Qk(T ), (14) k 2 kPh vk0,2,T ≤ kvk0,2,T ∀v ∈ L (T ).

LEMMA 5 Let T be an element of an aﬃne partition Th such that the corresponding family of partitions {Th}h>0 is locally quasiuniform. Then, for 1 ≤ l ≤ k + 1, l ∈ N, there exist constants C > 0 independent of hT and k such that

hl−r |v − P kv| ≤ C T |v| , 0 ≤ r ≤ l, h r,2,T ke(r,l) l,2,T hl−1/2−r 1 |v − P kv| ≤ C T |v| , 0 ≤ r < l − , (15) h r,2,E ke(r+1/2+ε,l) l,2,T 2 for all v ∈ W l,2(T ), where ε > 0 is arbitrarily small and

l + 1/2 − 2r, r ≥ 1, e(r, l) =: l − 3r/2, 0 ≤ r ≤ 1.

Proof: By [CQ82, Thm. 2.4], there exists a constant C > 0 independent of k such that, for r, l ∈ R+, r ≤ l, ˆk −e(r,l) l,2 ˆ kvˆ − Ph vˆkr,2,Tˆ ≤ Ck kvˆkl,2,Tˆ ∀vˆ ∈ W (T ). As in the proof of Lemma 3, this estimate together with (14) and the Deny-Lions lemma ([Cia91, Thm. 14.1]) implies that, for l ≤ k + 1, ˆk −e(r,l) l,2 ˆ |vˆ − Ph vˆ|r,2,Tˆ ≤ Ck |vˆ|l,2,Tˆ ∀vˆ ∈ W (T ), where C = C(d, l, Tˆ) > 0. In particular, the constant does not depend on the polynomial degree k. At the faces of Tˆ we make use of the following argument. Let Eˆ ⊂ ∂Tˆ be an arbitrary but fixed face. d−1 d−1 Without loss of generality we may consider it as a subset of R , where the elementsx ˆ ∈ R are characterized by xd = 0 and the elements of Tˆ satisfy the condition xd ≥ 0 (otherwise we apply a rotation and a translation of the coordinate system; both operations do not affect the differentiability properties of the elements of the function spaces under consideration).

8 Now, letw ˆ ∈ W s+1/2,2(Tˆ) for some s ∈ (0, l−1/2]. Then, by Kalyabin’s results on Stein’s extension s+1/2,2 ˆ operator ([Kal85, p. 185 and Thm. 1]), there exists a total extension operator ETˆ : W (T ) → s+1/2,2 d W (R ) such that kETˆwˆks+1/2,2,Rd ≤ Ckwˆks+1/2,2,Tˆ. The trace theorem ([AF03, Thm. 7.43 together with Rem. 7.33]) implies that

kETˆwˆks,2,Rd−1 ≤ CkETˆwˆks+1/2,2,Rd , so

kETˆwˆks,2,Eˆ ≤ kETˆwˆks,2,Rd−1 ≤ Ckwˆks+1/2,2,Tˆ. Note that for small s ∈ (0, 1), we have by a direct trace theorem for Lipschitz domains ([JK95, Thm. 3.1]) that

kwˆks,2,Eˆ ≤ kwˆks,2,∂Tˆ ≤ Ckwˆks+1/2,2,Tˆ. s,2 ˆ s+1/2,2 ˆ Therefore, ETˆwˆ|Eˆ =w ˆ|Eˆ in the sense of W (E) (consider, for l ≥ 2, wˆ ∈ W (T ) for s ∈ [1, l − 1/2] as an element of, say,w ˆ ∈ W 1,2(Tˆ)) and we ﬁnally arrive at the estimate

kwˆks,2,Eˆ ≤ kETˆwˆks,2,Rd−1 ≤ Ckwˆks+1/2,2,Tˆ, s ∈ (0, l − 1/2]. (16)

In summary, for r ∈ [0, l − 1/2) and ε > 0 arbitrarily small, we get

kwˆkr,2,Eˆ ≤ kwˆkr+ε,2,Eˆ ≤ Ckwˆkr+ε+1/2,2,Tˆ.

ˆk In particular, forw ˆ :=v ˆ − Ph v,ˆ we have ˆk ˆk kvˆ − Ph vˆkr,2,Eˆ ≤ Ckvˆ − Ph vˆkr+ε+1/2,2,Tˆ, and, by [CQ82, Thm. 2.4],

ˆk −e(r+ε+1/2,l) kvˆ − Ph vˆkr,2,Eˆ ≤ Ck kvˆkl,2,Tˆ. Thus, by (14) and the Deny-Lions lemma ([Cia91, Thm. 14.1]),

ˆk −e(r+ε+1/2,l) |vˆ − Ph vˆ|r,2,Eˆ ≤ Ck |vˆ|l,2,Tˆ.

The back-transformation to T runs analogously as in the proof of Lemma 3. J In contrast to the estimates of the projection error presented above, the next assertion can be proved without the use of the Deny-Lions lemma.

LEMMA 6 Let T be an element of an aﬃne partition Th such that the corresponding family l,2 of partitions {Th}h>0 is locally quasiuniform. Then, for 0 ≤ l and v ∈ W (T ) with 1 ≤ s ≤ min{k + 1, l}, there exist constants C > 0 independent of hT and k such that

h s |v − P kv| ≤ C T |v| , h 0,2,T k s,2,T hs−1 |v − P kv| ≤ C T |v| , h 1,2,T ks−3/2 s,2,T h s−1/2 |v − P kv| ≤ C T |v| . h 0,2,∂T k s,2,T

Proof: [Geo03, Cor. 3.15, (3.1)-(3.3)], [HSS02, Lemma 3.5], [SvdVvD06, Lemma 6.1, Rem. 6.2]. J

9 REMARK 2 For comparison, we mention the following special case of Lemma 5, (15):

hl−1/2 ε<1/6 hl−1/2 |v − P kv| ≤ C T |v| ≤ C T |v| . h 0,2,E kl−3/2(1/2+ε) l,2,T kl−1 l,2,T We see that this estimate is suboptimal of order k1/2. The reason lies in the behavior of the L2- projection error measured in the | · |r,2,T -norm for r > 0. Indeed, any proof of such an estimate which is based on a trace inequality will result in right-hand side bounds where the occuring norms cause a suboptimal result. To overcome this problem, a direct estimate of the L2-projection error is helpful, see [HSS02, Section 3.3].

4 Quadrature and lumping

In order to evaluate the occuring integrals approximately we will consider interpolatory quadrature rules (cf. [EG04, Def. 8.1]).

d DEFINITION 5 Let T ⊂ R be a nonempty, compact, connected subset with a Lipschitzian k boundary (cf. [AF03, Def. 4.9]). A quadrature rule on T with ndof nodes is characterized

k n J J o 1. by a set consisting of ndof real numbers ω1 , . . . , ω k called weights, and ndof k 2. by a set Q consisting of n points {x1, . . . , x k } ⊂ T, where xi 6= xj if i 6= j, called dof ndof quadrature nodes. The largest natural number k such that

nk Z dof X J p(x) dx = ωi p(xi) ∀p ∈ Qk(T ) T i=1 is called the degree of precision or the quadrature order of the quadrature rule.

From Z Z p(x) dx = p(FT (ˆx))| det JT | dxˆ T Tˆ Z X = p(FT (ˆxα))ϕ ˆα(ˆx)| det JT | dxˆ ˆ T d α∈N0, kαk`∞ ≤k Z X X J = ϕˆα(ˆx)| det JT | dxˆ p(xα) = ωα p(xα), xα ∈ Q, ˆ d T d α∈N0, kαk`∞ ≤k α∈N0, kαk`∞ ≤k we immediately get the weights Z Z J k ωi = ϕî(ˆx)| det JT | dxˆ = ϕi(x) dx, 1 ≤ i ≤ ndof , Tˆ T corresponding to the nodes xi := FT (ˆxi), whereϕ î is the Lagrange basis function tox î. For reasons of numerical robustness of the quadrature rule the following condition has to be satisfied:

J k ωi > 0 for 1 ≤ i ≤ ndof . The Deﬁnition 5 of the nodal quadrature rule can be used to deﬁne a discrete inner product. The induced discrete norm allows to derive norm equivalence estimates w.r.t. the Lp-norm for discrete

10 arguments such that the equivalence constants depend only on p, the polynomial degree k and the distribution of qudrature nodes. To do so, we start with the deﬁnition of control volumina and associated lumping operators. The discrete L2-norm is given by

2 X J 2 kvk0,2,T,h := (v, v)0,T,h := ωα v(xα) d α∈N0, kαk`∞ ≤k k k X X = ··· ωJ ··· ωJ v(x )2, x ∈ Q, α1 αd α α α1=0 αd=0

P J d p with d ω = |T |, ωα > 0, ∀α ∈ . The generalization to the L -norm for p ∈ [1, ∞) α∈N0, kαk`∞ ≤k α N0 is straightforward. The control volumina are introduced as follows:

α1−1 α1 ! αd−1 αd ! X J X J X J X J Ωα = ωi , ωi × · · · × ωi , ωi , i=0 i=0 i=0 i=0

P−1 J where we use the convention i=0 ωi := 0. As a consequence, for the d-dimensional Jordan measure of Ωα we have that |Ω | = ωJ ωJ ··· ωJ = ωJ , (17) α α1 α2 αd α and X |Ωα| = |T |. (18) d α∈N0, kαk`∞ ≤k ∞ The lumping operator Lh : C(T ) → L (T ) is deﬁned by

nk Xdof Lhv = v(xi)χΩi , xi ∈ Q, i=1 where 0, x∈ / G, χ (x) = G 1, x ∈ G is the indicator function of the set G. Integrating the p-th power of the lumping operator, from (17), (18) together with the aﬃne transformation of the reference element we see that

nk Z dof Z p p X p kLh(v)k0,p,T = |T | |Lh(v)| dxˆ = |Ωj| |Lh(v)| dxˆ ˆ ˆ T j=1 T p nk nk nk dof Z dof dof X X X p = v(xi)χΩ (x) dx = |Ωj||v(xj)| (19) i j=1 Ωj i=0 j=1 p = kvk0,p,T,h, xi ∈ Q. Furthermore, nk Z dof Z p N =Q X p k p kvk0,p,T,h = |v(xj)| ϕj(x) dx = Ih (|v| ) dx. T j=1 T

11 LEMMA 7 There exist constants CL1 ,CL2 > 0 independent of hT such that the following equivalence estimates are valid:

CL1 kvk0,p,T ≤ kLh(v)k0,p,T ≤ CL2 kvk0,p,T ∀v ∈ Qk(T ), p ∈ [2, ∞). (20) Proof: First we mention that, as a consequence of the substitution rule, the constants do not depend on hT for all elements T which result from an aﬃne transformation of the reference element. k k p Pndof p p Pndof p Since kLh(v)k0,p,T = j=1 |Ωj||v(xj)| and kvk`p = j=1 |v(xj)| , we obtain, by Nikolski’s lemma (cf. Lemma 17) and H¨older’sinequality for sums,

d p−2 d p−2 2 2p 2 2p ˆ 1/2 kvˆk0,p,Tˆ ≤ (3k ) kvˆk0,2,Tˆ ≤ (3k ) λmax(M) kvˆk`2 p−2 2 d 1/2 ≤ (3k (k + 1)) 2p λmax(Mˆ ) kvˆk`p !−1/p p−2 2 d 1/2 ≤ (3k (k + 1)) 2p λmax(Mˆ ) min |Ωˆ j| kLˆh(ˆv)k ˆ k 0,p,T 1≤j≤ndof and !1/p !1/p ˆ ˆ ˆ kLh(ˆv)k ˆ ≤ max |Ωj| kvˆk`p ≤ max |Ωj| kvˆk`2 0,p,T k k 1≤j≤ndof 1≤j≤ndof !1/p −1/2 ≤ λmin(Mˆ ) max |Ωˆ j| kvˆk ˆ k 0,2,T 1≤j≤ndof !1/p −1/2 (p−2)/2p ≤ λmin(Mˆ ) |Tˆ| max |Ωˆ j| kvˆk ˆ, k 0,p,T 1≤j≤ndof where Λmin(Mˆ ), Λmax(Mˆ ) denote the minimal and maximal eigenvalues, resp., of the corresponding ˆ R mass matrix with entries Mij := Tˆ ϕîϕˆj dx.ˆ J In the approximation of the L2-projection by means of quadrature rules, the choice of the positions of the quadrature nodes plays an essential role. On the other hand, there is also some freedom in the choice of the node set N in the case of k Lagrange basis polynomialsϕ î for 0 ≤ i ≤ k. In the case of coinciding node sets N and Q we see that the definitions of the L2-projection and of the Lagrange basis polynomials result in

k ndof ! k X J n k o 0 = (v − Ph v, w)0,T,h = ωi v(xi) − (Ph v)(xi) w(xi) ∀w ∈ Qk(T ) , i=1 k that is v(xi) = (Ph v)(xi) for all quadrature nodes. k k Pndof k 2 k Thus the representation (Ph v)(x) = j=1 (Ph v)jϕj(x) of the L -projection leads to (Ph v)i = 2 v(xi). Consequently, for the above approximation of the L -projection, we have the implication k k N = Q ⇒ Ph = Ih . The best accuracy can be reached by using the Gauss quadrature rules in the following sense. A (k + 1)d-node Gauss quadrature rule yields exact results for polynomials of maximum degree 2k + 1 (see, e.g., [EG04, Prop. 8.2], [BM97, Sect. 13,14] or [CHQZ07, pp. 448]). The quadrature points ˆk+1 w.r.t. the domain of integration I = [−1, 1] are the zeros of the Legendre polynomial ψk+1(ˆx), xˆ ∈ I, of degree k + 1. However, the use of Gauss quadrature rules does not lead to optimal estimates of the quadrature error w.r.t. the W l,2-norm, as the following result indicates. LEMMA 8 For all real numbers r and l, r ≤ l, l ≥ 1, there exists a constant C > 0 independent of k such that ˆk −e(r,l) l,2 kvˆ − IGvˆkr,2,I ≤ Ck kvˆkl,2,I ∀vˆ ∈ W (I), where e(r, l) is the function introduced in Lemma 5.

12 Proof: See [BM97, (13.15)]. J Alternatively, Gauss-Lobatto quadrature rules can be considered. Here, the quadrature points are the zeros of the polynomial ˆk 0 (x + 1)(x − 1)(ψk ) (x), x ∈ I. This quadrature rule is exact for polynomials of maximum degree 2k − 1 (see, e.g., [CHQZ07, pp. 448]). The boundary points are quadrature nodes. In contrast to the previous lemma, the following result is valid. LEMMA 9 For all real numbers r and l such that 2l > d+r and 0 ≤ r ≤ 1, there exists a constant C > 0 independent of k such that ˆk r−l l,2 ˆ kvˆ − IGLvˆkr,2,Tˆ ≤ Ck kvˆkl,2,Tˆ ∀vˆ ∈ W (T ). (21)

Proof: See [BM97, Thm. 14.2]. J In the application to quadrature, we have the following result (cf. Lemma 7). LEMMA 10 Let Q be the set of Gauss-Lobatto quadrature nodes. Then there exists a constant C > 0 independent of hT and k such that the following equivalence estimates are valid:

kvk0,2,T ≤ kLh(v)k0,2,T ≤ Ckvk0,2,T ∀v ∈ Qk(T ). (22) Proof: First we prove the statement for the reference element. On T,ˆ we have

kL (ˆv)k2 = kvˆk2 h 0,2,Tˆ 0,2,Tˆ ,GL by (19); the remaining part is a consequence of [CQ82, (3.9)]. The aﬃne back-transformation to the original element shows that the constants are independent of hT . J REMARK 3 Analogously to Lemma 7 we can conclude that the above inequalities (22) can be extended to the general case p ∈ [2, ∞). A further interesting property of Gauss-Lobatto quadrature nodes is related with the square sum of Lagrange polynomials.

LEMMA 11 Let ϕˆα(ˆx) be the Lagrange polynomials w.r.t. the Gauss-Lobatto quadrature nodes. Then we have the estimate X 2 ϕˆα(ˆx) ≤ 1 ∀xˆ ∈ T.ˆ d α∈N0, kαk`∞ ≤k Proof: The proof is a consequence of the tensor product representation together with [Fej32, § 2]:

k k X X X ϕˆ (ˆx)2 = ϕˆk (ˆx )2 ··· ϕˆk (ˆx )2 ≤ 1. α α1 1 αd d J d α =0 α =0 α∈N0, 1 d kαk`∞ ≤k | {z } | {z } ≤1 ≤1

5 Projection and interpolation errors w.r.t. Gauss-Lobatto quadrature nodes

The Legendre polynomials used in the representation of the discrete L2-projector possess two important properties: On the one hand, they form an orthogonal basis of Qk(T ), and, consequently, the corresponding mass matrix is diagonal. On the other hand, there exists a hierarchical decom- position of Qk(T ) in the following sense (cf. [EG04, Def. 1.18]).

13 DEFINITION 6 (Hierarchical modal basis) A family {Bk}k∈N0 , where Bk denotes a set of polynomials, is called hierarchical modal basis if, for all k ∈ N0, the following properties are satisﬁed:

1. Bk is a basis of Qk,

2. Bk ⊂ Bk+1. If the L2-projection is discretized by means of Lagrange polynomials for the node sets N = Q, then the discrete L2-orthogonality is conserved but, unfortunately, the corresponding Lagrange basis is not a hierarchical modal basis. This problem can be resolved by introducing the following notion. By N : B → N we denote that bijective function which assigns a set of polynomials B to the node set of the corresponding Lagrange basis.

DEFINITION 7 (embedded hierarchical nodal basis) A family {Bj} k , where Bj de- 1≤j≤ndof notes a set of polynomials of maximum degree k, is called embedded hierarchical nodal basis of degree K,K ∈ N0, if the following properties are satisﬁed: ˜ ˜ 1. BK with N(BK ) ⊂ N(Bk) is a basis of QK ,

2. BK ⊂ Bk,N(BK ) = N(B˜K ),

3. Bk is a basis of Qk.

EXAMPLE 1 (d = 1) (i) The Lagrange polynomials w.r.t. the Gauss-Lobatto quadrature nodes form an embedded hierarchical nodal basis of degree 1 and 2. This follows from the fact that N(B˜1) := {−1, 1} ⊂ N(Bk), ˜ k ∈ N, and N(B2) := {−1, 0, 1} ⊂ N(Bk), k = 2, 4, 6,... k (ii) The Lagrange polynomials w.r.t. the Gauss-Kronrod quadrature nodes {xi}i=1 =: N(Bk) form an embedded hierarchical nodal basis of degree K for k = 2K + 2,K ∈ N0. Given K + 1 Gauss K ˜ quadrature nodes {x˜i}i=0 =: N(BK ), Gauss-Kronrod quadrature rules are deﬁned by adding K + 2 nodes such that

N(B˜K ) ⊂ N(Bk), k Z X v dx = ωiv(xi) ∀v ∈ Q3K+4(I) I i=0 (see, e.g., [CGGR00]).

As an application we investigate some properties of the so-called ﬂuctuation operator, see the deﬁnition below. These results are important in the numerical analysis of discontinuous Galerkin methods for conservation laws.

DEFINITION 8 Let {Bj} k be an embedded hierarchical nodal basis of degree K,K ∈ 0. 1≤j≤ndof N K,k 0 K,k 0 For Qk(T ) = Vh (T ) ⊕ Vh(T ), where Vh (T ) := span BK and Vh(T ) := span{Bk \BK }, the K,k 2 K,k projector Ph : L (T ) → Vh (T ) is deﬁned by

K,k K,k (v − Ph v, w)0,T,h = 0 ∀w ∈ Vh (T ). (23)

0 K,k 2 2 The operator Ph(T ) := Id − Ph in L (T ), where Id denotes the identity in L (T ), is called ﬂuctuation operator.

14 Figure 1: Lagrange basis polynomials of Q4(I) for the Gauss-Lobatto nodes (left), corresponding 2,4 basis polynomials of Vh (I) (right)

LEMMA 12 Let {Bj} k be an embedded hierarchical nodal basis of degree K,K ∈ 0, 1≤j≤ndof N consisting of Lagrange polynomials and suppose N = Q. Then there is a constant C > 0 independent of hT such that

K,k l−r K |v − Ph v|r,2,T ≤ ChT |v|l,2,T + |Ph v|l,2,T , 0 ≤ r ≤ l, K,k l−1/2−r K |v − Ph v|r,2,E ≤ ChT |v|l,2,T + |Ph v|l,2,T , 0 ≤ r < l − 1/2 for all v ∈ W l,2(T ).

K,k Proof: Under the above assumptions we have, for all Lagrange basis polynomials ϕj ∈ Bk, that K,k K ˜ K,k ϕj (xi) = δij, 1 ≤ i, j ≤ ndof , xi ∈ N(BK ) = N(BK ). The deﬁnition of the projector Ph and the property K,k w ∈ Vh (T ) =⇒ w(xi) = 0, xi ∈ N(Bk) \ N(BK ) (24) imply that

k ndof ! K,k X n K,k o 0 = (v − Ph v, w)0,T,h = ωi v(xi) − (Ph v)(xi) w(xi) i=1 K ndof (24) X n K,k o = ωi v(xi) − (Ph v)(xi) w(xi) i=1 and, thus, K ndof ! X K,k K,k N =Q K,k K v(xi) = (Ph v)jϕj (xi) = (Ph v)i, 1 ≤ i ≤ ndof . (25) j=1 K As a consequence, if Ih denotes the Lagrange interpolation operator w.r.t. the nodes N(BK ), we obtain the relation

K K K ndof ndof ndof K K,k X K,k K X K,k K,k K X K,k K (25) K Ih (Ph v) = (Ph v)(xi)ϕi = (Ph v)jϕj (xi)ϕi = (Ph v)iϕi = Ih (v) . i=1 i,j=1 i=1

15 It can be used, together with Lemma 3, in the estimation of the projection error as follows:

LEMMA 13 Let {Bj} k be an embedded hierarchical nodal basis of degree K,K ∈ 0, 1≤j≤ndof N consisting of Lagrange polynomials and suppose N = Q. Then: K,k (i) Ph is a linear operator, K,k K,k (ii) kPh vk ≤ C(K, k, k · k)kvk0,∞,T for an arbitrary norm k · k on Vh (T ).

(iii) For m ∈ N, v ∈ Qk(T ), the following estimate is valid: 0 0 2m−1 (Ph(∇v),Ph(∇v ))0,T,h ≥ 0.

Proof: The assertion (i) easily follows from (25):

K ndof K,k X K,k K,k K,k Ph (αv + βw) = [αv(xi) + βw(xi)] ϕi = αPh v + βPh w . i=1 Item (ii) results from the estimates

K K ndof ndof K,k X K,k X K,k kPh vk ≤ kv(xi)ϕi k ≤ |v(xi)|kϕi k i=1 i=1  K  ndof X K,k ≤  kϕi k kvk0,∞,T ≤ C(K, k, k · k)kvk0,∞,T . i=1

To verify (iii), we mention the following elementary inequality on T (p := 2m): 4(p − 1) ∇v · ∇vp−1 = ∇vp/2 · ∇vp/2 ≥ 0 . p2 Then

0 0 p−1 (23) p−1 K,k K,k p−1 (Ph(∇v),Ph(∇v ))0,T,h = (∇v, ∇v )0,T,h − (Ph (∇v),Ph (∇v ))0,T,h k K ndof ndof (25) X J p−1 X J p−1 = ωi ∇v(xi) · ∇v (xi) − ωi ∇v(xi) · ∇v (xi) i=1 i=1 k ndof X J p−1 = ωi ∇v(xi) · ∇v (xi) K i=ndof +1 nk 4(p − 1) Xdof = ωJ ∇vp/2(x ) · ∇vp/2(x ) ≥ 0 . p2 i i i K i=ndof +1

0 K,k K,k K,k 2m−1 Interchanging the roles of Ph and Ph , we obtain analogously (Ph (∇v),Ph (∇v ))0,T,h ≥ 0. J Motivated by Lemma 12 and the error estimate (21) we investigate the local interpolation error of the Gauss-Lobatto interpolation operator.

16 LEMMA 14 Let T be an element of an aﬃne partition Th such that the corresponding family of partitions {Th}h>0 is locally quasiuniform. Assume that 1 ≤ l ≤ k + 1, l ∈ N, 0 ≤ r ≤ 1 k and 2l > d + r. Then, for the Lagrange interpolation operator IGL, there exist constants C > 0 independent of hT and k such that

h l−r |v − Ik v| ≤ C T |v| , (26) GL r,2,T k l,2,T h l−1/2−r |v − Ik v| ≤ C T |v| (27) GL r,2,E k l,2,T for all v ∈ W l,2(T ).

Proof: Using Lemma 9, the proof of (26) runs analogously to the proofs of Lemmata 3 and 5. To prove (27), we make use of the fact that, for tensor product elements, the restriction of the Gauss-Lobatto interpolant to a face E of T is a (d − 1)-dimensional Gauss-Lobatto interpolant. From Lemma 9 we see that ˆk r−˜l |vˆ − IGLvˆ|r,2,Eˆ ≤ Ck kvˆkl,2,Eˆ for allv ˆ ∈ W ˜l,2(Eˆ) and all real numbers r and ˜l such that 0 ≤ r ≤ 1, 2˜l > d − 1 + r. The trace theorem (cf. (16) in the proof of Lemma 5) leads to

kvˆk˜l,2,Eˆ ≤ Ckvˆk˜l+1/2,2,Tˆ.

If in addition l = ˜l + 1/2 ∈ N, the Deny-Lions lemma ([Cia91, Thm. 14.1]) implies that ˆk r−l+1/2 |vˆ − IGLvˆ|r,2,Eˆ ≤ Ck |vˆ|l,2,Tˆ.

Turning back to the original variables, we obtain (27). J COROLLARY 3 Let 1 ≤ l ≤ K + 1. Under the assumptions of Lemmata 12 and 14, there exist constants C > 0 independent of hT and k such that

h l−r |v − P K,kv| ≤ C T |v| + |P K v| , GL r,2,T K l,2,T GL l,2,T h l−1/2−r |v − P K,kv| ≤ C T |v| + |P K v| GL r,2,E K l,2,T GL l,2,T for all v ∈ W l,2(T ).

REMARK 4 As in the proof of Lemma 12, the error estimate will be reduced to an estimate of a suitable seminorm.

k In general, the operators ∇ and IGLdo not commute. An estimate of the commutation error is given in the next result.

LEMMA 15 Under the assumptions of Lemma 14, there exists a constant C > 0 independent of hT and k such that h l−1 kIk (∇v) − ∇Ik vk ≤ C T |v| . GL GL 0,2,T,GL k l,2,T

17 Proof:

(20) k k k k kIGL(∇v) − ∇IGLvk0,2,T,GL ≤ CkIGL(∇v) − ∇IGLvk0,2,T k k ≤ Ck∇(v − IGLv)k0,2,T + Ck∇v − IGL(∇v)k0,2,T (26) h l−1 h l−1 ≤ C T |v| + C T |∇v| k l,2,T k l−1,2,T h l−1 ≤ C T |v| . k l,2,T J

6 Inverse inequalities

Inverse inequalities are an important tool in the analysis of ﬁnite element methods. Using particular properties if the tensor product representation, in this section we derive sharp inverse inequalities such that, with exception of the diﬀerence of the derivative orders and the spatial dimension, all other information is known explicitly. It is well-known that in the Nikolski inequality the dependence of the degree of polynomials cannot be improved, as the example

1 − T 2(x)2 p(x) = n (28) 1 − x2 with Tn being the n-th Cebyˇsovˇ polynomial of the ﬁrst kind demonstrates, see [Tim63, p. 263] (although in other papers, e.g. [GNP08, Lemma 2.4] or [CB93, Lemma 1], diﬀerent statements can be found). The representation 1 h p p i T (x) = (x + x2 − 1)n + (x − x2 − 1)n , n 2 2 shows that 1 − Tn (x) has the zeros −1 and 1, thus (28) is really a polynomial. Now we present a generalization of Nikolski and Markov inequalities (see [Nik51], [HST37, Sect. III]) to the tensor product situation, and we include it into the proof of an inverse inequality given in [EG04, Lemma 1.138]. Throughout this section we restrict ourselves to the case k ≥ 1.

LEMMA 16 (local inverse inequality) Given a reference element {T,ˆ P,ˆ Σˆ} such that, for l ≥ ˆ l,∞ ˆ 0, the embedding P ⊂ W (T ) is satisﬁed. Let {Th}h∈(0,1] be a locally quasiuniform family of aﬃne partitions. If 0 ≤ m ≤ l, then there exist

1. for 1 ≤ p, q ≤ ∞ a constant C = C(l, m, p, q, d, σ0, T,Pˆ (Tˆ)) > 0 such that

1 1 m−l+d( p − q ) kvkl,p,T ≤ ChT kvkm,q,T ∀v ∈ P (T ), (29) ˆ ˆ 2. for 1 ≤ q ≤ p ≤ ∞ and P := Qk(T ) a constant C = C(l, m, p, d, σ0) > 0 such that

m−l d( 1 − 1 ) h h p q kvk ≤ C T T kvk ∀v ∈ P (T ). (30) l,p,T k2 2(q + 1)k2 m,q,T

The proof will be given after the presentation of the above mentioned Nikolski and Markov inequalities. ˆ LEMMA 17 (Nikolski) For 0 < q ≤ p ≤ ∞ and vˆ ∈ Qk(T ), the following estimate is valid:

2−d(1/p−1/q) kvˆk0,p,Tˆ ≤ (q + 1)k kvˆk0,q,Tˆ. (31)

18 Proof: Due to [DL93, Thm. 4.2.6], in the one-dimensional situation we have the following inequality: 2−(1/p−1/q) kvˆk0,p,I ≤ (q + 1)k kvˆk0,q,I ∀vˆ ∈ Qk(I), where, as above, I = [−1, 1]. Then, for 1 ≤ i ≤ d, we also have that

kvˆ(ˆx1,..., xî−1, ·, xî+1,..., xˆd)k0,∞,I 21/q ≤ (q + 1)k kvˆ(ˆx1,..., xî−1, ·, xî+1,..., xˆd)k0,q,I .

A successive application of this inequality leads to

It remains to make use of an Lp-type interpolation inequality (see, e.g., [EG04, Cor. B.7]):

q 1− q kvˆk ≤ kvˆk p kvˆk p 0,p,Tˆ 0,q,Tˆ 0,∞,Tˆ q q d (1− q ) 1− ≤ kvˆk p (q + 1)k2 q p kvˆk p 0,q,Tˆ 0,q,Tˆ 1 1 2−d( p − q ) ≤ (q + 1)k kvˆk0,q,Tˆ, 0 < q ≤ p ≤ ∞. J

LEMMA 18 (generalized Markov inequality) For v ∈ Qk(I) and p ≥ 1, the following estimate is valid: 0 2 kv k0,p,I ≤ CM (p)k kvk0,p,I , where 1 p k−1+1/p C = C (p) := 2(p − 1)1/p−1 p + 1 + . M,p M k kp − p + 1

For p = ∞, we even have that CM,∞ = 1 < lim CM (p) = 2e. Furthermore, ∀p ∈ : CM (p) ≤ p→∞ N 1+1/e CM := 6e .

Proof: For 1 < p < ∞, the assertion is proved in [HST37, Sect. III]. The estimate CM (p) ≤ CM = 1+1/e 6e for all p ∈ N can be found in [MMR94, p. 590]. For p = ∞, the result follows from [DL93, Thm. 4.1.4]. J

REMARK 5 The constants CM (p) can be improved. For instance, in [Bar98, Cor. 2.10] the existence of a constant C˜M (p) such that lim C˜M (p) = 1, p > 2 is demonstrated. p→∞

ˆ COROLLARY 4 For 1 ≤ p ≤ ∞ and vˆ ∈ Qk(T ), the following estimate is valid:

1 d + l p l kvˆk ≤ C k2 kvˆk (32) l,p,Tˆ l M,p 0,p,Tˆ

(with the constant CM,p as in Lemma 18).

19 Proof: Iterating over the order of derivatives, it is suﬃcient to verify the estimate

α 2 |α| k∂ vˆk0,p,Tˆ ≤ (CM,pk ) kvˆk0,p,Tˆ for |α| = 1. To do so, set α := ei, where ei denotes the i-th coordinate unit vector. For p < ∞ and 1 ≤ i ≤ d the generalized Markov inequality implies that

0 p kvˆ (ˆx1,..., xî−1, ·, xî+1,..., xˆd)k0,p,I p 2p p ≤ CM,pk kvˆ(ˆx1,..., xî−1, ·, xî+1,..., xˆd)k0,p,I , hence, on T,ˆ k∂αvˆkp = R ... R |∂ vˆ(ˆx)|p dxˆ dxˆ . . . dxˆ dxˆ . . . dxˆ 0,p,Tˆ I I i i 1 i−1 i+1 d ≤ Cp k2pkvˆkp . M,p 0,p,Tˆ For p = ∞, there is a pointy ˆ ∈ Tˆ such that

α k∂ vˆk0,∞,Tˆ = |∂ivˆ(ˆy)| = max |∂ivˆ(ˆy1,..., yî−1, xî, yî+1,..., yˆd)| xî∈I 2 ≤ CM,∞k kvˆ(ˆy1,..., yî−1, ·, yî+1,..., yˆd)k0,∞,I 2 ≤ CM,∞k kvˆk0,∞,Tˆ. Consequently,

kvˆkp = Pl P k∂αvˆkp ≤ Pl P C k2|α|p kvˆkp l,p,Tˆ i=0 |α|=i 0,p,Tˆ i=0 |α|=i M,p 0,p,Tˆ = Pl d+i−1 C k2ip kvˆkp i=0 i M,p 0,p,Tˆ ≤ d+l C k2lp kvˆkp , l M,p 0,p,Tˆ and α 2|α| kvˆkl,∞,Tˆ = max k∂ vˆk0,∞,Tˆ ≤ max CM,∞k kvˆk0,∞,Tˆ 0≤|α|≤l 0≤|α|≤l 2l ≤ CM,∞k kvˆk0,∞,Tˆ. J Now we are ready to prove the inverse estimates (29) and (30). Proof of Lemma 16: The ﬁrst estimate is proved along the lines of the proof of [EG04, Lemma 1.138]. In the particular case of a Lagrange reference element we use the same idea of proof but make use of the results prepared in this section. As usual, ﬁrst the assertion is proved for m = 0. By (32) and (31), on Tˆ we have that

1 d + l p l −d( 1 − 1 ) kvˆk ≤ C k2 (q + 1)k2 p q kvˆk . (33) l,p,Tˆ l M,p 0,q,Tˆ

In order to get the corresponding estimate for the transformed element T, we make use of (4), (5) and (6) for 0 ≤ j ≤ l : n op |v|p ≤ C kJ −1kj | det J |1/p |vˆ|p j,p,T j,d T `2 T j,p,Tˆ p √ j d ≤ C 2 dσ h−1 hT p kvˆkp j,d 0 T 2 j,p,Tˆ d p (33) 1 −j 1 1 p −d( − ) ≤ d+j p C˜(j, p, d, σ ) hT hT (q + 1)k2 p q kvˆkp , j 0 k2 2 0,q,Tˆ

20 √ j where C˜(j, p, d, σ0) = Cj,d 2 dσ0CM,p . By (3) with l = 0, the back-transformation yields

1 −j d( 1 − 1 ) d + j p h h p q |v| ≤ C˜(j, p, d, σ ) T T kvk . j,p,T j 0 k2 2(q + 1)k2 0,q,T

−1 Under the assumption hT ≥ 1 for 0 ≤ j ≤ l, the corresponding Sobolev norm can be estimated as p Pj p kvkj,p,T = s=0 |v|s,p,T −s d( 1 − 1 )p ≤ Pj d+s C˜(s, p, d, σ ) hT hT p q kvkp s=0 s 0 k2 2(q+1)k2 0,q,T (34) −j d( 1 − 1 )p d+1+j ˜ hT hT p q p ≤ j C(j, p, d, σ0) k2 2(q+1)k2 kvk0,q,T , and the assertion is proved for m = 0. Now, let 0 ≤ m ≤ l and α be a multiindex with 0 ≤ |α| ≤ l. In the case |α| ≤ l − m, the estimate (34) and the relation

α p X α p p k∂ vk0,p,T ≤ k∂ vk0,p,T = kvkl−m,p,T |α|≤l−m imply that

m−l d( 1 − 1 ) α hT hT p q k∂ vk0,p,T ≤ C(l − m, p, d, σ0) k2 2(q+1)k2 kvk0,q,T m−l d( 1 − 1 ) hT hT p q ≤ C(l − m, p, d, σ0) k2 2(q+1)k2 kvkm,q,T ,

d+1+j1/p ˜ where C(j, p, d, σ0) := j C(j, p, d, σ0). This inequality is also valid for l − m ≤ |α| ≤ l, because there exist two further multiindices β and γ such that α = β + γ, |β| = l − m and |γ| ≤ m. As in the ﬁrst case we get

α β γ k∂ vk0,p,T = k∂ (∂ v) k0,p,T m−l d( 1 − 1 ) hT hT p q γ ≤ C(l − m, p, d, σ0) k2 2(q+1)k2 k∂ vk0,q,T m−l d( 1 − 1 ) hT hT p q ≤ C(l − m, p, d, σ0) k2 2(q+1)k2 kvkm,q,T .

p Since this estimate is valid for 0 ≤ |α| ≤ l, the norm kvkl,p,T can be estimated by using its deﬁnition, and the statement follows with

√ l−m d + 1 + l − m1/pd + l1/p C(l, m, p, d, σ ) := C 2 dσ C . 0 l−m,d 0 M,p l − m l

The equality sign occurs in the case l = m = 0 and p = q. J

LEMMA 19 (global inverse inequality) Under the assumptions of Lemma 16, for a locally quasiuniform family of aﬃne partitions {Th}h∈(0,1] and for all v ∈ Wh, there exist

1. for 1 ≤ p, q ≤ ∞ a constant C = C(l, m, p, q, d, σ0, T,Pˆ (Tˆ),Cqu) > 0 such that

1 1 p 1 1 q X p m−l+min(0,d( p − q )) X q kvkl,p,T ≤ Ch kvkm,q,T , T ∈Th T ∈Th

21 ˆ ˆ 2. for 1 ≤ q ≤ p ≤ ∞ and P = Qk(T ) a constant C = C(l, m, p, d, σ0) > 0

1 1 1 1 m−l d( p − q ) X p Cquh Cquh X q kvkp ≤ C kvkq . l,p,T k2 2(q + 1)k2 m,q,T T ∈Th T ∈Th

Proof: The proof of the ﬁrst estimate can be found in [EG04, Cor. 1.141]. With

C = C(l, m, p, d, σ0), the second estimate follows from Lemma 16 and from the quasiuniformity of {Th}h>0:

p(m−l) pd( 1 − 1 ) X Cquh Cquh p q X kvkp ≤ Cp kvkp . l,p,T k2 2(q + 1)k2 m,q,T T ∈Th T ∈Th

Taking the p-th root, together with k · k`p ≤ k · k`q the statement follows for p, q 6= ∞. The proof in the case p = q = ∞ immediately follows from the deﬁnitions of the norms. J

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