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Subject Mathematics prtradpoeotmlcnegneo h ehdudrth findings. underline under analytical experiments method Numerical our regularity. the elliptic of injection of convergence stable assumption a optimal construct prove we and particular, In operator Poisson’s for levels. scheme all discretization on HDG equation same the uses it that sense Abstract. 28 .I sbsclyatolvlmto,weete“orespace” “coarse the where method, two-level a basically is It ]. EPIL,ADESRP,ADGIOKANSCHAT GUIDO AND RUPP, ANDREAS LU, PEIPEI , 29 7 n nttoay[ instationary and ] .Hwvr nyfwrslshv enahee o solv- for achieved been have results few only However, ]. eitoueahmgnosmlirdmto nthe in method multigrid homogeneous a introduce We 1. 21 Introduction n oandcmoiin[ decomposition domain and ] 25 tkssses otelocking-free the to systems, Stokes ] 1 51,6N0 65N50. 65N30, 65F10, 19 ,a ela opae [ plates to as well as ], C28/ - 2181/1 XC 18 DG Ger- (DFG, n. solvers ] e 11 12 9 , , for ] ore 24 14 rid on ds st e t l ] , 2 PEIPEILU,ANDREASRUPP,ANDGUIDOKANSCHAT order elements. In [22], similar results have been applied to hybridized mixed (e.g. Raviart–Thomas (RT)) elements. A BPX preconditioner for non-standard finite element methods, in- cluding hybridized RT, BDM, the weak Galerkin, and Crouzeix–Raviart methods, is analyzed in [26] and a two level for HDG methods for the diffusion problem is presented in [27]. All of the aforementioned methods utilize the piecewise linear conforming finite element space as the auxiliary space. Two-level analysis of HDG methods seems equally rare. A method for high order HDG discretizations is introduced in [30]. Here, the au- thors focus on showing that standard p-version domain decomposition techniques can be applied and prove condition number estimates on tetrahedral meshes polylogarithmic in p. More recently, a domain de- composition for a hybridized Stokes problem was presented in [1], but it does not discuss a coarse space. Moreover, [23] discusses local Fourier analysis of interior penalty based multigrid methods for tensor-product type polynomials in two spatial dimensions. The methods known so far are heterogeneous in the sense that the multigrid cycle is not performed on the HDG discretization itself, but on a surrogate scheme. In view of future generalization beyond second order elliptic problems, we decided to devise a homogeneous method which uses the same discretization scheme on all levels, and thus only employs a single finite element method. The analysis uses the abstract framework developed for noninherited forms in [16]. It is based on arguments found in [20, 31]. Nevertheless, the coarse grid operator is genuinely HDG and of the same type as the fine grid operator. Only the injection operator from coarse to fine level uses continuous interpolation in an intermediate step. Since we focus on new coarse spaces and intergrid operators, we rely on standard smoothers and analyze a standard Poisson problem with elliptic regularity to present the basic ideas of our method. However, the regularity assumtion could be weakened utilizing the ideas of [4], of [5] (which analyzes nonconforming multigrid methods), of [21] (which deals with multigrid methods for DG), and of [22] at the cost of more technicalities within our proofs. As for robust smoothers for higher order methods, this will be subject to further research. The remainder of this paper is structured as follows: In Section 2, we review the HDG method for elliptic PDEs. Furthermore, an overview over the used function spaces, scalar products, and operators is given. Section 3 is devoted to the definition of the multigrid and states its main convergence result. Sections 4 and 5 verify the assumptions of the main convergence result, while Section 6 underlines its validity by numerical experiments. Short conclusions wrap up the paper. HMG — HOMOGENEOUS MULTIGRID FOR HDG 3

2. Model equation and discretization We consider the standard diffusion equation in mixed form defined on a polygonally bounded Lipschitz domain Ω ⊂ Rd with boundary ∂Ω. We assume homogeneous Dirichlet boundary conditions on ∂Ω. Thus, we approximate solutions (u, q) of ∇·q = f in Ω, (2.1a) q + ∇u = 0 in Ω, (2.1b) u =0 on ∂Ω, (2.1c) for a given function f. In the analysis, we will assume elliptic regularity, namely u ∈ H2(Ω) if f ∈ L2(Ω), such there is a constant c> 0 for which holds

|u|H2(Ω) ≤ ckfkL2(Ω). (2.2) Here, and in the following, L2(Ω) denotes the space of square integrable functions on Ω with inner product and norm

2 (u, v)0 := uv dx, and kuk0 := (u,u)0. (2.3) ZΩ The space Hk(Ω) is the Sobolev space of k-times weakly differentiable functions with derivatives in L2(Ω). We note that the assumption of homogeneous boundary data was introduced for simplicity of presen- tation and can be lifted by standard arguments.

2.1. Spaces for the HDG multigrid method. Starting out from a subdivision T0 of Ω into simplices, we construct a hierarchy of meshes Tℓ for ℓ = 1,...,L recursively by refinement, such that each cell of Tℓ−1 is the union of several cells of mesh Tℓ. We assume that the mesh is regular, such that each facet of a cell is either a facet of another cell or on the boundary. Furthermore, we assume that the hierarchy is shape regular and thus the cells are neither anisotropic nor otherwise distorted. We call ℓ the level of the quasi-uniform mesh Tℓ and denote by hℓ the characteristic length of its cells. We assume that refinement from one level to the next is not too fast, such that there is a constant cref > 0 with

hℓ ≥ crefhℓ−1. (2.4)

By Fℓ we denote the set of faces of Tℓ. The subset of faces on the boundary is

D Fℓ := {F ∈Fℓ : F ⊂ ∂Ω}. (2.5)

T Moreover, we define Fℓ := {F ∈ Fℓ : F ⊂ ∂T } as the set of faces of 2 a cell T ∈ Tℓ. On the set of faces, we define the space L (Fℓ) as the 4 PEIPEILU,ANDREASRUPP,ANDGUIDOKANSCHAT space of square integrable functions with the inner product

hhλ,µiiℓ = λµ dσ, (2.6) ∂T TX∈Tℓ Z 2 and its induced norm |||µ|||ℓ = hhµ,µiiℓ. Note that interior faces appear twice in this definition such that expressions like hhu,µiiℓ with possi- 1 2 bly discontinuous u|T ∈ H (T ) for all T ∈ Tℓ and µ ∈ L (Fℓ) are defined without further ado. Additionally, we define an inner product commensurate with the L2-inner product in the bulk domain, namely |T | ∼ hλ,µiℓ = λµ dσ = hF λµ dσ. (2.7) |∂T | ∂T F TX∈Tℓ Z FX∈Fℓ Z 2 Its induced norm is kµkℓ = hµ,µiℓ. Let p ≥ 1 and Pp be the space of (multivariate) polynomials of degree up to p. Then, we define the space of piecewise polynomials on the skeleton by λ ∈ P ∀F ∈F M := λ ∈ L2(F ) |F p ℓ . (2.8) ℓ ℓ λ =0 ∀F ∈F D  |F ℓ 

The HDG method involves a local solver on each mesh cell T ∈

Tℓ, producing cellwise approximations uT ∈ VT and qT ∈ W T of the functions u and q in equation (2.1), respectively. We choose VT = d Pp. Then, choosing W T = Pp yields the so called hybridizable local discontinuous Galerkin (LDG-H) scheme. Our current analysis is in fact limited to this case and other choices require a modification of Lemma 4.7. We will also use the concatenations of the spaces VT and W T , respectively, as a function space on Ω, namely 2 Vℓ := v ∈ L (Ω) v|T ∈ VT , ∀T ∈ Tℓ , (2.9) 2 Rd W ℓ := q ∈ L (Ω; ) q|T ∈ W T , ∀T ∈ Tℓ .

2.2. Hybridizable discontinuous for the dif- fusion equation. The HDG scheme for (2.1) on a mesh Tℓ consists of a local solver and a global coupling equation. The local solver is defined cellwise by a weak formulation of (2.1) in the discrete spaces VT × W T and defining suitable numerical traces and fluxes. Namely, given λ ∈ Mℓ find uT ∈ VT and qT ∈ W T , such that

qT · pT dx − uT ∇·pT dx = − λpT · ν dσ ZT ZT Z∂T (2.10a)

− qT ·∇vT dx + (qT · ν + τℓuT )vT dσ = τℓ λvT dσ (2.10b) ZT Z∂T Z∂T hold for all vT ∈ VT , and all pT ∈ W T , and for all T ∈ Tℓ. Here, ν is the outward unit normal with respect to T and τℓ > 0 is the penalty coefficient. While the local solvers are implemented cell by cell, it is HMG — HOMOGENEOUS MULTIGRID FOR HDG 5 helpful for the analysis to combine them by concatenation. Thus, the local solvers define a mapping

Mℓ → Vℓ × W ℓ (2.11) λ 7→ (Uℓλ, Qℓλ), where for each cell T ∈ Tℓ holds Uℓλ = uT and Qℓλ = qT . In the same 2 way, we define operators Uℓf and Qℓf for f ∈ L (Ω), where now the local solutions are defined by the system

qT · pT dx − uT ∇·pT dx = 0 (2.12a) ZT ZT

− qT ·∇vT dx + (qT · ν + τℓuT )vT dσ = fvT dx. (2.12b) ZT Z∂T ZT Once λ has been computed, the HDG approximation to (2.1) on mesh Tℓ will be computed as

uℓ = Uℓλ + Uℓf (2.13) qℓ = Qℓλ + Qℓf The global coupling condition is derived through a discontinuous Galerkin version of mass balance and reads: Find λ ∈ Mℓ, such that for all µ ∈ Mℓ

(qℓ · ν + τℓ(uℓ − λ)) µ dσ =0. (2.14) T F T ∈Tℓ D X F ∈FXℓ \Fℓ Z

In [12], it is shown that λ ∈ Mℓ is the solution of the coupled system from (2.10) to (2.14) if and only if it is the solution of

aℓ(λ,µ)= bℓ(µ) ∀µ ∈ Mℓ, (2.15a) with

aℓ(λ,µ)= QℓλQℓµ dx + τℓ(Uℓλ − λ)(Uℓµ − µ) dσ, (2.15b) Ω T ∈Tℓ ∂T Z X Z

bℓ(µ)= Uℓµf dx. (2.15c) ZΩ Furthermore, the bilinear form aℓ(λ,µ) is symmetric and positive defi- nite. Thus, it induces a norm

2 kµkaℓ = aℓ(µ,µ), (2.16)

We close this subsection by associating an operator Aℓ : Mℓ → Mℓ with the bilinear form aℓ(·, ·) by the relation

hAℓλ,µiℓ = aℓ(λ,µ) ∀µ ∈ Mℓ. (2.17) 6 PEIPEILU,ANDREASRUPP,ANDGUIDOKANSCHAT

2.3. The injection operator Iℓ. The difficulty of devising an “in- jection operator” Iℓ : Mℓ−1 → Mℓ originates from the fact that the finer mesh has edges which are not refinements of the edges of the coarse mesh. In [31], several possible injection operators are discussed, but turn out to be unstable. In order to assign reasonable values to these edges, we construct the injection operator in three steps. First, introduce the continuous finite element space c 1 Vℓ := u ∈ H0 (Ω) u|T ∈ Pp(T ) ∀T ∈ Tℓ . (2.18) We assume that the shape function basis on each mesh cell T is defined through a Lagrange interpolation condition with respect to support points x located on vertices, edges, and in the interior of the cell. c Thus, a function in Vℓ is uniquely determined by the values in these support points. We now define the continuous extension operator c c Uℓ : Mℓ → Vℓ , (2.19) by the interpolation conditions λ(x) if x is on the boundary of a face, c [Uℓ λ](x)= λ(x) if x is in the interior of a face, (2.20) [Uℓλ](x) if x is in the interior of a cell.

Here, λ is the arithmetic mean of the values attained by λ on different faces meeting in x. c The spaces Vℓ are nested, such that there is a natural injection operator Ic : V c → V c ℓ ℓ−1 ℓ (2.21) u 7→ u. c On Vℓ the trace on edges is well defined, such that we can write c γℓ : V → Mℓ ℓ (2.22) u 7→ γℓu.

Using these, we define the injection operator Iℓ as the concatenation of extension, natural injection, and trace, namely

Iℓ : Mℓ−1 → Mℓ c c (2.23) λ 7→ γℓIℓ Uℓ−1λ.

By its definition, Iℓ is the operator such that this diagram commutes: c Mℓ Vℓ γℓ

c Iℓ Iℓ

U c ℓ−1 c Mℓ−1 Vℓ−1 HMG — HOMOGENEOUS MULTIGRID FOR HDG 7

−1 Moreover, if τℓ ∼ hℓ , the following Lemma can be proved similarly to [8, Thms. 3.6 and 3.8]:

Lemma 2.1 (Boundedness). The injection operator Iℓ is bounded in the sense that

aℓ(Iℓλ,Iℓλ) . aℓ−1(λ,λ) ∀λ ∈ Mℓ−1. (2.24) 2.4. Operators for the multigrid method and analysis. After the discrete operator Aℓ and the injection operator Iℓ have been de- fined, we introduce the remaining operators here. First, there are two 2 operators from Mℓ to Mℓ−1, which replace the L projection and the Ritz projection of conforming methods, respectively. They are Πℓ−1 and Pℓ−1 defined by the conditions

Πℓ−1 : Mℓ → Mℓ−1, hΠℓ−1λ,µiℓ−1 = hλ,Iℓµiℓ ∀µ ∈ Mℓ−1. (2.25)

Pℓ−1 : Mℓ → Mℓ−1, aℓ−1(Pℓ−1λ,µ)= aℓ(λ,Iℓµ) ∀µ ∈ Mℓ−1, (2.26)

The operator Πℓ−1 is used in the implementation, while Pℓ−1 is key to the analysis. The multigrid operator for preconditioning Aℓ will be defined in Sec- tion 3.1. It will be referred to as

Bℓ : Mℓ → Mℓ. (2.27) It relies on a smoother

Rℓ : Mℓ → Mℓ, (2.28) which can be defined in terms of Jacobi or Gauss-Seidel iterations, † respectively. Denote by Rℓ the adjoint operator of Rℓ with respect to i h·, ·iℓ and define Rℓ by

i Rℓ if i is odd, Rℓ = † (2.29) (Rℓ if i is even. At this point, we have defined HDG versions of all operators involved in standard multigrid analysis. Additionally, we define the averaging operator avg c Iℓ : Vℓ → Vℓ . (2.30) Analog to (2.20), it is defined by interpolation in the support points x c of the shape functions of the space Vℓ , namely avg [Iℓ u](x)= u(x). (2.31) Here u is the arithmetic mean of the values u(x) from all mesh cells avg meeting at x. For x ∈ ∂Ω, we let [Iℓ u](x) = 0. A summary of the different operators connecting the spaces can be found in Figure 1. 8 PEIPEILU,ANDREASRUPP,ANDGUIDOKANSCHAT

3. Multigrid method and main convergence result We consider a standard (symmetric) V-cycle multigrid method for (2.15) (cf. [4, 16]) for which we conduct convergence analysis (as done in [16]). Thus, Section 3.1 is devoted to illustrating the multigrid method (cf. [16] where the algorithm is also taken from), while Section 3.2 states the main convergence result. Let us begin citing an estimate for eigenvalues and condition numbers of the matrices Aℓ.

Lemma 3.1. Suppose that Tℓ is quasiuniform. Then, there are positive constants C1 and C2 independent of ℓ such that 2 −2 2 C1kλkℓ ≤ aℓ(λ,λ) ≤ βℓC2hℓ kλkℓ , ∀λ ∈ Mℓ, (3.1) 2 where βℓ := 1 + (τℓhℓ) . Proof. This is a Corollary of Theorem 3.2 in [10].  This implies that for the stiffness matrix, we can bound the condition number κℓ by −2 κℓ . βℓhℓ (3.2) −1 which implies that for all choices of τℓ satisfying τℓ . hℓ the condition −2 number grows like hℓ . Here and in the following, . has the meaning of smaller than or equal to up to a constant only dependent on the regularity constant of the mesh family and cref.

Aℓ, Bℓ, † Rℓ, Rℓ c Mℓ Vℓ Vℓ Uℓ

γℓ

c Pℓ−1, Πℓ−1 Iℓ Iℓ

c Uℓ−1

Iavg Uℓ−1 ℓ−1 c Mℓ−1 Vℓ−1 Vℓ−1

Figure 1. Sketch of the different operators connecting function spaces of refinement levels ℓ and ℓ−1. Here, the operators needed to implement the multigrid precondi- tioner Bℓ are depicted red, while operators only appear- ing in the analysis are in blue and spaces are black. The c avg dashed arrows commute, while in general Uℓ−1 =6 Iℓ−1 ◦ Uℓ−1. HMG — HOMOGENEOUS MULTIGRID FOR HDG 9

3.1. Multigrid algorithm. Let m ∈ N \{0} be the number of fine- level smoothing steps. We recursively define the multigrid operator of the refinement level ℓ Bℓ : Mℓ → Mℓ, (3.3) −1 0 by the following steps. Let B0 = A0 . For ℓ > 0, let x = 0 ∈ Mℓ. Then for µ ∈ Mℓ, i (1) Define x ∈ Mℓ for i =1,...,m by i i−1 i i−1 x = x + Rℓ(µ − Aℓx ). (3.4) 0 m (2) Set y = x + Iℓq, where q ∈ Mℓ−1 is defined as m q = Bℓ−1Πℓ−1(µ − Aℓx ). (3.5) i (3) Define y ∈ Mℓ for i =1,...,m as i i−1 i+m i−1 y = y + Rℓ (µ − Aℓy ). (3.6) m (4) Let Bℓµ = y . 3.2. Main convergence result. The analysis of the multigrid method is based on the framework introduced in [16]. There, convergence is A traced back to three assumptions. Let λℓ be the largest eigenvalue of Aℓ, and † −1 Kℓ := 1 − (1 − RℓAℓ)(1 − RℓAℓ) Aℓ . (3.7) Then, there exists constants C1,C2,C3 > 0 independent of the mesh level ℓ, such that there holds • Regularity approximation assumption: 2 kAℓλkℓ |aℓ(λ − IℓPℓ−1λ,λ)| ≤ C1 A ∀λ ∈ Mℓ. (A1) λℓ

• Stability of the “Ritz quasi-projection” Pℓ−1 and injection Iℓ :

kλ − IℓPℓ−1λkaℓ ≤ C2kλkaℓ ∀λ ∈ Mℓ. (A2) • Smoothing hypothesis: 2 kλkℓ A ≤ C3hKℓλ,λiℓ. (A3) λℓ Theorem 3.1 in [16] reads Theorem 3.2. Assume that (A1), (A2), and (A3) hold. Then for all ℓ ≥ 0, |aℓ(λ − BℓAℓλ,λ)| ≤ δaℓ(λ,λ), (3.8) where C C δ = 1 3 (3.9) m − C1C3 with the number of smoothing steps m> 2C1C3. Thus, in order to prove uniform convergence of the multigrid method, we will now set out to verify these assumptions. 10 PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT

4. Proof of (A1) The main statement of this section is ∼ −1 Theorem 4.1. If (2.1) has elliptic regularity and τℓ = hℓ , then (A1) is satisfied. 4.1. Preliminaries. We begin the analysis with some basic results on the injection operator Iℓ and the “Ritz quasi-projection” Pℓ−1. After- wards, we deal with “quasi-orthogonality”, before we close the section analyzing the “reconstruction approximation”.

Lemma 4.2 (Stability). The “Ritz quasi-projection” Pℓ−1 : Mℓ → Mℓ−1 is stable in the sense that for all λ ∈ Mℓ, we have

kPℓ−1λkaℓ−1 . kλkaℓ . (4.1) Proof. From (2.26), we can deduce that 2 kPℓ−1λkaℓ−1 = aℓ(λ,IℓPℓ−1λ) ≤kλkaℓ kIℓPℓ−1λkaℓ . kλkaℓ kPℓ−1λkaℓ−1 , (4.2) where we used the Cauchy–Schwarz inequality for aℓ(., .) and the bound- edness of Iℓ from Lemma 2.1.  Lemma 4.3. The DG reconstructions of the injection operator admits the estimate

kUℓ−1µ − UℓIℓµk0 . hℓkµkaℓ−1 , ∀µ ∈ Mℓ−1. (4.3) Proof. We introduce intermediate approximations such that

Uℓ−1µ − UℓIℓµ avg avg c c = Uℓ−1µ − Iℓ−1Uℓ−1µ + Iℓ−1Uℓ−1µ − Uℓ−1µ + Uℓ−1µ − UℓIℓµ, (4.4)

=:Ξ1 =:Ξ2 =:Ξ3 and use| triangle{z inequality.} | For Ξ1,{z one utilizes} | the average{z operator} approximation property [2, (2.28)] and the definition of k·kaℓ to obtain 2 2 2 kµkaℓ−1 kΞ1k0 . hℓ−1|||Uℓ−1µ − µ|||ℓ−1 ≤ hℓ−1 . (4.5) τℓ−1 For the second term, similar to [8, Lem. 3.1] we have

1/2 1/2 kµkaℓ−1 kΞ2k0 . hℓ−1|||Uℓ−1µ − µ|||ℓ−1 ≤ hℓ−1 1/2 . (4.6) τℓ−1

If p = 1, Ξ3 = 0, since Iℓµ is the restriction of a continuous function c and Uℓ simply recovers Uℓ−1µ in this case. If p ≥ 2, we can use [8, Theo. 3.8] to conclude 2 c c kΞ3k0 . hℓ k∆Uℓ−1µk0 . hℓk∇Uℓ−1µk0 . hℓkµkaℓ−1 , (4.7) where the last inequality is [8, Lem. 3.4].  HMG — HOMOGENEOUS MULTIGRID FOR HDG 11

In the following lemmas, we use the continuous linear finite element space c V ℓ := {u ∈ C(Ω) : u|T ∈ P1(T ) ∀T ∈ Tℓ and u =0 on ∂Ω} (4.8) to show quasi-orthogonality and a reconstruction approximation prop- erty of the method.

Lemma 4.4 (Quasi-orthogonality). For all λ ∈ Mℓ, we have that c (∇w, Qℓλ − Qℓ−1Pℓ−1λ)0 =0, ∀w ∈ V ℓ−1. (4.9) c Proof. For w ∈ V ℓ−1 define µ := γℓ−1w ∈ Mℓ−1. By definition, we immediately obtain

Iℓµ = γℓw,

Qℓ−1µ = QℓIℓµ = −∇w, (4.10)

Uℓ−1µ = UℓIℓµ = w.

From the definitions of aℓ and aℓ−1 we obtain

aℓ−1(Pℓ−1λ,µ) =(Qℓ−1Pℓ−1λ, Qℓ−1µ)0

+ τℓ−1hhUℓ−1Pℓ−1λ − Pℓ−1λ, Uℓ−1µ − µiiℓ−1, (4.11) =0 a (λ,I µ)=(Q λ, Q I µ) + τ hhU λ − λ, U I µ − I µii . ℓ ℓ ℓ ℓ ℓ 0 ℓ ℓ | {zℓ ℓ } ℓ ℓ =0 (4.12) | {z } By definition of Pℓ−1 in (2.26), these two terms are equal and thus we obtain the claimed result as their difference.  Lemma 4.5 (Reconstruction approximation). If (2.1) has elliptic reg- ∼ −1 ularity and τℓ = hℓ , then for all λ ∈ Mℓ there exists an auxiliary c function u ∈ V ℓ−1 such that

kQℓλ + ∇uk0 + kQℓ−1Pℓ−1λ + ∇uk0 . hℓkAℓλkℓ. (4.13) The proof of this lemma is based on an explicit construction of u based on the techniques in [10]. It is conducted in the following sub- section.

4.2. Proof of reconstruction approximation. Following [10], we construct several auxiliary quantities in order to define u in Lemma 4.5. First, we define extension operators ST ;i on Mℓ for each cell T and each of its faces Fi into Pp+1 on T by the interpolation conditions

hST ;iλ, ηiFi = hλ, ηiFi ∀η ∈ Pp+1(Fi), (4.14)

(ST ;iλ, v)T =(Uλ, v)T ∀v ∈ Pp(T ). (4.15) 12 PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT

These are used to define the auxiliary inner product T #Fℓ ♥ 1 hλ,µiℓ = T ST ;iλST ;iµ dx, (4.16) #F T T ∈Tℓ ℓ i=1 X X Z and its corresponding norm k·k♥ℓ. Then, for λ ∈ Mℓ (which is fixed in this section, while u andu ¯ depend on it) let φλ ∈ Mℓ be defined by ♥ hφλ,µiℓ = aℓ(λ,µ)= hAℓλ,µiℓ, µ ∈ Mℓ. (4.17)

Thus, φλ represents Aℓλ in Mℓ. Similarly, fλ = Uℓφλ ∈ Vℓ represents Aℓλ on the whole domain. Based on these representations, we define 1 u˜ ∈ H0 (Ω) by 1 (∇u,˜ ∇v)0 =(fλ, v)0, ∀v ∈ H0 (Ω), (4.18) and λ˜ℓ ∈ Mℓ by

aℓ(λ˜ℓ,µ)=(fλ, Uℓµ)0 ∀µ ∈ Mℓ. (4.19)

In the remainder of this subsection, we show that u = P ℓ−1λ˜ℓ can be used in Lemma 4.5. Here, the Ritz quasi-projection P ℓ−1 to the cellwise c linear coarse space V ℓ−1 is defined by c c P ℓ−1 : Mℓ → V ℓ−1, (∇P ℓ−1λ, ∇w)0 = aℓ(λ,γℓw) ∀w ∈ V ℓ−1. (4.20) ˜ We begin with an approximation result for λℓ:

Lemma 4.6. If (2.1) has elliptic regularity, for all λ ∈ Mℓ, we have ˜ kλ − λℓkaℓ . hℓkAℓλkℓ and kfλk0 . kAℓλkℓ. (4.21) Proof. This is [31, Lemma 5.10].  We now prove the estimate for the second norm in Lemma 4.5. To this end, we introduce the auxiliary function λ˜ℓ−1 ∈ Mℓ−1, which is defined like λ˜ℓ in equation (4.19), but on level ℓ − 1. Then, ˜ Qℓ−1Pℓ−1λ + ∇P ℓ−1λℓ

= Qℓ−1Pℓ−1λ − Qℓ−1Pℓ−1λ˜ℓ + Qℓ−1Pℓ−1λ˜ℓ − Qℓ−1λ˜ℓ−1

=:Ξ1 =:Ξ2 ˜ ˜ | {z+ Qℓ−1λℓ−1 +} ∇|u˜ +(−∇u˜ +{z∇P ℓ−1λℓ)}. (4.22) =:Ξ3 =:Ξ4

To obtain the result, we| now have{z to bound} | all four{z norms}kΞ1k0 to kΞ4k0. First, ˜ ˜ kΞ1k0 ≤kPℓ−1λ − Pℓ−1λℓkaℓ−1 . kλ − λℓkaℓ . hℓkAℓλkℓ, (4.23) where the first inequality follows directly from the definition of k·kaℓ−1 , the second inequality is Lemma 4.2, and the last inequality is Lemma HMG — HOMOGENEOUS MULTIGRID FOR HDG 13

4.6. Using the definition of λ˜ℓ−1, (2.26), and (4.19), we obtain for ˜ ˜ eℓ−1 := Pℓ−1λℓ − λℓ−1 that

aℓ−1(eℓ−1,µ)=(fλ, UℓIℓµ − Uℓ−1µ)0 ∀µ ∈ Mℓ−1. (4.24)

Choosing µ = eℓ−1, reusing that kQℓ−1 ·k0 ≤k·kaℓ−1 and exploiting the Cauchy–Schwarz inequality, we get

2 kΞ2k0 ≤ aℓ(eℓ−1, eℓ−1)=(fλ, UℓIℓeℓ−1 − Uℓ−1eℓ−1)0 (4.25)

≤kfλk0kUℓ−1eℓ−1 − UℓIℓeℓ−1k0. (4.26)

The correct bound for kΞ2k0 can now be deduced, since Lemma 4.3 implies that

kUℓ−1eℓ−1 − UℓIℓeℓ−1k0 . hℓkeℓ−1kaℓ−1 (4.27) which allows to extract

2 keℓ−1kaℓ−1 . hℓkfλk0keℓ−1kaℓ−1 (4.28) from (4.25). Dividing this inequality by keℓ−1kaℓ−1 yields

2 2 2 2 2 2 kΞ2k0 ≤keℓ−1kaℓ−1 . hℓ kfλk0 . hℓ kAℓλkℓ , (4.29) where the last inequality follows from Lemma 4.6. For kΞ3k0, we utilize the convergence properties of the HDG method and that Qℓ−1λ˜ℓ−1 + Qℓ−1fλ approximates q˜ = −∇u˜. Hence, we can deduce that ˜ kQℓ−1λℓ−1 + Qℓ−1fλ + ∇u˜k0 . hℓ−1|u˜|H2(Ω) . hℓ−1kfλk0 . hℓkAℓλkℓ. (4.30a) Here, the elliptic regularity of (2.1) enters together with Lemma 4.6 — which is also needed to deduce

kQℓ−1fλk0 . hℓ−1kfλk0 . hℓkAℓλkℓ, (4.30b) where the first inequality is [8, Lem. 3.7]. (4.30) implies the desired properties for kΞ3k0. For the last term, we observe that

(∇P ℓ−1λ˜ℓ, ∇w)0 = aℓ(λ˜ℓ,γℓw)=(fλ, Uℓγℓw)0 =(fλ,w)0 (4.31) c for all w ∈ V ℓ−1. Thus, we can exploit the approximation property of (continuous) linear finite elements to obtain the result for kΞ4k0 similar to (4.30a). The inequality for the first term can be obtained analogously substi- tuting (4.22) by

Qℓλ + ∇P ℓ−1λ˜ℓ = Qℓλ − Qℓλ˜ℓ + Qℓλ˜ℓ + ∇u˜ −∇u˜ + ∇P ℓ−1λ˜ℓ. (4.32) Additionally, we use similar techniques as above to prove the following lemma which will be used in the proof of the main convergence result. 14 PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT

Lemma 4.7. If for all ℓ, τℓhℓ ≤ c for some c > 0, we have for all λ ∈ Mℓ, µℓ ∈ Mℓ, and µℓ+1 ∈ Mℓ+1 that 2 2 τℓ|||Uℓλ − λ|||ℓ . kQℓλ + ∇P ℓ−1µℓk0 2 2 (4.33) τℓ|||Uℓλ − λ|||ℓ . kQℓλ + ∇P ℓµℓ+1k0.

Proof. Obviously, we have UℓγℓP ℓ−1µℓ = P ℓ−1µℓ and QℓγℓP ℓ−1µℓ = −∇P ℓ−1µℓ. Thus

2 2 τℓ|||Uℓλ − λ|||ℓ = τℓ|||Uℓ(λ − γℓPµℓ) − (λ − γℓP ℓ−1µℓ)|||ℓ 2 2 . kQℓ(λ − γℓP ℓ−1µℓ)k0 = kQℓλ + ∇P ℓ−1µℓk0, (4.34) where the inequality is [31, p. 68]. The second inequality can be ob- tained analogously.  4.3. Proof of regularity approximation. With these preliminaries given, we can now state the proof of Theorem 4.1. At first, we recognize that it suffices to show that 2 2 |aℓ(λ − IℓPℓ−1λ,λ)| . hℓ kAℓλkℓ , (4.35) since this is sufficient for (A1) to hold (cf. [31, Thm. 3.6]). Using the bilinearity of aℓ and (2.26), we immediately obtain

aℓ((λ − IℓPℓ−1λ,λ)= aℓ(λ,λ) − aℓ−1(Pℓ−1λ,Pℓ−1λ)

=(Qℓλ, Qℓλ)0 − (Qℓ−1Pℓ−1λ, Qℓ−1Pℓ−1λ)0 (T1) 2 2 + τℓ|||Uℓλ − λ|||ℓ − τℓ−1|||Uℓ−1Pℓ−1λ − Pℓ−1λ|||ℓ−1, (T2) where the second equation is due to the definition of the bilinear forms aℓ and aℓ−1. Now, we estimate both terms (T1) and (T2) separately. For the first, we use binomial factoring and quasi-orthogonality to ob- tain

(T1) =(Qℓλ + Qℓ−1Pℓ−1λ, Qℓλ − Qℓ−1Pℓ−1λ)0 (4.36)

=(Qℓλ +2∇u + Qℓ−1Pℓ−1λ, Qℓλ − Qℓ−1Pℓ−1λ)0. (4.37) Here, u is from Lemma 4.5. Thus,

(T1) ≤ kQℓλ + ∇uk0 + k∇u + Qℓ−1Pℓ−1λk0 kQℓλ − Qℓ−1Pℓ−1λk0.   (4.38) Due to Lemma 4.5, we can further estimate

(T1) . hℓkAℓλkℓ kQℓλ + ∇uk0 + k∇u + Qℓ−1Pℓ−1λk0 (4.39) Application of Lemma 4.5 gives the desired result for (T1). For (T2), we exploit that both summands have exactly the same form and can be treated analogously. Thus, we only demonstrate the procedure for the second summand: 2 2 2 2 τℓ−1kUℓ−1Pℓ−1λ − Pℓ−1λkℓ−1 . kQℓ−1Pℓ−1λ + ∇uk0 . hℓ kAℓλkℓ , (4.40) HMG — HOMOGENEOUS MULTIGRID FOR HDG 15

Figure 2. Coarse grid (level 0) for numerical experi- ments. Meshes on higher levels are generated by uniform refinement. where the first inequality is Lemma 4.7 and the second inequality is Lemma 4.5.

5. Proof of (A2) and (A3)

The proof of (A2) is a simple consequence of Lemma 2.1 with Pℓ−1λ instead of λ i.e., we have

aℓ(λ−IℓPℓ−1λ,λ − IℓPℓ−1λ) (5.1)

=aℓ(λ,λ) − 2aℓ(λ,IℓPℓ−1λ)+ aℓ(IℓPℓ−1λ,IℓPℓ−1λ) (5.2)

≤aℓ(λ,λ) −2aℓ−1(Pℓ−1λ,Pℓ−1λ) +C aℓ−1(Pℓ−1λ,Pℓ−1λ), (5.3)

≤0 2 .kλkaℓ by Lemma 4.2 For the proof of| (A3), we{z heavily rely} on| [3] (where{z (A3) is} denoted (2.11)). Theorems 3.1 and 3.2 of [3] ensure that (A3) holds if the subspaces satisfy a “limited interaction property” which holds, because each degree of freedom (DoF) only “communicates” with other DoFs which are located on the same face as the DoF or on the other faces of the two adjacent elements.

6. Numerical experiments For the numerical evaluation of our multigrid method for HDG, we consider the following Poisson problem on the unit square Ω = (0, 1)2:

−∆u = f in Ω, (6.1a) u =0 on ∂Ω, (6.1b) where f is chosen as one. The first mesh is shown in Figure 2 and it is successively refined in our experiments. The implementation is based on the FFW toolbox from [6] and employs the Gauss–Seidel smoother. It uses a Lagrange basis and the Euclidean inner product in the coefficient space instead of the inner product h., .iℓ. These two 2 inner products are equivalent up to a factor of hℓ . Supposing that the 16 PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT

smoother one step two steps mesh level 123456 123456 1 τ = h 33 39 38 36 35 35 17 20 19 19 18 18 =1 τ =1 33 39 36 35 34 33 17 19 18 18 17 17 p 1 τ = h 13 12 11 10 10 09 08 07 07 06 06 05 =2 τ =1 13 12 11 10 10 09 08 07 07 06 06 05 p 1 τ = h 24 25 25 25 25 25 15 15 15 15 15 15 =3 τ =1 24 25 25 25 25 25 15 15 15 15 15 15 p Table 1. Numbers of iterations with one and two smoothing steps for f ≡ 1. The polynomial degree of the HDG method is p. matrix form of (6.1) is Ax = b we stop the iteration for solving the linear system of equations if kb − Ax k iter 2 < 10−6. (6.2) kbk2 The initial value x on mesh level ℓ is the solution on level ℓ − 1, thus we perform a nexted iteration. The numbers of iteration steps needed are shown in Table 1 for one and two pre- and post-smoothing steps on each level, respectively. Clearly, these numbers are independent of the mesh level, as predicted by our analysis. Additionally, the numbers are fairly small, such that we can conclude that we actually have an efficient method. If we employ two smoothing steps instead of one, the number of steps is almost divided by two, such that both options will result in similar numerical effort. Finally, we see that the choice 1 of τ ∈{ h , 1} does not significantly influence the number of iterations. We ran experiments for polynomial degrees one, two, and three, where iteration counts remain well bounded; nevertheless, we expect rising counts for higher degrees, as we use a point smoother. Additionally, we tested the correctness of our implementation by em- ploying a right hand side leading to the solution u = sin(2πx) sin(2πy). The estimated orders of convergence (EOC) of the primary unknown u computed as

ku − u k 2 EOC = log ℓ−1 L (Ω) / log(2), (6.3) ku − u k 2  ℓ L (Ω)  and the secondary unknown q of the HDG method are reported in Table 2; they coincide well with the orders predicted in [13]. Iteration counts are almost identical to those in Table 1, such that we do not 1 report them here. We see that the choice τ = h is suboptimal as compared to τ = 1, as the error in the secondary unknown q converges HMG — HOMOGENEOUS MULTIGRID FOR HDG 17

mesh 2 34 56 7 EOC u q u q u q u q u q u q 1 τ = h 1.4 1.2 1.9 1.8 2.0 1.7 2.0 1.4 2.0 1.2 2.0 1.0 =1 τ =1 1.5 1.3 2.0 1.9 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 p 1 τ = h 3.4 3.0 3.1 2.8 3.0 2.6 3.0 2.4 3.0 2.1 3.0 2.0 =2 τ =1 3.4 3.1 3.1 2.9 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 p 1 τ = h 3.9 2.8 4.4 3.7 4.2 3.5 4.1 3.2 4.0 3.1 4.0 3.0 =3 τ =1 2.8 2.8 3.9 3.9 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 p Table 2. Estimated orders of convergence (EOC) for primary unknown u and secondary unknown q when the polynomial degree of the HDG method is p and the exact solution is u = sin(2πx) sin(2πy). slower by one order. This is why we included results for τ = 1 in Table 1 albeit a theoretical justification is still missing.

7. Conclusions We proposed a homogeneous multigrid method for HDG. We proved analytically that this method converges independently of the mesh size. Numerical examples have shown that the condition numbers are not only independent of the mesh size but also reasonably small. As as consequence, we have been enabled to efficiently solve linear systems of equations arising from HDG discretizations of arbitrary order. Our proofs apply to stabilization terms τ ∼ h−1, but numerical experiments suggest optimal convergenca also for τ ∼ 1.

References 1. G. Barrenechea, M. Bosy, V. Dolean, F. Nataf, and P.-H. Tournier, Hybrid discontinuous Galerkin discretisation and domain decomposition precondition- ers for the Stokes problem, Comput. Methods Appl. Math. 19 (2018), no. 4, 703–722. 2. A. Bonito and R. H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal. 48 (2010), no. 2, 734– 771. 3. J.H. Bramble and J.E. Pasciak, The analysis of smoothers for multigrid algo- rithms, Math. Comput. 58 (1992), no. 198, 467–488. 4. J.H. Bramble, J.E. Pasciak, and J. Xu, The analysis of multigrid with nonnested spaces or noninherited quadratic forms, Math. Comput. 56 (1991), no. 193, 1–34. 5. S. Brenner, Convergence of nonconforming multigrid methods without full el- liptic regularity, Math. Comput. 68 (1999), no. 225, 25–53. 6. A. Byfut, J. Gedicke, D. G¨unther, J. Reininghaus, and S. Wiedemann, FFW documentation, https://github.com/project-openffw/openffw. 18 PEIPEI LU, ANDREAS RUPP, AND GUIDO KANSCHAT

7. F. Celiker, B. Cockburn, and K. Shi, Hybridizable discontinuous Galerkin meth- ods for Timoshenko beams, J. Sci. Comput. 44 (2010), no. 1, 1–37. 8. H. Chen, P. Lu, and X. Xu, A robust multilevel method for hybridizable dis- continuous Galerkin method for the Helmholtz equation, J. Comput. Phys. 264 (2014), 133–151. 9. B. Cockburn, B. Dong, and J. Guzm´an, A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems, J. Sci. Comput. 40 (2009), no. 1-3, 141–187. 10. B. Cockburn, O. Dubois, J. Gopalakrishnan, and S. Tan, Multigrid for an HDG method, IMA J. Numer. Anal. 34 (2013), no. 4, 1386–1425. 11. B. Cockburn and J. Gopalakrishnan, The derivation of hybridizable discontinu- ous Galerkin methods for Stokes flow, SIAM J. Numer. Anal. 47 (2009), no. 2, 1092–1125. 12. B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of dis- continuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. 13. B. Cockburn, J. Gopalakrishnan, and F.J. Sayas, A projection-based error anal- ysis of HDG methods, Math. Comput. 79 (2010), no. 271, 1351–1367. 14. B. Cockburn, N. C. Nguyen, and J. Peraire, A comparison of HDG methods for Stokes flow, J. Sci. Comput. 45 (2020), no. 1, 215–237. 15. B. Cockburn and F.-J. Sayas, Divergence-conforming HDG methods for Stokes flows, Math. Comput. 83 (2014), no. 288, 1571–1598. 16. H.Y. Duan, S.Q. Gao, R.C.E. Tan, and S. Zhang, A generalized BPX multi- grid framework covering nonnested V-cycle methods, Math. Comput. 76 (2007), no. 257, 137–152. 17. H. Egger and Ch. Waluga, hp analysis of a hybrid DG method for Stokes flow, IMA J. Numer. Anal. 33 (2013), no. 2, 687–721. 18. X. Feng and O. Karakashian, Two-level non-overlapping Schwarz methods for a discontinuous Galerkin method, SIAM J. Numer. Anal. 39 (2001), no. 4, 1343–1365. 19. G. Fu, Ch. Lehrenfeld, A. Linke, and T. Streckenbach, Locking free and gradient robust H(div)-conforming HDG methods for , arXiv preprint arXiv:2001.08610 (2020). 20. J. Gopalakrishnan, A Schwarz preconditioner for a hybridized mixed method, Comput. Methods Appl. Math. 3 (2003), no. 1, 116–134. 21. J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numerische Mathematik 95 (2003), 527–550. 22. J. Gopalakrishnan and S. Tan, A convergent multigrid cycle for the hybridized mixed method, Numer. Lin. Alg. Appl. 16 (2009), 689–714. 23. Y. He, S. Rhebergen, and H. De Sterck, Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods, arXiv preprint arXiv:2006.11433 (2020). 24. Jianguo Huang and Xuehai Huang, A hybridizable discontinuous Galerkin method for Kirchhoff plates, J. Sci. Comput. 78 (2019), no. 1, 290–320. 25. Ch. Lehrenfeld and J. Sch¨oberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Computer Methods Appl. Mech. Engrg. 307 (2016), 339 – 361. 26. B. Li and X. Xie, BPX preconditioner for nonstandard finite element methods for diffusion problems, SIAM Journal on 54 (2016), no. 2, 1147–1168. 27. B. Li, X. Xie, and S. Zhang, Analysis of a two-level algorithm for HDG methods for diffusion problems, Comm. Comput. Phys. 19 (2016), no. 5, 1435–1460. HMG — HOMOGENEOUS MULTIGRID FOR HDG 19

28. N.C. Nguyen, J. Peraire, and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Computer Methods Appl. Mech. Engrg. 199 (2010), no. 9, 582–597. 29. I. Oikawa, Analysis of a reduced-order HDG method for the Stokes equations, J. Sci. Comput. 67 (2016), no. 2, 475–492. 30. J. Sch¨oberl and Ch. Lehrenfeld, Domain decomposition preconditioning for high order hybrid discontinuous Galerkin methods on tetrahedral meshes, Advanced Finite Element Methods and Applications (Berlin, Heidelberg) (Th. Apel and O. Steinbach, eds.), Lecture Notes in Applied and Computational Mechanics, vol. 66, Springer, 2013, pp. 27–56. 31. S. Tan, Iterative solvers for hybridized finite element methods, Ph.D. thesis, University of Florida, 2009.

Department of Mathematics Sciences, Soochow University, Suzhou, 215006, China Email address: [email protected]

Interdisciplinary Center for Scientific Computing (IWR), Heidel- berg University, Mathematikon, Im Neuenheimer Feld 205, 69120 Hei- delberg, Germany Email address: [email protected], [email protected]

Interdisciplinary Center for Scientific Computing (IWR) and Math- ematics Center Heidelberg (MATCH), Heidelberg University, Mathe- matikon, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany Email address: [email protected]