Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations

Vol. 40, N^o 5, 2006, pp. 923–937 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006040

MAXIMUM-NORM RESOLVENT ESTIMATES FOR ELLIPTIC FINITE ELEMENT OPERATORS

ON NONQUASIUNIFORM TRIANGULATIONS

Nikolai Yu. Bakaev

, Michel Crouzeix

and Vidar Thom´ ee

Abstract. In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under weaker conditions on the triangulations than quasiuniformity. In the two-dimensional case, the bound for the resolvent contains a logarithmic factor.

Mathematics Subject Classification. 65M06, 65M12, 65M60.

Received: May 2, 2006.

Introduction

Consider the initial-value problem

u_t−∆u= 0 in Ω andu= 0 on∂Ω, fort >0, withu(·,0) =v in Ω, (0.1) where Ω is a domain in R², and denote byE(t) the solution operator related to this problem and defined by u(t) =E(t)v. Then it is a special case of a result of Stewart [11] that if∂Ω is smooth, then E(t) is an analytic semigroup on C0( ¯Ω) ={v∈ C( ¯Ω) : v= 0 on∂Ω}generated by ∆. This follows from the resolvent estimate

(λI+ ∆)⁻¹vC ≤ C

1+|λ|vC, forv ∈ C0( ¯Ω) andλ /∈Σ_δ ={λ: |argλ| ≤δ}, (0.2)

Keywords and phrases.Resolvent estimates, stability, smoothing, maximum-norm, elliptic, parabolic, finite elements, nonquasi- uniform triangulations.

1 Department of Economic Dynamics, EAI, Moscow Engineering Physics Institute (State University), Kashirskoe Shosse 31, Moscow 115409, Russia. [email protected]

2 IRMAR, Universit´e de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France. [email protected]

3 Department of Mathematics, Chalmers University of Technology, 41296 G¨oteborg, Sweden. [email protected] c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006040 Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006040

wherevC = sup_x∈Ω|v(x)|and whereδ∈(0,¹₂π) is arbitrary. In addition to the stability estimateE(t)vC ≤ vC, which follows by the maximum-principle, this entails the smoothing estimate

E(t)vC ≤C

tvC, forv∈ C0( ¯Ω).

Such a result is valid also under lesser regularity requirements on∂Ω,cf. Ouhabaz [8].

In this paper, we are interested in maximum-norm estimates for spatially semidiscrete approximations of parabolic problems such as (0.1) based on continuous, piecewise polynomial finite elements of degreer−1≥1.

LetTh={τ}denote a family of closed face-to-face triangles in ¯Ω with mutually disjoint interiors, with diameter h_τ, and seth= max_τ∈Th diam (τ). Let Ω_hbe the interior of the set∪{τ:τ∈ Th}and assume that Ω_h⊆Ω. If Ω is a polygonal domain it is natural to chooseTh so that Ω_h= Ω.

We consider, in fact, a whole family of such triangulations{Th} and assume that this is a regular family of triangulations in the sense that h_τ/d_τ ≤C for allτ ∈ Th,where d_τ is the radius of the largest disc contained in τ. We associate withTh the finite dimensional spaces

S_h={χ∈ C( ¯Ω) : χ|τ ∈P_r−1 forτ ∈ Th, χ= 0 on∂Ω∪(Ω\Ω_h)}, whereP_k denotes the set of polynomials of degreek.

The semidiscrete finite element problem associated with (0.1) is then to findu_h(t)∈S_h fort >0 such that, withv_h∈S_hgiven,

(u_h,t, χ) + (∇u_h,∇χ) = 0 forχ∈S_h, t >0, (0.3) u_h(·,0) =v_h in Ω, where (v, w) =

Ωv(x)w(x) dx.

With−∆h:S_h→S_h defined by

−(∆hψ, χ) = (∇ψ,∇χ), ∀ψ, χ∈S_h, this problem may also be written

u_h,t−∆_hu_h= 0, fort >0, withu_h(0) =v_h.

The solution operator of this problem, defined by u_h(t) = E_h(t)v_h, is the semigroup E_h(t) = e^∆ht in S_h generated by ∆_h. The issue is then to show that this semigroup is analytic inS_h, equipped with the maximum- norm, and this may be expressed either as a resolvent estimate for−∆h or as the stability and a smoothing property ofE_h(t).

In Schatz et al. [9] it was thus shown in the case of a convex domain Ω with smooth boundary, and for quasiuniform piecewise linear finite elements (r= 2) that, with_h= max(1,log(1/h)),

E_h(t)v_hC+tE_h(t)v_hC ≤C_hv_hC, forv_h∈S_h. (0.4) Using semigroup theory this shows the resolvent estimate (cf. [12], Lem. 8.7)

(λI+ ∆_h)⁻¹v_hC ≤ C²_h

1+|λ|vh_C, forλ /∈Σ_δh, whereδ_h=¹₂π−c⁻²_h . (0.5) In Schatzet al. [10] the logarithmic factor in (0.4) was removed, which implies that the resolvent estimate (0.5) holds without a logarithmic factor as well, and for λ∈Σ_δ,for some δ∈(0,¹₂π) independent of h. In Bakaev et al. [3] a direct proof was given that this resolvent estimate holds for any angleδ∈(0,¹₂π). The result in [3]

holds for Ω inR^dwithd≥2 arbitrary, with∂Ω smooth. In Chatzipantelidiset al. [4] such a resolvent estimate, with a logarithmic factor, was shown when Ω is a plane polygonal domain, which may be nonconvex. For some earlier workcf., e.g., [1, 2, 7].

In all these results quoted the family of triangulations is required to be quasiuniform, which is a somewhat undesirable restriction. Our purpose in this paper is therefore to weaken this condition. The technique of proof will depend heavily on Crouzeix and Thom´ee [5], where the stability of theL₂-projection ontoS_h was studied under milder assumptions on the triangulations than quasiuniformity.

An earlier attempt to treat this problem was made in Crouzeix and Thom´ee [6] where a resolvent estimate of the desired type, with a logarithmic factor, was shown for a modified discrete Laplacian, defined by

−(∆hψ, χ)_h= (∇ψ,∇χ), ∀ψ, χ∈S_h,

where (·,·)_h denotes a simple quadrature approximation of the L₂-inner product, and for triangulations of Delaunay type, not required to be quasiuniform.

We now introduce some notation. Following [5], givenτ₀∈ Th, we letQ_j(τ₀) denote the set of triangles which are “j triangles away fromτ₀”, defined by settingQ₀(τ₀) =τ₀and then, recursively, forj ≥1,Q_j(τ₀) to be the union of the closed triangles τ which are not in

i<jQ_i(τ₀), but which have at least one vertex in Q_j−1(τ₀).

We further set l(τ₀, τ) =j forτ∈Q_j(τ₀) and denote byn_j(τ₀) the number of triangles inQ_j(τ₀).

In what follows we shall use the following auxiliary result from [5] showing the exponential decay property of theL₂-projectionP_h which was used to show the maximum-norm stability of this operator:

Lemma 0.1. There exist C >0 andγ=γ_r∈(0,1)such that, for all τ, τ₀∈ Th andv∈L₂, with supp v∈τ₀, Phv_L2(τ)≤Cγ^l(τ,τ0)vL2.

In [5] it was shown that one can choose,e.g.,γ₂= 0.318, γ₃= 0.376, γ₄= 0.353.

We now make the assumption that the family {Th}of triangulations satisfies, with some α≥1 andβ≥1, h_τ/h_τ0≤Cα^l(τ,τ0), for allτ, τ₀∈ Th, (0.6) and

n_j(τ)≤Cβ^j, j≥1, for allτ ∈ Th. (0.7) For quasiuniform triangulations this holds with α= 1 andβ any number >1, and if (0.6) holds with α >1, we may chooseβ=α⁴in (0.7).

Under these assumptions we show that if the above conditions on{Th} hold, with (0.6) and (0.7), and if

α²βγ <1, (0.8)

withγ as in Lemma 0.1, then, for any fixedδ∈(0,¹₂π), we have (λI+ ∆_h)⁻¹χ_C ≤ C^1/2_h

1 +|λ|, ∀χ∈S_h, λ /∈Σ_δ. (0.9) Here and below we write _h = max(1,log(1/h_min)), whereh_min = min_τ∈Thh_τ. For example, for r= 2, with β = α⁴, the condition (0.8) requires α < γ₂^−1/6 = (0.318)^−1/6 ≈ 1.21, which permits a substantial degree of nonquasiuniformity.

We note that the L₂-projection P_h : L₂ → S_h is stable in maximum-norm if αβγ < 1, thus in particular when condition (0.8) holds. This was shown in [5] in the case of a polygonal domain Ω, with Ω_h= ¯Ω, but the proof is valid under our present assumptions.

It follows from (0.9) by standard semigroup theory that, under our present assumptions onTh, the solution operatorE_h(t) of (0.3) satisfies the stability and smoothing estimate (0.4), with the factor_h replaced by^1/2_h .

The resolvent estimate (0.9) will be shown in Section 3 below, in which the Laplacian is replaced by a more general second order elliptic operator. We begin in the next Section 2 by considering a spatially one-dimensional elliptic operator. In this case we shall show the corresponding resolvent estimate without the logarithmic factor.

1. The one-dimensional case

In this section we consider the one-dimensional elliptic operator

Au=−(au)+bu+cu, in Ω = (0,1),

witha, b, cbounded real-valued functions, witha(x)≥a₀>0 on Ω. We introduce the sesquilinear form A(u, w) =

₁

0 (auw¯+buw¯+cuw) dx.¯ (1.1) It is then an easy matter to show that there exist constantsc₀>0, c₁, c₂, c₃∈Rsuch that

c₀w²−c₁w²≤ReA(w, w)≤c₂w² and |ImA(w, w)| ≤c₃w w, ∀w∈H₀¹. (1.2) Here . denotes the usual L₂-norm on Ω. With the sesquilinear form (1.1) we associate its numerical range W(A)⊂Cdefined by

W(A) ={A(w, w); w∈H₀¹, w= 1}. (1.3)

From the previous assumptions we may write A(w, w) = x+ iy for w = 1, where x ≥ c₀w²−c₁ and

|y| ≤c₃w. Therefore

W(A)⊂ P={z=x+iy∈C; x≥c₀c⁻²₃ y²−c₁}, (1.4) e.g., the numerical range ofAis included in the horizontal parabolic domainP.

We consider now a closed subset Σ⊂Cof the complex plane such that

d(λ,P)≥c(1+|λ|), for allλ∈Σ, where c >0. (1.5) For instance, we can choose for Σ the complement of any open sector containing P. When A is selfadjoint positive definite,P is a subset of the positive real axis, and Σ may be chosen as the complement of any sector Σ_δ as defined in (0.2).

Let 0 =x₀< x₁<· · ·< x_N+1= 1 be a partition of Ω into subintervalsI_j = (x_j, x_j+1) and leth_j =x_j+1−xj. We assume

h_i/h_j≤Cα^|i−j|, withα≥1. (1.6) LetS_h={χ∈ C₀(Ω) : χ|Ij ∈P_r−1, j= 0, . . . , N}, whereP_k denotes the set of polynomials of degree≤k, and defineA_h: S_h→S_hby

(A_hψ, χ) =A(ψ, χ), ∀ψ, χ∈S_h. The following is then the main result in this section.

Theorem 1. Under the above assumptions, with 1≤α < r, we have (λI−Ah)⁻¹v_hC ≤ C

1+|λ|vhC, ∀λ∈Σ, v_h∈S_h. Proof. We introduce, forx∈Ω, the adjoint discrete Green’s function

G^x_h(y,¯λ) = ((¯λI−A^∗_h)⁻¹δ^x_h)(y), forλ∈Σ,

whereδ_h^x∈S_his the discrete delta-function defined by

(χ, δ_h^x) =χ(x), ∀χ∈S_h. It is easy to see that

((λI−Ah)⁻¹χ)(x) = (χ, G^x_h(., λ)), ∀χ∈S_h,

and in order to prove Theorem 1 it suffices to show that, withC independent ofxand λ, G^x_h(.,λ)¯ L1 ≤ C

1+|λ|, forλ∈Σ. (1.7)

The following will be a basic tool.

Lemma 1.1. There is a constant C=C_Σ such that, for v∈H₀¹ andλ∈Σ,

if λv²−A(v, v) =F, then (1+|λ|)v²+v²≤C|F|.

Proof. We first note that, sinceA(v, v)/v²∈W(A), we have

d(λ,P)v²≤λ−A(v, v)/v²v²=|F|.

By (1.5) this shows

(1+|λ|)v²≤C|F|.

The conclusion of the lemma follows since, by (1.2) and the triangle inequality,

c₀v²≤ReA(v, v) +c₁v²≤ |F|+ (c₁+ Reλ)v²≤C|F|. We note that, with G=G^x_h(·,λ) for¯ x∈Ω, λ∈Σ, we have

λ(χ, G)−A(χ, G) = (χ, δ_h^x) =χ(x), ∀χ∈S_h. (1.8) Choosingχ=Gand using Lemma 1.1 we obtain

(1+|λ|)G²+G²≤CG_C ≤CG^1/2G^1/2. (1.9) Using the inequalityxy≤¹₄x⁴+³₄y^4/3 to bound the right hand side, we find

(1+|λ|)G²+G²≤ ¹₂G²+CG^2/3, (1.10) and hence

G ≤ C

(1+|λ|)^3/4 and G ≤ C

(1+|λ|)^1/4· (1.11)

SinceGL1 ≤ Gthis implies (1.7) for λbounded.

For treating large values ofλ∈Σ we use the weight function

ρ(y) =ρ^x_h(y) = ((x−y)²+h²_x)^1/2, whereh_x:=h_j ifx∈[x_j, x_j+1).

We consider the expression

λρG²−A(ρG, ρG) =λ(ρ²G, G)−A(ρ²G, G)−R(G, G),

where

R(G, G) =A(ρG, ρG)−A(ρ²G, G), or, after subtraction of (1.8) withχ=P_h(ρ²G),

λρG²−A(ρG, ρG) =F, where

F =−A(ρ²G−P_h(ρ²G), G)−(ρ²G, δ)−R(G, G). (1.12) By Lemma 1.1 this implies

(1+|λ|)ρG²+(ρG)²≤C|F|. (1.13)

The proof of the bound needed for the right hand side will be based on several lemmas. The first one is a one-dimensional analogue of Lemma 0.1.

Lemma 1.2. There existsC >0 such that, for allv∈L₂ with supp(v)∈I_l,

PhvL2(Ij)≤Cγ^|j−l|v, for all j, l, whereγ=γ_r= 1/r. (1.14) Proof. We recall some material from [5]. First we introduce the spacesS_h²={χ∈S_h; χ(x_j) = 0, j= 1, . . . , N} andS_h¹, the orthogonal complement ofS_h² inS_h with respect to the inner product inL₂(Ω). Forr= 2 we have S_h²={0}andS_h¹=S_h. We also introduce the orthogonal projectionπ_j ontoS_h^j, j= 1,2, and obtain at once

P_h=π₁+π₂. Recall that π₂ is determined locally on eachI_j by the equations

(π₂w, q)_L2(Ij)= (w, q)_L2(Ij), for allq∈P_r−1 withq(x_j) =q(x_j+1) = 0.

Thus, since supp(v)⊂I_l, we have, since thenπ₂v|Ij = 0, that

Phv_L2(Ij)=π1v_L2(Ij) ifj=l, and also

Phv_L2(Il)≤ Phv ≤ v.

To show (1.14) it therefore suffices to consider the casej=l.

We now consider the functionsψ_i, i= 1, . . . , N,defined byψ_i ∈S_h¹andψ_i(x_j) =δ_ij forj= 1, . . . , N. Recall from Lemma 2 of [5] that supp(ψ_j) =I_j−1∪I_j, and that these functions constitute a basis for S_h¹with

ψ_i+1²_L2(Ii)=ψ_i²_L2(Ii)= h_i

r²−1, ψ_i²=h_i−1+h_i

r²−1 and (ψ_i, ψ_i+1) = (−1)^r h_i r(r²−1)· Now if we setπ₁v=_N

i=1w_iψ_i, we have, withw₀=w_N+1= 0,

(ψ_i−1, ψ_i)w_i−1+ψi²w_i+ (ψ_i+1, ψ_i)w_i+1= (v, ψ_i), fori= 1, . . . , N.

After division of theith equation byψ_i², this linear system can be written as

(I+K)W =F := (f₁, . . . , f_N)^T, withf_i = (v, ψ_i)/ψ_i², (1.15)

where W = (w₁, . . . , w_N)^T and where we note that f_i = 0 fori = l, l+ 1. Here K = (k_ij) is the tridiagonal N×N matrix with diagonal entriesk_ii= 0 and bidiagonal elements

k_i,i−1=(ψ_i, ψ_i−1)

ψ_i² = (−1)^r r

h_i−1

h_i−1+h_i and k_i,i+1= (ψ_i, ψ_i+1)

ψ_i² = (−1)^r r

h_i h_i−1+h_i· We now introduce the norms

Wp= ^N

i=1

(h_i+1+h_i)|wi|^p_1/p

, for 1≤p <∞, withW_∞= max

i |wi|,

and also denote by · p the matrix operator norms induced by these vector norms. In particular we have K_∞= max_i

j|kij|= 1/r, and noticing thatDKD⁻¹=K^T, whereD= diag(h₀+h₁, h₁+h₂, . . . , h_N−1+h_N), we then also obtainK1= 1/r. From the Riesz-Thorin interpolation theorem we deduce thatKp≤1/rfor allpwith 1≤p≤ ∞.

We now introduce the projection P_j : C^N → C^N defined by (P_jW)_i =w_i ifi = j−1 or i = j, and = 0 otherwise. Using (1.15) and the (2s+1)-diagonal character ofK^s we find

P_jW =

s≥|j−l|−1

(−1)^sP_jK^sF,

and therefore

PjW₂≤

s≥|j−l|−1

K^s₂F₂≤ 1 r^|j−l|

r² r−1F₂. Simple calculations using (1) give

π₁v²_L2(Ij)≤ |wj|²ψj²_L2(Ij)+|w_j+1|²ψ_j+1²_L2(Ij)+ 2|wj||w_j+1||(ψ_j+1, ψ_j)|

≤ h_j r²−1

1 +1

r (|wj|²+|w_j+1|²)≤ 1

r(r−1)PjW². To boundF2, we note that

|fi|²≤ v²ψ_i²_L2(Il)

ψi⁴ = (r²−1)v² h_l

(h_i−1+h_i)², fori=l, l+ 1, and hence

F²₂= (h_l−1+h_l)|fl|²+ (h_l+h_l+1)|fl+1|²≤(r²−1)v² h_l

h_l−1+h_l+ h_l+1 h_l+h_l1

≤2(r²−1)v².

Altogether we obtain

π₁v_L2(Ij)≤(r(r−1))^−1/2PjW ≤r^−|j−l|r^3/2(r−1)^−3/2F₂

≤r^−|j−l|r^3/2(r+ 1)^1/2(r−1)⁻¹√

2v=C_rr^−|j−l|v,

which completes the proof.

A version of the following Lemma was shown in the quasiuniform case in [13].

Lemma 1.3. Under the assumption(1.6) we have

ρ δ^x_h ≤Ch^1/2_x , for x∈Ω.

Proof. Letx∈[x_j, x_j+1), recall that h_x =h_j. Then for any ϕ∈ C₀^∞(I_l) with ϕ= 1 we have, using a local inverse estimate onI_j and Lemma 1.2,

(δ^x_h, ϕ) = (δ_h^x, P_hϕ) = (P_hϕ)(x)≤Ch^−1/2_j Phϕ_L2(Ij)≤Ch^−1/2_j r^−|l−j|. Hence

δ^x_hL2(Il)= sup

ϕ∈C₀^∞(Il) ϕ=1

(δ^x_h, ϕ)≤Ch^−1/2_j r^−|l−j|. Fory∈I_lwe also have, by (1.6),

ρ(y)²=|y−x|²+h²_j ≤ C

|l−j|

s=0

α^sh_{j 2}

+h²_j ≤C(|l−j|+ 1)²α^2|l−j|h²_j. (1.16) Hence, sinceα/r <1,

ρδ_h^x2≤ N l=1

sup

Il

ρ(y)²δ^x_h²_L2(Ij)≤C N

l=1

(|l−j|+ 1)²α^2|l−j|h_jr^−2|l−j|≤C h_j

s≥0

(s+1)²(α/r)^2s=Ch_j,

which shows the Lemma.

Lemma 1.4. Under the assumptions from the beginning of this section we have

|R(G, G)| ≤ C

(1+|λ|)^3/2 + C(ρG)

(1+|λ|)^3/4, for x∈Ω, λ∈Σ.

Proof. We find at once

R(G, G) =A(ρG, ρG)−A(ρ²G, G) = (aρG, ρG)−(bρG, ρG)−2 Im (aρG,(ρG)), and hence, since|ρ| ≤1,

|R(G, G)| ≤CG²+CG (ρG).

The Lemma now follows by (1.10).

Lemma 1.5. Under the assumptions from the beginning of this section we have

|A(ρ²G−P_h(ρ²G), G)| ≤ C

(1+|λ|)^1/4ρG, for x∈Ω, λ /∈Σ_δ.

Proof. We setζ=ρ²G−R_h(ρ²G) whereR_h is theH₀¹-projection ontoS_h. Thenρ²G−P_h(ρ²G) = (I−P_h)ζ.

We have from Theorem 2 in [5]

(ρ²G−P_h(ρ²G))=((I−P_h)ζ) ≤Cζ. Therefore

|A(ρ²G−P_h(ρ²G), G)| ≤C(ρ²G−P_h(ρ²G)) G ≤ C

(1+|λ|)^1/4ζ. (1.17)

It is well known that, since we are in the one-dimensional case, R_hu(x_i) =u(x_i) for all i. We consider now a subintervalI_j and setρ_j=ρ(x_j). Noting that ρ/ρ_j is bounded above and below onI_j we have

ζ_L2(Ij)=((I−R_h)((ρ²−ρ²_j)G))_L2(Ij)≤Cρ_jG_L2(Ij)+Cρ_jh_jG_L2(Ij)≤Cρ G_L2(I). Taking square and summing, this shows

ζ ≤CρG.

In view of (1.17) this completes the proof.

To continue the proof of Theorem 1 we setµ= (1+|λ|)^−1/2 and obtain, using Lemmas 1.3, 1.4 and 1.5, for F defined in (1.12),

|F| ≤ |A(ρ²G−P_h(ρ²G), G)|+ρG ρδ+|R(G, G)| ≤C(µ^1/2+h^1/2_x )ρ G+Cµ³+Cµ^3/2(ρG). Using (1.13) we deduce

µ⁻²ρ G²+(ρG)²≤C(µ^1/2+h^1/2_x )ρ G+Cµ³+Cµ^3/2(ρG)

≤¹₂µ⁻²ρ G²+Cµ²(µ+h_x) +Cµ³+(ρG)². Therefore

ρ G ≤Cµ²(h_x+µ)^1/2. Using the estimate (1.11) we obtain

(ρ+µ)G ≤Cµ²(h_x+µ)^1/2. Noting that

(ρ+µ)⁻¹²≤2 ₁

0

dy

y²+h²_x+µ² ≤C(h_x+µ)⁻¹, we finally have

GL1≤ (ρ+µ)⁻¹ (ρ+µ)G ≤Cµ²=C(1+|λ|)⁻¹,

which completes the proof.

As a consequence of Theorem 1 we may conclude that−Ah generates an analytic semigroupE_h(t) = e^−Aht, the solution operator of the semidiscrete problem

(u_h,t, χ) +A(u_h, χ) = 0 forχ∈S_h, t >0, withu_h(·,0) =v_h in Ω,

associated with the parabolic equation with elliptic operatorA, and that stability and smoothing estimates as in (0.4) hold, this time without the logarithmic factor_h, but with an exponentially growing factor e^c1tifc₁>0 in (1.2).

2. The two-dimensional case

In this section we consider the elliptic operator

Au=−div (a∇u) +b· ∇u+cu, in Ω⊂R², (2.1) witha, b, cbounded real-valued, anda(x)≥a₀>0 in Ω. This time we set

A(u, w) =

Ω(a∇u· ∇w¯+b· ∇uw¯+cuw) dx,¯

and note that there arec₀>0, c₁, c₂, c₃∈Rsuch that

c₀∇w²−c₁w²≤ReA(w, w)≤c₂∇w² and |ImA(w, w)| ≤c₃∇w w, ∀w∈H₀¹.

The numerical range W(A) is defined as in (1.3), and again (1.4) holds. As earlier we choose a closed subset Σ⊂Csuch that Σ∩ P=∅andd(λ,P)≥c(1+|λ|) forλ∈Σ.

We now consider triangulationsThand the corresponding finite dimensional spacesS_hconsisting of piecewise polynomials of degreer−1≥1, as defined in Introduction. We shall show the following resolvent estimate for the discrete versionA_h: S_h→S_h of the operatorA in (2.1).

Theorem 2. Let the conditions onΩ and{Th} from the introduction hold, in particular (0.6) and (0.7)with someα, β≥1, and let

α²βγ <1, (2.2)

with γ=γ_r as in Lemma0.1. Then we have

(λI−A_h)⁻¹v_hC≤ C^1/2_h

1 +|λ|vh_C, ∀ v_h∈S_h, λ∈Σ, (2.3) where, as above, _h= max(1,log(1/h_min))with h_min = min_τ∈Thh_τ.

Forx∈Ω fixed we will use the adjoint discrete Green’s function

G^x_h(y,¯λ) = ((¯λI−A^∗_h)⁻¹δ_h^x)(y) forλ∈Σ, whereδ_h^x∈S_his the discrete delta-function defined by

(χ, δ^x_h) =χ(x), ∀χ∈S_h. As in Section 1 we have

((λI−Ah)⁻¹χ)(x) = (χ, G^x_h(., λ)), ∀χ∈S_h, and to prove the Theorem it suffices to show

G^x_h(.,¯λ)L1≤ C^1/2_h

1+|λ|, forλ∈Σ, x∈Ω. (2.4)

We obtain in the same way as for Lemma 1.1.

Lemma 2.1. There is a constant C=C_Σ such that, for v∈H₀¹ andλ∈Σ,

if λv²−A(v, v) =F, then (1+|λ|)v²+∇v²≤C|F|.

We note that, writing for brevityG=G^x_h(·,¯λ) forx∈Ω, λ∈Σ,

λ(χ, G)−A(χ, G) = (χ, δ_h^x) =χ(x), ∀χ∈S_h. (2.5) Choosingχ=Gand using Lemma 2.1 we obtain

(1+|λ|)| G²+∇G²≤C|G(x)| ≤CGC ≤C^1/2_h ∇G. (2.6) This yields, withµ:= (1+|λ|)^−1/2,

∇G ≤C^1/2_h and G ≤Cµ ^1/2_h . (2.7)

Remark. It appears that the continuous Green’s functiong=g^x(.,¯λ) satisfies∇g=∞andg ≤C µ. For A=−∆ this can be shown by an argument which starts with an explicit formula for the Green’s function when Ω =R², and the conclusion should hold also for more general operatorsA. Thus the first estimate in (2.7) may be considered as satisfying, but an improvement of the second estimate to G ≤Cµ might be possible and would show Theorem 2 without a logarithmic factor.

Now we will deduce the L₁ estimate (2.4) from the estimate of G. For λ bounded this follows directly from the inequality GL1 ≤CG. We now turn to larger values ofλ∈Σ. With the given pointx∈Ω_h we associate a triangle τ (arbitrarily ifxis on an edge) such thatx∈τ and seth_x=h_τ. We then use the weight function

ρ(y) =ρ^x_h(y) = (|x−y|²+h²_x)^1/2, and note thatρ²is a quadratic polynomial. We have

G_L1 ≤ (ρ²+µ²)⁻¹ (ρ²+µ²)G ≤ C

h_x+µ(ρ²+µ²)G, (2.8)

where the second inequality follows from

(ρ²+µ²)⁻¹²≤2π _∞

0

rdr

(r²+h²_x+µ²)² = π h²_x+µ²· We shall show that

ρ²G ≤Cµ²(h_x+µ+G), (2.9)

and therefore

G_L1 ≤Cµ(µ+G).

Using the second inequality in (2.7), this completes the proof of (2.4) and hence of the theorem.

For the proof of (2.9) we consider the expression

λρ^mG²−A(ρ^mG, ρ^mG) =λ(ρ^2mG, G)−A(ρ^2mG, G)−R_m(G, G), wherem= 1 or 2, and

R_m(G, G) =A(ρ^mG, ρ^mG)−A(ρ^2mG, G).

After subtraction by (2.5) withχ=P_h(ρ^2mG), this yields

λρ^mG²−A(ρ^mG, ρ^mG) =F_m, (2.10)

where

F_m=−A(ρ^2mG−P_h(ρ^2mG), G)−(ρ^2mG, δ^x_h)−R_m(G, G). (2.11) By Lemma 2.1 it follows from (2.10) that

µ⁻²ρ^mG²+∇(ρ^mG)²≤C|Fm|. (2.12)

To show (2.9) we will use this first form = 1 and then form = 2, together with the appropriate bounds for F₁ andF₂. The bounds needed for these functions will require the following Lemmas, which are analogous to those used in the one-dimensional case.

Lemma 2.2. Under the assumptionα²βγ <1 we have

ρ δ^x_h ≤C and ρ²δ_h^x ≤Ch_x, for x∈Ω.

Lemma 2.3. Under the assumptions of Theorem 2 we have, for x∈Ω, χ∈S_h.

|A(ρ²χ−P_h(ρ²χ), χ)| ≤C(∇(ρχ) χ+χ²) (2.13) and

|A(ρ⁴χ−P_h(ρ⁴χ), χ)| ≤C(∇(ρχ)+χ)ρ²χ. (2.14) Assuming that these lemmas have been proved, we are now ready for the proof of our main result. We first remark that

R_m(G, G) =m²(a ρ^m−1G∇ρ, ρ^m−1G∇ρ)−m(b· ∇ρ ρ^m−1G, ρ^mG) + 2mIm (a∇(ρ^mG), ρ^m−1G∇ρ).

Using that |∇ρ| ≤1 in Ω, this implies

|Rm(G, G)| ≤C(∇(ρ^mG) ρ^m−1G+ρ^m−1G²). (2.15) We now takem= 1 in (2.12) and use (2.11) to obtain

µ⁻²ρG²+∇(ρG)²≤C|F₁| ≤C

|A(ρ²G−P_h(ρ²G), G)|+|(ρG, ρδ^x_h)|+|R₁(G, G)|

. (2.16)

Using Lemma 2.2, (2.13), and (2.15), we get µ⁻²ρG²+∇(ρG)²≤C

∇(ρG) G+G²+ρG

≤ ¹₂(∇(ρG)²+µ⁻²ρG²) +C(µ²+G²), which shows

µ⁻²ρG²+∇(ρG)²≤C(µ+G)², and hence

ρG ≤Cµ(µ+G) and ∇(ρG) ≤C(µ+G). (2.17) We now takem= 2 in (2.12) to find

µ⁻²ρ²G²+∇(ρ²G)²≤C|F2| ≤C

|A(ρ⁴G−P_h(ρ⁴G), G)|+|(ρ²G, ρ²δ_h^x)|+|R2(G, G)|

.

Hence, using Lemma 2.2, (2.14), and (2.15) withm= 2, µ⁻²ρ²G²+∇(ρ²G)²≤C

∇(ρG)+G+h_x

ρ²G+∇(ρ²G) ρG+ρG²

≤ ¹₂µ⁻²ρ²G²+∇(ρ²G)²+C µ²(∇(ρG)+G+h_x)²+CρG². Using now (2.17) this yields

µ⁻²ρ²G²≤Cµ²(h_x+µ+G)²,

and completes the proof of (2.9). It now only remains to prove Lemmas 2.2 and 2.3.

Proof of Lemma 2.2. Letτ∈Q_j(τ₀). Then for anyϕ∈ C₀^∞(τ) withϕ= 1 we have, using Lemma 0.1, (δ^x_h, ϕ) = (δ_h^x, P_hϕ) = (P_hϕ)(x)≤Ch⁻¹_τ0Phϕ_L2(τ0)≤Ch⁻¹_τ0 γ^j.

Hence

δ^x_hL2(τ)= sup

ϕ∈C₀^∞(τ) ϕ=1

(δ_h^x, ϕ)≤Ch⁻¹_τ0 γ^j.