Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

HAL Id: hal-01428790

https://hal.inria.fr/hal-01428790

Submitted on 6 Jan 2017

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of

Hamilton–Jacobi–Bellman equations

Iain Smears

To cite this version:

Iain Smears. Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin ap-

proximations of Hamilton–Jacobi–Bellman equations. Journal of Scientific Computing, Springer Ver-

lag, 2017, �10.1007/s10915-017-0428-5�. �hal-01428790�

for discontinuous Galerkin approximations of Hamilton–Jacobi–Bellman equations

Iain Smears

Received: date / Accepted: date

Abstract We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approx- imation of fully nonlinear Hamilton–Jacobi–Bellman (HJB) partial differential equations.

These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix A. In this work, we construct a nonoverlapping domain decomposition preconditioner P, that is based on A, and we then show that the effectiveness of the preconditioner for solving the nonsymmetric problems can be studied in terms of the condition number κ(P

⁻¹

A). In particular, we establish the bound κ(P

⁻¹

A) . 1 + p

⁶

H

³

/q

³

h

³

, where H and h are respectively the coarse and fine mesh sizes, and q and p are respectively the coarse and fine mesh polynomial degrees. This repre- sents the first such result for this class of methods that explicitly accounts for the dependence of the condition number on q; our analysis is founded upon an original optimal order ap- proximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these meth- ods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under h-refinement for moderate polynomial degrees.

Keywords domain decomposition · preconditioners · GMRES · discontinuous Galerkin · finite element methods · approximation in discontinuous spaces · Hamilton–Jacobi–Bellman equations

Mathematics Subject Classification (2000) 65F10 · 65N22 · 65N55 · 65N30 · 35J66

1 Introduction

In [20–22], discontinuous Galerkin finite element methods (DGFEM) were introduced for the numerical solution of linear nondivergence form elliptic equations and fully nonlinear Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. In these applications,

Iain Smears

Inria Paris, 2 Rue Simone Iff, 75589, Paris, France. E-mail: iain.smears@inria.fr

the appropriate norm on the finite element space is a broken H

²

-norm with penalization of the jumps in values and in first derivatives across the faces of the mesh. As a result, it is typical for the condition number of the discrete problems to be of order p

⁸

/h

⁴

, where h is the mesh size and p is the polynomial degree. The purpose of this work is to study the application of a commonly used class of nonoverlapping domain decomposition preconditioners to these problems.

Nonoverlapping domain decomposition methods, along with their overlapping counter- parts, have been successfully developed for a range of applications of DGFEM by many authors [2–4, 6, 10, 11, 14]. In order to solve a problem on a fine mesh T

_h

, these methods combine a coarse space solver, defined on a coarse mesh T

H

, with local fine mesh solvers, defined on a subdomain decomposition T

S

of the domain Ω. The discontinuous nature of the finite element space leads to a significant flexibility in the choice of the decomposition T

_S

, which can either be overlapping or nonoverlapping. As explained in the above references, these preconditioners possess many advantages in terms of simplicity and applicability, as they allow very general choices of basis functions, nonmatching meshes and varying ele- ment shapes, and are naturally suited for parallelization. It has been pointed out by various authors, such as Lasser and Toselli in [14, p. 1235], that nonoverlapping methods feature re- duced inter-subdomain communication burdens, thus representing an advantage in parallel computations.

For problems involving H

¹

-type norms, such as divergence form second-order elliptic PDE, nonoverlapping additive Schwarz preconditioners for h-version methods [10] lead to condition numbers of order 1 + H/h, where H is the coarse mesh size, while overlapping methods lead to a condition number of order 1 + H/δ, where δ is the subdomain overlap.

For problems in H

²

-type norms such as the biharmonic equation, the h-version analysis [11]

leads to condition numbers of order 1 + H

³

/h

³

. We remark that the analysis in these works leaves the polynomial degree implicit inside the generic constants. However an analysis that keeps track of all parameters is important in practice for determining their effect on the performance of the preconditioners, even if robustness of the condition number cannot be guaranteed. Antonietti and Houston [4] were the first to keep track of the dependence on the polynomial degrees for this class of preconditioners for problems in H

¹

-norms, where they showed a condition number bound of order 1 + p

²

H/h. However, their numerical experiments lead them to conjecture the improved bound of order 1 + p

²

H/qh, where q is the coarse space polynomial degree. This conjecture was recently proved in [5] using ideas first developed in this work.

As can be seen from the theoretical analysis in the above references, the effectiveness of the preconditioner depends in an essential way on the approximation properties between the coarse and fine spaces. In the analysis of h-version DGFEM, it is sufficient to consider low-order projection operators from the fine space to the coarse space; for example, coarse element mean-value projections are employed in [10] and local first-order elliptic projec- tions are used in [11]. However, low-order projections lead to suboptimal bounds for the condition number with respect to polynomial degrees. This work resolves this suboptimality through an original optimal order approximation result between coarse and fine spaces.

There are further classes of preconditioners for p-version and hp-version methods for

problems in H

¹

-norms that achieve condition numbers either independent or depending

only polylogarithmically on the polynomial degree, such as Neumann–Neumann and FETI

methods, see [17, 24] and the many references therein. We are aware of one work on gen-

eralising these methods to H

²

-norm problems: Brenner and Wang [8] considered iterative

substructuring methods for the h-version C

⁰

interior penalty discretizations of the bihar-

monic equation. They show that the usual choices of orthogonalised basis functions required

by these algorithms do not extend to the H

²

-norm context, and that different basis functions must be used on different elements of the mesh. In comparison, the overlapping and non- overlapping methods described above generalise straightforwardly to the H

²

-norm context without additional difficulties. Moreover, a comparison of the computations in [7] and [8]

suggests that the substructuring algorithms only yield a similar performance in practice to the two-level additive Schwarz methods for these problems.

1.1 Main results

The numerical scheme of [21] for fully nonlinear HJB equations leads to a discrete non- linear problem that can be solved iteratively by a semismooth Newton method. The linear systems obtained from the Newton linearization are generally nonsymmetric but coercive with respect to a discrete H

²

-type norm. In section 3, we apply existing GMRES conver- gence theory for SPD preconditioners [9, 15] to these nonsymmetric systems, leading to a guaranteed minimum convergence rate, with a contraction factor expressed in terms of the condition number κ(P

⁻¹

A), where A is the matrix of a related symmetric bilinear form that is spectrally equivalent to a discrete H

²

-type norm, and where P is an arbitrary symmet- ric positive definite preconditioner. Thus, the construction and analysis of preconditioners for a symmetric problem can be used for the solution of the nonsymmetric systems appear- ing in applications to HJB equations [21]. A further benefit is that the preconditioner does not require re-assembly at each new semismooth Newton iteration.

Section 4 presents the specific construction of a nonoverlapping additive Schwarz pre- conditioner P based on A, and sections 5 and 6 show the condition number bound

κ(P

⁻¹

A) . 1 + p

²

H

q h + p

⁶

H

³

q

³

h

³

. (1.1)

In comparison to the existing literature, this is the first bound for this class of preconditioners that explicity accounts for the coarse mesh polynomial degree. Unfortunately, (1.1) implies that this standard class of preconditioners cannot be expected to be robust with respect to the polynomial degree. Nevertheless, our result shows that the coarse space polynomial degree can contribute significantly to reducing the condition number.

The central original result underpinning our analysis is Theorem 4 of section 5, which shows that for any v

h

∈ V

h,p

, there is a function v ∈ H

²

(Ω) ∩ H

0¹

(Ω) such that

kv

h

− vk

_L²_(Ω)

+ h

p kv

h

− vk

_H¹(Ω;T_h)

. h

²

p

²

|v

h

|

J,h

, kvk

_H²_(Ω)

. kv

h

k

2,h

, (1.2) where the piecewise Sobolev norms k·k

_H^s(Ω;T_h)

, jump seminorm |·|

J,h

, and the discrete H

²

-type norm k·k

2,h

are defined in section 2. This result is a natural converse to classical direct approximation theory, since, here, the nonsmooth function from the discrete space V

h,p

is approximated by a smoother function from an infinite dimensional space. It follows from (1.2) that there exists a function v

H

in the coarse space V

H,q

, of polynomials of degree q on T

_H

, such that

kv

_h

− v

H

k

_Hk(Ω;Th)

. H

^2−k

q

^2−k

kv

_h

k

_2,h

, k ∈ {0, 1, 2}, (1.3)

thus yielding an approximation between coarse and fine meshes that is optimal in the orders

of the mesh size and the polynomial degree. The approximation result is used to show the

stable decomposition property for the additive Schwarz preconditioner in section 6, thereby leading to the spectral bound (1.1).

The first numerical experiment, in section 7.1, confirms that (1.1) is sharp with respect to the orders in the polynomial degrees. The experiment of section 7.2 compares non- overlapping methods with their overlapping counterparts, where it is found that they are competitive in both iteration counts and computational cost. Despite the polynomial de- gree suboptimality of these preconditioners, in section 7.3 we show computationally that for h-refinement, nonoverlapping methods can be efficient and competitive in challenging applications to fully nonlinear HJB equations.

2 Definitions

For real numbers a and b, we shall write a . b to signify that there is a positive constant C such that a ≤ Cb, where C is independent of the quantities of interest, such as the element sizes and polynomial degrees, but possibly dependent on other quantities, such as the mesh regularity parameters.

Let Ω ⊂ R

^d

, d ∈ {2, 3}, be a bounded convex polytopal domain. Note that convexity of Ω implies that the boundary ∂Ω of Ω is Lipschitz [13]. Let {T

_h

}

_h

be a sequence of shape-regular meshes on Ω, consisting of simplices or parallelepipeds. For each element K ∈ T

_h

, let h

K

:= diam K. It is assumed that h = max

K∈Th

h

K

for each mesh T

_h

. Let F

_hⁱ

denote the set of interior faces of the mesh T

_h

and let F

_h^b

denote the set of boundary faces. The set of all faces of T

_h

is denoted by F

_h^i,b

:= F

_hⁱ

∪ F

_h^b

. Since each element has piecewise flat boundary, the faces may be chosen to be flat. For K ∈ T

_h

or F ∈ F

_h^i,b

, we use h·, ·i

_K

, respectively h·, ·i

_F

, to denote the L

²

-inner product over K , respectively F , of scalar functions, vector fields, and higher-order tensors.

Mesh conditions The meshes are allowed to be nonmatching, i.e. there may be hanging nodes. We assume that there is a uniform upper bound on the number of faces composing the boundary of any given element; in other words, there is a c

F

> 0, independent of h, such that

K∈T

max

h

card{F ∈ F

_h^i,b

: F ⊂ ∂K} ≤ c

F

∀K ∈ T

h

. (2.1) It is also assumed that any two elements sharing a face have commensurate diameters, i.e.

there is a c

T

≥ 1, independent of h, such that

max(h

K

, h

K⁰

) ≤ c

T

min(h

K

, h

K⁰

), (2.2) for any K and K

⁰

in T

_h

that share a face. For each h, let p := (p

_K

: K ∈ T

_h

) be a vector of positive integers; note that this requires p

K

≥ 1 for all K ∈ T

_h

. We make the assumption that p has local bounded variation: there is a c

P

≥ 1, independent of h, such that

max(p

K

, p

K⁰

) ≤ c

P

min(p

K

, p

K⁰

), (2.3)

for any K and K

⁰

in T

_h

that share a face.

Function spaces For each K ∈ T

_h

, let P

_pK

(K ) be the space of all real-valued polynomials in R

^d

with either total or partial degree at most p

K

. In particular, we allow the combination of spaces of polynomials of fixed total degree on some parts of the mesh with spaces of polynomials of fixed partial degree on the remainder. We also allow the use of the space of polynomials of total degree at most p

K

even when K is a parallelepiped. The discontinuous Galerkin finite element spaces V

h,p

are defined by

V

h,p

:= n

v ∈ L

²

(Ω) : v|

_K

∈ P

pK

(K), ∀K ∈ T

h

o

. (2.4)

Let s := (s

K

: K ∈ T

h

) denote a vector of non-negative real numbers. The broken Sobolev space H

^s

(Ω; T

h

) is defined by

H

^s

(Ω; T

h

) := n

v ∈ L

²

(Ω) : v|

_K

∈ H

^sK

(K), ∀K ∈ T

h

o

. (2.5)

For s ≥ 0, we set H

^s

(Ω; T

_h

) := H

^s

(Ω; T

_h

), where s

K

= s for all K ∈ T

_h

. The norm k·k

_H^s(Ω;T_h)

and semi-norm |·|

_H^s(Ω;T_h)

are defined on H

^s

(Ω; T

h

) as

kvk

_H^s_(Ω;Th)

:= X

K∈Th

kvk

²_HsK(K)

!

¹₂

, |v|

_H^s_(Ω;Th)

:= X

K∈Th

|v|

²_HsK(K)

!

¹₂

. (2.6) For a function v

h

∈ V

h,p

, the element-wise gradient ∇v

_h

|

_K

and the Hessian D

²

v

h

|

_K

are well-defined for all K ∈ T

h

since v

h

is smooth on K . Thus expressions such as hD

²

u

h

, D

²

v

h

i

K

are well-defined for all K ∈ T

_h

and all u

h

, v

h

∈ V

h,p

.

Jump and average operators For each face F ∈ F

_h^i,b

, let n

F

∈ R

^d

denote a fixed choice of a unit normal vector to F . Since F is flat, n

F

is constant over F . Let K be an element of T

_h

for which F ⊂ ∂K; then n

F

is either inward or outward pointing with respect to K.

Let τ

F

: H

^s

(K ) → H

^s−1/2

(F ), s > 1/2, denote the trace operator from K to F , and let τ

F

be extended componentwise to vector-valued functions.

For each face F , define the jump operator _J · _K and the average operator {·} by J φ _K := τ

F

φ|

_K

ext

− φ|

_K

int

, {φ} := 1 2 τ

F

φ|

_K

ext

+ φ|

_K

int

, if F ∈ F

_hⁱ

, J φ _K := τ

F

φ|

_K

ext

, {φ} := τ

F

φ|

_K

ext

, if F ∈ F

_h^b

, where φ is a sufficiently regular scalar or vector-valued function, and K

ext

and K

int

are the elements to which F is a face, i.e. F = ∂K

ext

∩ ∂K

int

. Here, the labelling is chosen so that n

F

is outward pointing with respect to K

ext

and inward pointing with respect to K

int

. Using this notation, the jump and average of scalar-valued functions, resp. vector-valued, are scalar-valued, resp. vector-valued.

Tangential differential operators For F ∈ F

_h^i,b

, let H

_T^s

(F ) denote the space of H

^s

-regular tangential vector fields on F , thus H

_T^s

(F ) := {v ∈ H

^s

(F )

^d

: v · n

F

= 0 on F }. We define the tangential gradient ∇

_T

: H

^s

(F ) → H

_T^s−1

(F ) and the tangential divergence div

T

: H

_T^s

(F ) → H

^s−1

(F ), where s ≥ 1, following [13]. Let {t

i

}

^d−1_i=1

⊂ R

^d

be an orthonormal coordinate system on F . Then, for u ∈ H

^s

(F ) and v ∈ H

_T^s

(F ) such that v = P

d−1

i=1

v

i

t

i

, with v

i

∈ H

^s

(F ) for i = 1, . . . , d − 1, we define

∇

T

u :=

d−1

X

i=1

t

i

∂u

∂t

i

, div

T

v :=

d−1

X

i=1

∂v

i

∂t

i

. (2.7)

Mesh-dependent norms In the following, we let u

h

and v

h

denote functions in V

h,p

. For face-dependent positive real numbers µ

F

and η

F

, let the jump stabilization bilinear form J

h

: V

h,p

× V

h,p

be defined by

J

h

(u

h

, v

h

) := X

F∈F_hⁱ

µ

F

h _J ∇u

_h

· n

F

K , _J ∇v

_h

· n

F

K i

F

+ X

F∈F_h^i,b

µ

F

h _J ∇

_T

u

h

K , _J ∇

_T

v

h

K i

_F

+ η

F

h _J u

h

K , _J v

h

K i

_F

. (2.8)

Define the jump seminorm |·|

_J,h

and the mesh-dependent norm k·k

_2,h

on V

h,p

by

|v

_h

|

²_J,h

:= J

h

(v

h

, v

h

), kv

_h

k

²_2,h

:= X

K∈Th

kv

_h

k

²_H2(K)

+ |v

_h

|

²_J,h

. (2.9)

For each face F ∈ F

_h^i,b

, define

˜ h

F

:=

(

min(h

K

, h

K⁰

), if F ∈ F

_hⁱ

,

h

K

, if F ∈ F

_h^b

, p ˜

F

:=

(

max(p

K

, p

K⁰

), if F ∈ F

_hⁱ

, p

K

, if F ∈ F

_h^b

,

(2.10) where K and K

⁰

are such that F = ∂K ∩ ∂K

⁰

if F ∈ F

_hⁱ

or F ⊂ ∂K ∩ ∂Ω if F ∈ F

_h^b

. The assumptions on the mesh and the polynomial degrees, in particular (2.2) and (2.3), show that if F is a face of an element K, then h

K

≤ c

T

h ˜

F

and p ˜

F

≤ c

P

p

K

. Henceforth, it is assumed that the parameters µ

F

and η

F

in (2.8) are given by

µ

F

:= c

µ

˜ p

²_F

˜ h

F

, η

F

:= c

η

˜ p

⁶_F

˜ h

³_F

∀ F ∈ F

_h^i,b

, (2.11) where c

µ

and c

η

are fixed positive constants independent of h and p.

Approximation Under the hypothesis of shape-regularity of {T

_h

}, for any function u ∈ H

^s

(Ω; T

h

), there exists an approximation Π

h

u ∈ V

h,p

, such that for each element K ∈ T

h

,

ku − Π

h

uk

_Hr(K)

. h

^min(s_K ^{K, pK+1)−r}

p

^s_K^K−r

kuk

_HsK(K)

∀ r, 0 ≤ r ≤ s

K

, (2.12a) and, if s

K

> 1/2,

kD

^α

(u − Π

h

u)k

_L2(∂K)

. h

^{min(sK, pK}+1)−|α|−1/2 K

p

^s_K^{K−|α|−1/2}

kuk

_HsK(K)

∀α, |α| ≤ k,

(2.12b)

where k is the greatest non-negative integer strictly less than s

K

− 1/2. The constants

in (2.12a) and (2.12b) do not depend on u, K, p

K

, h

K

or r, but depend possibly on

max

K∈Th

s

K

. Vector fields can be approximated componentwise.

3 HJB equations

We consider fully nonlinear HJB equations of the form sup

α∈Λ

[L

^α

u − f

^α

] = 0 in Ω, u = 0 on ∂Ω,

(3.1) where Ω is a bounded convex domain, Λ is a compact metric space, and the operators L

^α

are given by

L

^α

v := a

^α

: D

²

v + b

^α

· ∇v − c

^α

v, v ∈ H

²

(Ω), α ∈ Λ. (3.2) For simplicity of presentation here, we restrict our attention to the case b

^α

≡ 0, c

^α

≡ 0, and refer the reader to [21] for the general case. The matrix-valued function a and the scalar function f are assumed to be continuous on Ω×Λ, and a is assumed to be uniformly elliptic, uniformly over Ω×Λ. The PDE in (3.1) is fully nonlinear in the sense that the Hessian of the unknown solution appears inside the nonlinear term in (3.1). As a result of the nonlinearity, no weak form of the equation is available: this has constituted a long-standing difficulty in the development of high-order methods for this class of problems.

However, provided that the coefficients of L

^α

satisfy the Cordes condition, which, in the case of pure diffusion, requires that there exist ε ∈ (0, 1] such that

|a

^α

(x)|

²

(Tr a

^α

(x))

²

≤ 1

d − 1 + ε ∀ α ∈ Λ, ∀ x ∈ Ω. (3.3)

then the boundary-value problem (3.1) has a unique solution in H

²

(Ω) ∩ H

₀¹

(Ω), see [21, Theorem 3]. Observe that for problems in two spatial dimensions, condition (3.3) is equiva- lent to uniform ellipticity.

Defining the operator F

γ

[u] := sup

_α∈Λ

[γ

^α

(L

^α

u − f

^α

)], where γ

^α

= Tr a

^α

/|a

^α

|

²

, the numerical scheme of [21] for solving (3.1) associated to a homogeneous Dirichlet bound- ary condition is to find u

h

∈ V

h,p

such that

A

_h

(u

h

; v

h

) = 0 ∀ v

h

∈ V

h,p

, (3.4) where the nonlinear form A

_h

is defined in [21, Eq. (5.3)], and can be equivalently given as

A

h

(u

h

; v

h

) := X

K∈Th

hF

γ

[u

h

], ∆v

h

i

K

+ 1

2 a

h

(u

h

, v

h

) − X

K∈Th

h∆u

h

, ∆v

h

i

K

+ J

h

(u

h

, v

h

)

! , (3.5) where the bilinear form a

h

: V

h,p

× V

h,p

→ R is defined by

a

h

(u

h

, v

h

) := X

K∈Th

hD

²

u

h

, D

²

v

h

i

_K

+ J

h

(u

h

, v

h

)

+ X

F∈F_hⁱ

[hdiv

_T

∇

_T

{u

_h

} , _J ∇v

_h

· n

F

K i

_F

+ hdiv

_T

∇

_T

{v

_h

} , _J ∇u

_h

· n

F

K i

_F

]

− X

F∈F_h^i,b

[h∇

_T

{∇u

_h

· n

F

} , _J ∇

_T

v _K i

_F

+ h∇

_T

{∇v

_h

· n

F

} , _J ∇

_T

v

h

K i

_F

] . (3.6)

3.1 Semismooth Newton method

In [21, Section 8], it is shown that the discretized nonlinear problem (3.4) can be solved by a semismooth Newton method, which leads to a sequence of nonsymmetric but positive definite linear systems to be solved at each iteration. We summarize here the essential ideas on the semismooth Newton, and refer the reader to [21] for the complete analysis.

For x ∈ Ω and M ∈ R

^d×dsym

, define F

γ

(x, M ) := sup

_α∈Λ

[γ

^α

(x) (a

^α

(x):M − f

^α

(x))], and let Λ(x, M ) denote the set of all α ∈ Λ that attain the supremum in F

γ

(x, M); note that Λ(x, M) is always a non-empty subset of Λ due to the compactness of Λ and the continuity of the functions a, f and γ over Ω × Λ. This defines a set-valued mapping (x, M) 7→ Λ(x, M ). For a function v ∈ H

²

(Ω; T

h

), let Λ[v] denote the set of all Lebesgue measurable mappings α(·) : Ω → Λ that satisfy α(x) ∈ Λ(x, D

²

v(x)) for almost every x ∈ Ω; in [21, Theorem 10], it is shown that Λ[v] is non-empty for any v ∈ H

²

(Ω; T

_h

).

The semismooth Newton method is now defined as follows. Start by choosing an initial iterate u

⁰_h

∈ V

h,p

. Then, for each nonnegative integer j, given the previous iterate u

^j_h

∈ V

h,p

, choose an α

j

∈ Λ[u

^j_h

]. Next, the function f

^αj

: Ω → R is defined by f

^αj

: x 7→

f

^αj(x)

(x); the functions a

^αj

and γ

^αj

are defined in a similar way. Note that the measur- ability of the mappings α

j

ensures the measurability of f

^αj

, a

^αj

and γ

^αj

. Then, find the solution u

^j+1_h

∈ V

h,p

of the linearized system

B

_h^j

(u

^j+1_h

, v

h

) = X

K∈Th

hγ

^αj

f

^αj

, ∆v

h

i

K

∀ v

h

∈ V

h,p

, (3.7)

where the bilinear form B

_h^j

: V

h,p

× V

h,p

→ R is defined by B

_h^j

(w

h

, v

h

) := X

K∈T_h

hγ

^αj

a

^αj

:D

²

w

h

, ∆v

h

i

_K

+ 1

2 a

h

(u

h

, v

h

) − X

K∈Th

h∆u

_h

, ∆v

h

i

_K

+ J

h

(u

h

, v

h

)

! , (3.8)

In [21, Theorem 11], it was shown that u

^j_h

→ u as j → ∞ for a sufficiently close initial guess u

⁰_h

, and moreover that the convergence is superlinear. It was also shown that the bilinear forms B

^j_h

are uniformly bounded and coercive in an H

²

-type norm, with constants independent of the iterates. Since the preconditioners of this work take advantage of the coercivity of the B

_h^j

, we summarize the relevant results in the following lemma.

Lemma 1 Let Ω be a bounded convex polytopal domain and let {T

h

}

h

be a shape-regular sequence of meshes satisfying (2.1). Let the bilinear forms B

_h^j

be defined by (3.8). Then, there exist positive constants c

_µ

and c

_η

such that if c

µ

≥ c

_µ

and c

η

≥ c

_η

, then the bilinear forms a

h

and B

_h^j

are uniformly coercive: for all v

h

, w

h

∈ V

h,p

, we have

kv

_h

k

²_2,h

. a

h

(v

h

, v

h

), |a

_h

(v

h

, w

h

)| . kv

_h

k

_2,h

kw

_h

k

_h,2

, (3.9)

kv

_h

k

²_2,h

. B

_h^j

(v

h

, v

h

), |B

^j_h

(v

h

, w

h

)| . kv

_h

k

_2,h

kw

_h

k

_2,h

, (3.10)

where the constants are independent of the sequence {u

^j_h

}

^∞_j=0

and of the choice of the

mappings α

j

∈ Λ[u

^j_h

] for each j ≥ 0.

Proof First we prove (3.9). The continuity bound in (3.9) is a straightforward consequence of the trace and inverse inequalities. To show the coercivity bound, we first show that

kv

_h

k

²_2,h

. X

K∈T_h

kD

²

v

h

k

²_L2(K)

+ |v

_h

|

²_J,h

=: |v

_h

|

²_h,2

. (3.11) For any v

h

∈ V

h,p

, integration by parts gives

X

K∈Th

k∇v

_h

k

²_L2(K)

= X

K∈Th

hv

_h

, −∆v

_h

i

K

+ X

F∈F_h^i,b

h _J v

h

K , {∇v

_h

· n

F

}i

F

+ X

F∈F_hⁱ

h{v

_h

} , _J ∇v

_h

· n

F

K i

_F

. (3.12) Hence, the trace and inverse inequalities imply that

X

K∈Th

k∇v

h

k

²_L²_(K)

. kv

h

k

_L²_(Ω)

|v

h

|

h,2

. (3.13) We recall the broken Poincaré inequality

kv

_h

k

²_L2(Ω)

. X

K∈Th

k∇v

_h

k

²_L2(K)

+ X

F∈F_h^i,b

1

˜ h

F

k _J v

h

K k

²_L2(F)

. (3.14)

Therefore it follows from (3.13) that we have P

K∈Th

kv

_h

k

²_H1(K)

. |v

_h

|

²_h,2

, from which we deduce (3.11). The proof of (3.9) is now completed by noting that [20, Lemma 7] implies that there exist c

_µ

and c

_η

such that |v

_h

|

²_h,2

. a

h

(v

h

, v

h

), whenever c

µ

≥ c

_µ

and c

η

≥ c

_η

, since a

h

equals the bilinear form denoted by B

_DG(1)

in the notation of [20, Lemma 7]. The continuity bound for B

_h^j

in (3.10) can also be shown straightforwardly through the Cauchy–

Schwarz inequality with the trace and inverse inequalities, where we note that the functions γ

^αj

, respectively a

^αj

, appearing in (3.8) are uniformly bounded in L

^∞

by kγk

_C(Ω×Λ)

, respectively by kak

_C(Ω×Λ;Rd×d)

; this implies that the continuity constants in (3.10) can be taken to be independent of the {u

^j_h

}

^∞_j=0

. The coercivity bound in (3.10) was shown in [20, Theorem 8] and [21, Eq. (8.5)], where it is seen that the coercivity constant is independent of the iteration count j, but otherwise may depend on the constant ε from (3.3) and the choice

of the penalty parameters c

µ

and c

η

. u t

3.2 Iterative solution by the preconditioned GMRES method

Each step of the semismooth Newton method requires the solution of (3.7). These linear systems have a common general form, which consists of finding u ˜

h

∈ V

h,p

such that

B

h

(˜ u

h

, v

h

) = `

h

(v

h

) ∀ v

h

∈ V

h,p

, (3.15) where we shall henceforth omit to denote the dependence of the bilinear form B

h

and of the right-hand side `

h

on the iteration number of the semismooth Newton method. It follows from Lemma 1 that there exist positive constants c

B

and C

B

such that, for any v

h

and w

h

∈ V

h,p

,

a

h

(v

h

, v

h

) ≤ 1 c

B

B

h

(v

h

, v

h

), |B

_h

(v

h

, w

h

)| ≤ C

B

p a

h

(v

h

, v

h

) p

a

h

(w

h

, w

h

),

(3.16)

where c

B

and C

B

are independent of the iteration count of the semismooth Newton method and the discretization parameters. Therefore the sequence of linearisations of (3.4) are uni- formly bounded and coercive with respect to the norm defined by the bilinear form a

h

.

The coercivity and boundedness of B

h

imply that an efficient preconditioner for a

h

can also be used effectively as a preconditioner for the GMRES algorithm applied to (3.15).

Indeed, assume that P is an SPD preconditioner for the matrix A := (a

h

(φ

i

, φ

j

)) that satisfies

0 < c

P

≤ v

^>

Av

v

^>

Pv ≤ C

P

∀ v ∈ R

^dimVh,p

\ {0}, (3.17) where we assume that c

P

and C

P

are the best possible constants in (3.17). Thus the condi- tion number κ(P

⁻¹

A) = C

P

/c

P

. Let the matrix B := (B

h

(φ

j

, φ

i

)). Then, the precondi- tioner P can be used in either the right or left preconditioned GMRES method [18, 19] for solving (3.15) as follows. First, we define the norms k·k

_P

and k·k

_P−1

on R

^dimVh,p

by

kvk

²_P

:= v

^>

P v, kvk

²_P−1

:= v

^>

P

⁻¹

v ∀ v ∈ R

^dimVh,p

. (3.18) Applying k-steps of the right preconditioned GMRES method in the P

⁻¹

-inner product computes u

k

as the solution of

u

k

= P

⁻¹

w

k

, w

k

= argmin

˜

wk∈Kk(BP⁻¹,r0)+w0

kBP

⁻¹

(w − w ˜

k

)k

_P−1

, (3.19) where r

0

denotes the initial residual, w := Pu, w

0

:= Pu

0

, and where K(BP

⁻¹

, r

0

) denotes the k-dimensional Krylov subspace generated by BP

⁻¹

and r

0

. It is well-known that (3.19) is equivalent to

u

k

= argmin

˜

uk∈Kk(P⁻¹B,P⁻¹r0)+u0

kP

⁻¹

B (u − u ˜

k

)k

P

, (3.20) which is obtained after k-steps of the left-preconditioned GMRES algorithm in the P-inner product, see [18]. For a discussion of the implementation of the preconditioned GMRES method in the P- and P

⁻¹

-inner products, we refer the reader to [18, p. 269].

It follows from (3.16) and the hypothesis (3.17) that B is also coercive and bounded in the norm defined by P: for any v and w ∈ R

^dimVh,p

, we have

kvk

²_P

≤ 1 c

P

c

B

v

^>

Bv, |v

^>

Bw| ≤ C

P

C

B

kvk

_P

kwk

_P

.

This enables us to appeal to the following well-known bound from GMRES convergence theory [9].

Theorem 1 Let u ∈ R

^dimVh,p

be the vector representing the solution of (3.15). For each k ≥ 1, let u

k

be defined by (3.19) or equivalently by (3.20), with associated residual r

k

. Then

kr

_k

k

_P−1

kr

0

k

_P−1

= kP

⁻¹

r

k

k

_P

kP

⁻¹

r

0

k

_P

≤

1 − c

²_P

c

²_B

C

_P²

C

_B²

k/2

=

1 − 1

κ(P

⁻¹

A)

²

c

²_B

C

_B²

k/2

. (3.21) The bound (3.21) and the coercivity of B imply the following bound for the error:

ku − u

k

k

²_P

≤ 1 c

P

c

B

(u − u

k

)

^>

r

k

≤ 1 c

P

c

B

ku − u

k

k

_P

kr

_k

k

_P−1

,

thereby implying that

ku − u

k

k

_P

≤ kr

0

k

_P−1

c

P

c

B

1 − c

²_B

C

_B²

κ(P

⁻¹

A)

²

k/2

. (3.22)

The bound (3.22) gives a guaranteed minimum convergence rate for GMRES in the P-norm, which is equivalent to the a

h

-norm up to the condition number κ(P

⁻¹

A). We recall that a

h

defines a norm equivalent to k·k

_2,h

, which is the norm of interest, as shown by Lemma 1.

This strongly suggests that the P

⁻¹

-norm, as opposed to the Euclidean norm, of the residual is a natural objective to be minimized by GMRES, as in (3.19) and (3.20).

The conclusion from (3.21) and (3.22) is that, if P is a robust preconditioner for A in the sense of yielding uniformly bounded condition numbers with respect to the parameters being varied, then P will also be a robust preconditioner for the nonsymmetric problems arising from linearizations of HJB equations. In section 4, we construct a specific symmet- ric positive definite preconditioner P, based on a nonoverlapping domain decomposition method, that will be used to solve (3.7).

Remark 1 The general preconditioning strategy proposed here was largely motivated by the analysis in [15]. It is well-known [26] that convergence bounds for GMRES, such as (3.21), need not be descriptive of the convergence rate obtained in practice, i.e. GMRES may perform significantly better than what is predicted by (3.21) alone. In particular this is observed in some of the experiments of section 7.3 below. This implies that the efficiency of the preconditioners must generally be assessed from computations.

3.3 Condition number of the unpreconditioned problem

The condition number of the matrix A := (a

_h

(φ

i

, φ

j

)) depends on the choice of basis for V

h,p

. However, in practice, the basis is often chosen to be either a nodal basis or a mapped orthonormal basis. For example, let us assume that each basis function φ

i

of V

h,p

has support in only one element, and is mapped from a member of a set of functions that are L

²

-orthonormal on a reference element. Then, arguments that are similar to those in [4]

show that the `

²

-norm condition number κ(A) of the matrix A := (a

_h

(φ

i

, φ

j

)) satisfies κ (A) . max

K∈Th

p

⁸_K

h

⁴_K

max

K∈T_h

h

^d_K

min

K∈Th

h

^d_K

, (3.23)

where it is recalled that d is the dimension of the domain Ω.

4 Domain decomposition preconditioners

Let Ω be partitioned into a set T

_S

:= {Ω

_i

}

^N_i=1

of nonoverlapping Lipschitz polytopal sub- domains Ω

i

. The partition T

_S

is assumed to be conforming. A coarse simplicial or par- allelepipedal mesh T

_H

is associated to each fine mesh T

_h

. Let H

D

:= diam D for each D ∈ T

H

and suppose that H := max

D∈TH

H

D

. It is required that the sequence of meshes {T

_H

}

_H

satisfy the mesh conditions of section 2. Furthermore, the partitions T

_S

, T

_H

and T

_h

are assumed to be nested, in the sense that no face of T

_S

, respectively T

_H

, cuts the interior of an element of T

_H

, respectively T

_h

. Hence, each element D ∈ T

_H

satisfies D = S

K,

where the union is over all elements K ∈ T

_h

such that K ⊂ D.

For each mesh T

_H

, let q := (q

D

: D ∈ T

_H

) be a vector of positive integers; so q

D

≥ 1 for each element D ∈ T

H

. Assume that q satisfies the bounded variation property of (2.3), and that q

D

≤ min

K⊂D

p

K

for all D ∈ T

H

. For each D ∈ T

H

, define the sets

T

h

(D) := {K ∈ T

h

: K ⊂ D} , F

_hⁱ

(D) :=

n

F ∈ F

_hⁱ

: F ⊂ D o

, F

_hⁱ

(∂D) := n

F ∈ F

_hⁱ

: F ⊂ ∂D o

, F

_h^i,b

(∂D) := {F ∈ F

_h^i,b

: F ⊂ ∂D}.

(4.1)

Although the sets F

_hⁱ

(D) and F

_h^i,b

(D) are not disjoint, the above assumptions on the meshes imply that F

_h^i,b

= S

D

F

_hⁱ

(D) ∪ F

_h^i,b

(∂D) and that F

_hⁱ

= S

D

F

_hⁱ

(D) ∪ F

_hⁱ

(∂D).

Define the function spaces V

h,pⁱ

:= n

v ∈ L

²

(Ω

i

) : v|

_K

∈ P

pK

(K ) ∀ K ∈ T

_h

, K ⊂ Ω

i

o

, 1 ≤ i ≤ N, (4.2a) V

H,q

:= n

v ∈ L

²

(Ω) : v|

_D

∈ P

_qD

(D) ∀ D ∈ T

_H

o

. (4.2b)

For convenience of notation, let V

_h,p⁰

:= V

H,q

. It follows from the above conditions on the meshes that every function v

H

∈ V

H,q

also belongs to V

h,p

, so let I

0

: V

H,q

→ V

h,p

denote the natural imbedding map. For 1 ≤ i ≤ N, let I

i

: V

_h,pⁱ

→ V

h,p

denote the natural injection operator defined by

I

i

v

i

:=

( v

i

on Ω

i

,

0 on Ω − Ω

i

, ∀ v

i

∈ V

h,pⁱ

. (4.3) Then, any function v

h

∈ V

h,p

can be decomposed as v

h

= P

N

i=1

I

i

v

h

|

_Ω

i

. Let the bilinear forms a

ⁱ_h

: V

_h,pⁱ

× V

_h,pⁱ

→ R , 0 ≤ i ≤ N, be defined by

a

ⁱ_h

(u

i

, v

i

) := a

h

(I

i

u

i

, I

i

v

i

) ∀ u

i

, v

i

∈ V

_h,pⁱ

. (4.4) It is clear that the bilinear forms a

ⁱ_h

are symmetric and coercive on V

_h,pⁱ

× V

_h,pⁱ

. For each 0 ≤ i ≤ N, let A

i

denote the matrix that corresponds to the bilinear form a

ⁱ_h

and let I

i

denotes the matrix corresponding to the injection operator I

i

. Therefore, for each 0 ≤ i ≤ N, the matrix A

i

has dimension dim V

_h,pⁱ

× dim V

_h,pⁱ

, and the matrix I

i

has dimension dim V

h,p

× dim V

_h,pⁱ

. Then, we define P

⁻¹_i

:= I

i

A

⁻¹_i

I

^>_i

, which therefore has dimension dim V

h,p

× dim V

h,p

. The additive Schwarz preconditioner P is defined in terms of its inverse by

P

⁻¹

:=

N

X

i=0

P

⁻¹_i

. (4.5)

Thus P

⁻¹

defines a symmetric positive definite preconditioner P that may be used as ex- plained in section 3. Further preconditioners, such as multiplicative, symmetric multiplica- tive and hybrid methods, are presented in [23, 25] and the references therein. The general theory of Schwarz methods [23, 25] simplifies the analysis of these preconditioners to the verification of three key properties.

Property 1 Suppose that there exists a positive constant c

0

such that each v

h

∈ V

h,p

admits a decomposition v

h

= P

N

i=0

I

i

v

i

, with v

i

∈ V

_h,pⁱ

, for each 0 ≤ i ≤ N, with

N

X

i=0

a

ⁱh

(v

i

, v

i

) ≤ c

0

a

h

(v

h

, v

h

). (4.6)

Property 2 Assume that there exist constants ε

ij

∈ [0, 1], such that

|a

_h

(I

i

v

i

, I

j

v

j

)| ≤ ε

ij

p a

h

(I

i

v

i

, I

i

v

i

) a

h

(I

j

v

j

, I

j

v

j

), (4.7) for all v

i

∈ V

_h,pⁱ

and all v

j

∈ V

_h,p^j

, 1 ≤ i, j ≤ N. Let ρ(E) denote the spectral radius of the matrix E := (ε

ij

).

Property 3 Suppose that there exists a constant ω ∈ (0, 2), such that

a

h

(I

i

v

i

, I

i

v

i

) ≤ ω a

ⁱh

(v

i

, v

i

) ∀ v

i

∈ V

h,pⁱ

, 0 ≤ i ≤ N. (4.8) Properties 1–3 are sometimes referred to respectively as the stable decomposition prop- erty, the strengthened Cauchy–Schwarz inequality, and local stability.

The following theorem from the theory of Schwarz methods is quoted from [25].

Theorem 2 If Properties 1–3 hold, then the condition number κ(P

⁻¹

A) obtained by the additive Schwarz preconditioner satisfies

κ(P

⁻¹

A) ≤ c

0

ω (ρ(E) + 1) . (4.9) Remark 2 With the above choices of bilinear forms a

ⁱ_h

and with the arguments presented in [4], it is seen that (4.8) holds in fact with equality for ω = 1. Also, in (4.7), we can take ε

ij

= 1 if ∂Ω

i

∩ ∂Ω

j

6= ∅, and ε

ij

= 0 otherwise. Therefore, as explained in [4], ρ (E) ≤ N

c

+ 1, where N

c

is the maximum number of adjacent subdomains that a given subdomain might have. Therefore, Properties 2 and 3 hold, and it remains to verify Property 1.

The following theorem determines a bound on the constant appearing in (4.6), which can be used in conjunction with Theorem 2 to analyse the properties of the preconditioners.

The proof of this result is given in the following sections.

Theorem 3 Let Ω ⊂ R

^d

, d ∈ {2, 3}, be a bounded convex polytopal domain, and let T

_S

, {T

_H

}

_H

and {T

_h

}

_h

be successively nested shape-regular sequences of meshes, with T

_S

conforming, and {T

_H

}

_H

and {T

_h

}

_h

satisfying (2.1), (2.2) and (2.3). Let µ

F

and η

F

satisfy (2.11) for each face F , with c

µ

and c

η

chosen to satisfy the hypothesis of Lemma 1. Then, each v

h

∈ V

h,p

admits a decomposition v

h

= P

N

i=0

I

i

v

i

, with v

i

∈ V

_h,pⁱ

, 0 ≤ i ≤ N, such that

N

X

i=0

a

ⁱ_h

(v

i

, v

i

) . ˜ c

0

a

h

(v

h

, v

h

), (4.10) where the constant c ˜

0

is given by

˜

c

0

:= 1 + max

D∈TH

q

D

H

D

K∈T

max

_h(D)

p

²_K

h

K

D∈T

max

H

H

_D²

q

_D²

+ max

D∈TH

q

D

H

D

K∈T

max

h(D)

p

⁶_K

h

³_K

D∈T

max

H

H

_D⁴

q

⁴_D

. (4.11) It follows from Theorems 2 and 3 that the condition number satisfies

κ(P

⁻¹

A) . ˜ c

0

(N

c

+ 2) , (4.12)

where c ˜

0

is given in (4.11) above, and N

c

is the maximum number of adjacent subdomains

that a given subdomain from T

_S

might have. Thus the condition number does not depend on

the number N of subdomains, but may depend on the maximum number of neighbours any

subdomain possesses, denoted by N

c

in (4.12). If the sequence of coarse spaces {V

_H,q

}

_H

satisfy the assumption that H

D

/q

D

. min

D∈TH

H

D

/q

D

for all D ∈ T

H

, then the constant

˜

c

0

in the above proposition simplifies to

˜

c

0

' 1 + max

D∈TH

H

D

q

D

K∈T

max

h(D)

p

²_K

h

K

+ H

_D³

q

_D³

max

K∈Th(D)

p

⁶_K

h

³_K

. (4.13)

Moreover, if the sequences of meshes {T

H

}

H

and {T

h

}

h

are quasiuniform, and if the poly- nomial degrees are also quasiuniform in the sense that q := max

D

q

D

. q

D

for all D ∈ T

H

and p := max

_K

p

K

. p

K

for all K ∈ T

_h

, then the condition number of the preconditioned system satisfies the bound

κ(P

⁻¹

A) . (N

c

+ 2)

1 + p

²

H

q h + p

⁶

H

³

q

³

h

³

. (4.14)

It is well-known that the above bound is optimal in terms of the powers of H and h, see [7, 11]. The numerical experiments of section 7 show that the bound (4.14) is also sharp in terms of the orders of p and q. Choosing the coarse space such that H ' h and q ' p implies that κ(P

⁻¹

A) . p

³

, which shows that the preconditioner is robust with respect to h but not with respect to p. The explicit dependence of our bound on q shows nonetheless a significant improvement over the condition number of order p

⁸

/h

⁴

for the unpreconditioned matrix.

The preconditioner P can therefore be used to precondition the nonsymmetric systems (3.7) of the semismooth Newton method, where the convergence is guaranteed by Theorem 1 in combination with (4.14).

5 Approximation of discontinuous functions

As explained in the introduction, the optimal bound for the condition numbers, as given by Theorem 3, rests upon the optimality of approximation properties between coarse and fine spaces. Therefore, in this section, we first determine how closely a function in V

h,p

can be approximated by functions in H

²

(Ω) ∩ H

0¹

(Ω). This leads to an approximation result for functions in V

h,p

by functions in V

H,q

that is of optimal order in both the coarse mesh size and polynomial degree.

5.1 Lifting operators

Let V

_h,p^d

denote the space of d-dimensional vector fields with components in V

h,p

. Let r

h

: L

²

(F

_h^i,b

) → V

_h,p^d

and r

h

: L

²

(F

_hⁱ

) → V

h,p

be defined by

X

K∈Th

hr

h

(w), v

h

i

K

= X

F∈F_h^i,b

hw, {v

h

· n

F

}i

F

∀ v

h

∈ V

h,p^d

, (5.1) X

K∈Th

hr

_h

(w), v

h

i

_K

= X

F∈F_hⁱ

hw, {v

_h

}i

_F

∀ v

h

∈ V

h,p

. (5.2)

The following result is well-known; for instance, see [4] for a proof.

Lemma 2 Let Ω be a bounded Lipschitz domain and let {T

_h

}

_h

be a shape-regular sequence of meshes satisfying (2.1), (2.2) and (2.3). Then, the lifting operators satisfy the following bounds:

X

K∈Th

kr

h

(w)k

²_L2(K)

. X

F∈F_h^i,b

˜ p

²_F

˜ h

F

kwk

²_L2(F)

∀ w ∈ L

²

(F

_h^i,b

), (5.3a)

X

K∈Th

kr

_h

(w)k

²_L2(K)

. X

F∈F_hⁱ

˜ p

²_F

˜ h

F

kwk

²_L2(F)

∀ w ∈ L

²

(F

_hⁱ

). (5.3b)

For v

h

∈ V

h,p

and v

h

∈ V

_h,p^d

, define G

h

(v

h

) ∈ V

_h,p^d

and D

h

(v

h

) ∈ V

h,p

element-wise by

G

h

(v

h

)|

_K

:= ∇v

_h

|

_K

− r

h

( _J v

h

K )|

_K

, (5.4a) D

h

(v

h

)|

_K

:= div v

h

|

_K

− r

h

( _J v

h

· n

F

K )|

_K

, (5.4b) for all K ∈ T

_h

. Observe that D

h

(v

h

) belongs to L

²

(Ω) for any v

h

∈ V

_h,p^d

.

Lemma 3 Let Ω be a bounded Lipschitz polytopal domain, and let {T

_h

}

_h

be a shape- regular sequence of meshes satisfying (2.1), (2.2) and (2.3). Let η

F

and µ

F

satisfy (2.11) for all F ∈ F

_h^i,b

. Then, for any v

h

∈ V

h,p

, we have

X

K∈Th

p

⁴_K

h

²_K

kr

h

( _J v

h

K )k

²_L2(K)

. |v

h

|

²_J,h

, (5.5a) X

K∈Th

|r

h

( _J v

h

K )|

²_H¹_(K)

+ X

F∈F_hⁱ

µ

F

k _J r

h

( _J v

h

K ) · n

F

K k

²_L²_(F)

. |v

h

|

²_J,h

. (5.5b) Proof The definition of the lifting operator gives

X

K∈T_h

p

⁴_K

h

²_K

kr

_h

( _J v

h

K )k

²_L2(K)

= X

F∈F_h^i,b

Z

F

J v

h

K p

⁴

h

²

r

h

( _J v

h

K ) · n

F

ds

.



 X

F∈F_h^i,b

˜ h

³_F

˜ p

⁶_F

˜ p

⁸_F

˜ h

⁴_F

kr

_h

( _J v

h

K )k

²_L2(F)





1 2

|v

_h

|

_J,h

.

The trace and inverse inequalities then yield X

K∈Th

p

⁴_K

h

²_K

kr

_h

( _J v

h

K )k

²_L2(K)

. X

K∈Th

p

⁴_K

h

²_K

kr

_h

( _J v

h

K )k

²_L2(K)

!

¹

2

|v

_h

|

_J,h

,

which implies (5.5a). The bound (5.5b) then follows from (5.5a) as a result of the trace and

inverse inequalities. u t

Corollary 1 Under the hypotheses of Lemma 3, every v

h

∈ V

h,p

satisfies X

K∈Th

|G

_h

(v

h

)|

²_H1(K)

+ X

F∈F_hⁱ

µ

F

k _J G

h

(v

h

) · n

F

K k

²_L2(F)

. kv

_h

k

²_2,h

. (5.6)

We also have kD

_h

(G

h

(v

h

))k

_L2(Ω)

. kv

_h

k

_2,h

for every v

h

∈ V

h,p

.

Proof Inequality (5.6) is an easy consequence of the definition of G

h

in (5.4a) and of Lemma 3. For v

h

∈ V

h,p

and K ∈ T

h

, we have

D

h

(G

h

(v

h

))|

_K

= [∆v

h

− div r

h

( _J v

h

K ) − r

h

( _J ∇v

_h

· n

F

K ) + r

h

( _J r

h

( _J v

h

K ) · n

F

K )] |

_K

. (5.7) In view of (5.3b), it is apparent that the global L

²

-norms over Ω of the first and third terms on the right-hand side of (5.7) are bounded by kv

_h

k

_2,h

, whilst the bounds on the L

²

-norms

of the second and fourth terms follow from (5.5b). u t

5.2 Approximation by H

²

-regular functions

The first step towards the aforementioned approximation result is to consider the discrete analogue of the orthogonality of Helmholtz decompositions. In this section, we shall view the element-wise gradient of a function v

h

∈ V

h,p

as an element of L

²

(Ω)

^d

, and thus we denote it by ∇

_h

v

h

.

Lemma 4 Let Ω ⊂ R

^d

, d ∈ {2, 3}, be a bounded Lipschitz polytopal domain, and let {T

_h

}

_h

be a shape-regular sequence of meshes satisfying (2.1), (2.2) and (2.3). If µ

F

and η

F

satisfy (2.11) for every face F ∈ F

_h^i,b

, then, for any v

h

∈ V

h,p

and any ψ ∈ H

¹

(Ω)

^2d−3

, we have

Z

Ω

G

h

(v

h

) · curl ψ dx

+ Z

Ω

∇

_h

v

h

· curl ψ dx . max

K∈Th

h

K

p

^3/2_K

|v

_h

|

_J,h

kψk

_H¹_(Ω)

. (5.8) Proof It follows from (5.5a) that k∇

_h

v

h

−G

_h

(v

h

)k

_L²_(Ω)

. max

K

h

K

/p

²_K

|v

_h

|

_J,h

, so it is enough to show that (5.8) is satisfied by G

h

(v

h

). Consider momentarily ψ ∈ H

²

(Ω)

^2d−3

; then, integration by parts yields

Z

Ω

G

h

(v

h

) · curl ψ dx = X

F∈F_h^i,b

h _J v

h

K , {curl ψ · n

F

}i

_F

− X

K∈Th

hr

_h

( _J v

h

K ), curl ψi

_K

. Therefore, the definitions of the lifting operators r

h

and r

h

imply that

Z

Ω

G

h

(v

h

) · curl ψ dx = X

F∈F_h^i,b

h _J v

h

K , {curl(ψ − Π

h

ψ) · n

F

}i

_F

− X

K∈Th

hr

h

( _J v

h

K ), curl(ψ − Π

h

ψ)i

K

. Thus, if ψ ∈ H

²

(Ω)

^2d−3

, it is seen from the approximation bounds of (2.12) and from the lifting bound (5.5a) that

Z

Ω

G

h

(v

h

) · curl ψ dx . max

K∈Th

h

²_K

p

³_K

|v

_h

|

_J,h

kψk

_H2(Ω)

. (5.9) Now, let ψ ∈ H

¹

(Ω)

^2d−3

. We apply [1, Thm. 5.33] to the components of ψ: for each ε > 0, there exists a ψ

ε

∈ C

^∞

( R

^d

)

^2d−3

such that

kψ − ψ

ε

k

_L²_(Ω)

+ εkψ − ψ

ε

k

_H¹_(Ω)

. ε|ψ|

_H¹_(Ω)

, (5.10a)

kψ

_ε

k

_H2(Ω)

. ε

⁻¹

kψk

_H1(Ω)

, (5.10b)