Evaluation of the condition number in linear systems arising in finite element approximations

Vol. 40, N^o 1, 2006, pp. 29–48 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006006

EVALUATION OF THE CONDITION NUMBER IN LINEAR SYSTEMS ARISING IN FINITE ELEMENT APPROXIMATIONS

Alexandre Ern

and Jean-Luc Guermond

Abstract. This paper derives upper and lower bounds for the ^p-condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsizehare obtained. The theoretical results are applied to finite element approximations of elliptic PDE’s in variational and in mixed form, and to first-order PDE’s approxi- mated using the Galerkin–Least Squares technique or by means of a non-standard Galerkin technique inL¹(Ω). Numerical simulations are presented to illustrate the theoretical results.

Mathematics Subject Classification. 65F35, 65N30.

Received: March 7, 2005. Revised: July 6, 2005.

1. Introduction

The finite element method provides an extremely powerful tool to approximate partial differential equations arising in engineering sciences. Since the linear systems obtained with this technique are generally very large and sparse, the most practical way to solve them is to resort to iterative methods. Estimates for the convergence rate of iterative methods usually depend on the condition number of the system matrix (see, e.g., [11, 14]).

Although in general it is the distribution of eigenvalues rather than the condition number that controls the convergence rate of iterative methods, a study of the condition numberper se is still of interest. In particular, it is important to assess this quantity as a function of the meshsize used in the finite element method.

It is well-known that second-order elliptic equations,e.g., a Laplacian, yield stiffness matrices whose Euclidean condition number explodes as the reciprocal of the square of the meshsize; see, e.g., [3]. More generally, let p∈[1,+∞] and denote by · pthe^p-norm inR^N,i.e., for allW ∈R^N, set

Wp= _N

i=1

|Wi|^p _p¹

, (1)

Keywords and phrases. Finite elements, condition number, partial differential equations, linear algebra.

1 CERMICS, ´Ecole nationale des ponts et chauss´ees, Champs sur Marne, 77455 Marne la Vall´ee Cedex 2, France.

ern@cermics.enpc.fr

2 Dept. Math, Texas A&M, College Station, TX 77843-3368, USA and LIMSI (CNRS-UPR 3152), BP 133, 91403, Orsay, France. guermond@math.tamu.edu

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006006

if 1≤p <+∞andW∞= max_1≤i≤N|Wi|. Use a similar notation for the associated matrix norm overR^N,N. Define the^p-condition number of an invertible matrixA ∈R^N,N by

κ_p(A) =ApA⁻¹p. (2)

Recent work on the conditioning of finite element matrices has focused on upper bounds for the Euclidean condition number in the case of locally refined meshes; see,e.g., [1, 3]. The objective of the present paper is to give upper and lower bounds onκ_p(A) forp∈[1,+∞] whenAis the stiffness matrix associated with the finite element approximation of a linear, abstract model problem posed in Banach spaces. The analysis is restricted to finite element bases that are localized in space, i.e., nodal bases. The case of hierarchical and modal bases is not discussed. Technical aspects related to locally refined meshes are not addressed either.

This paper is organized as follows. Section 2 collects preliminary results. Necessary and sufficient conditions for wellposedness of an abstract model problem are stated, and the finite element setting for the approximation of this problem is introduced. Section 3 contains the main results of the paper. Section 4 presents various applications to finite element approximations of PDE’s. Elliptic PDE’s either in variational or in mixed form are first considered. Then, first-order PDE’s approximated using either the Galerkin–Least Squares (GaLS) technique or a non-standard Galerkin technique in L¹(Ω) are analyzed. For most of the examples (with the exception of the last one where the casep= 1 is considered), the analysis in Section 4 focuses on the casep= 2.

Numerical illustrations are reported in Section 5. Finally, Appendix A collects technical results concerning norm equivalence constants and the existence of large-scale discrete functions in finite element spaces.

2. Preliminaries 2.1. Wellposedness

Let W and V be two real Banach spaces equipped with some norms, say · W and · V, respectively.

Consider a linear bounded operator

A:W −→V. (3)

Recall that as a consequence of the Open Mapping Theorem and the Closed Range Theorem [15], the following holds:

Lemma 2.1. The following statements are equivalent:

(i) A is bijective.

(ii) There exists a constant α >0 such that

∀w∈W, AwV ≥αwW, (4)

∀v∈V, (A^Tv = 0) =⇒ (v= 0). (5)

Another way of interpretingAconsists of introducing the bilinear forma∈ L(W ×V;R) such that

∀(w, v)∈W ×V, a(w, v) = v, AwV,V, (6) where ·,·V,V denotes the duality paring. Owing to a standard corollary of the Hahn–Banach Theorem, for allf ∈V and for allw∈W,Aw=f if and only if a(w, v) = v, fV,V for allv∈V. Then, a reformulation of Lemma 2.1, henceforth referred to as the BNB Theorem [2, 8, 13], is the following:

Theorem 2.2 (Banach–Neˇcas–Babuˇska). The following statements are equivalent:

(i) For all f ∈V, the problem

Seek u∈W such that

a(u, v) = v, fV,V, ∀v∈V, (7) is well-posed.

(ii) There exists a constant α >0 such that

w∈Winf sup

v∈V

a(w, v) wWvV

≥α, (8)

∀v∈V, (∀w∈W, a(w, v) = 0) =⇒ (v = 0). (9) IfV is reflexive, the above setting is unchanged ifV is substituted by V andV byV. As an illustration of a nonreflexive situation, the reader may think of W =W^1,1(Ω), V =L¹(Ω), V =L^∞(Ω), andA: W u−→

u+u_x∈V.

Remark 2.3. When supremun and/or infimum over sets of functions are considered, it is always implicitly understood that the zero function is excluded from the set in question whenever it makes sense. This convention is meant to alleviate the notation.

2.2. The finite element setting

Let Ω be an open domain inR^d. Let n be a positive integer. In the sequel, we assume thatW andV are Banach spaces of Rⁿ-valued functions on Ω. For p ∈ [1,+∞], equip [L^p(Ω)]ⁿ with the norm wL^p(Ω) = (

Ω

_n

i=1|wi|^p)^1p if p = ∞ and for p = ∞, set wL^∞(Ω) = max_1≤i≤ness sup_Ω|wi|. Let (w, v)_L2(Ω) =

Ω

_n

i=1w_iv_i denote the [L²(Ω)]ⁿ-inner product. Likewise, for a measurable subset K⊂Ω, set (w, v)_L2(K)=

K

_n

i=1w_iv_i.

To construct an approximate solution to (7), we introduce a family of meshes of Ω that we denote by{Th}h>0. The parameterhrefers to the maximum meshsize,i.e.,h= max_K∈Thh_Kwhereh_K= diam(K). LetW_handV_h be finite-dimensional approximation spaces based on the meshTh. These spaces are meant to approximateW andVrespectively; henceforth,W_his referred to as the solution space andV_has the test space. Letp∈[1,+∞] and denote by p its conjugate, i.e., ¹_p + _p¹ = 1 with the convention that p = 1 if p= +∞ andp = +∞if p= 1. We assume hereafter that dim(W_h) = dim(V_h) and that there isp∈[1,+∞] such thatW_h ⊂[L^p(Ω)]ⁿ andV_h⊂[L^p(Ω)]ⁿ. The spacesW_handV_hare equipped with some norms, say · Wh and · Vh, respectively.

LetA:W →V be an isomorphism. Problem (7) is approximated by replacing the spacesW andV by their finite-dimensional counterparts, yielding the approximate problem:

Seeku_h∈W_h such that

a_h(u_h, v_h) = vh, fh, ∀vh∈V_h. (10) Problem (10) involves a consistent approximationa_h of the bilinear form aand a consistent approximation of the linear form in the right-hand side. The way vh, fhis defined is not important for the present investigation.

Henceforth, we assume

whinf∈Wh

sup

vh∈Vh

a_h(w_h, v_h) whWhvhVh

>0. (11)

This, together with the fact that dim(W_h) = dim(V_h), implies that the discrete problem (10) has a unique solution.

LetN = dim(W_h) = dim(V_h). Assume we are given a basis forV_h, say{ϕ₁, . . . , ϕ_N}. The elements in this basis are hereafter referred to as the global shape functions ofV_h. Likewise let{ψ₁, . . . , ψ_N}be the global shape functions in W_h. For a function v_h ∈V_h, denote byV ∈R^N the coordinate vector of v_h relative to the basis {ϕ₁, . . . , ϕ_N},i.e.,v_h=_N

i=1Viϕ_i ∈V_h. Denote by C_Vh :V_h−→R^N the linear operator that maps vectors in V_hto their coordinate vectors inR^N,i.e.,C_Vhv_h=V. Similarly, denote byC_Wh :W_h−→R^N the operator that maps vectors inW_h to their coordinate vectors in R^N. It is clear that both C_Vh and C_Wh are isomorphisms.

Denote by (·,·)_N the Euclidean scalar product inR^N.

Define the so-called stiffness matrixA with entries

a_h(ψ_j, ϕ_i)

1≤i,j≤N. This definition is such that for all (w_h, v_h)∈W_h×V_h, (C_Vhv_h,ACWhw_h)_N =a_h(w_h, v_h). The discrete problem (10) yields the linear system:

Seek U ∈R^N such that

AU =F, (12)

where the entries ofF areFi= ϕi, fhfor 1≤i≤N. The solutionu_h to (10) is thenu_h=C_W⁻¹

hU.

2.3. Norm equivalence constants

SinceW_handV_hare finite-dimensional and sinceC_WhandC_Vh are isomorphisms, it is legitimate to introduce the following positive constants

m_s,p,h= inf

wh∈Wh

w_hL^p(Ω)

CWhw_hp

, M_s,p,h= sup

wh∈Wh

w_hL^p(Ω)

CWhw_hp

, (13)

m_t,p,h= inf

vh∈Vh

v_hLp(Ω)

C_Vhv_hp, M_t,p,h= sup

vh∈Vh

v_hLp(Ω)

C_Vhv_hp· (14) Here, the subscriptssandtrefer to the solution space and the test space, respectively. The constants introduced in (13)–(14) are such that

∀wh∈W_h, m_s,p,hWp≤ whL^p(Ω)≤M_s,p,hWp, (15)

∀vh∈V_h, m_t,p,hVp≤ vh_Lp(Ω)≤M_t,p,hVp, (16) withW=CWhw_h andV =CVhv_h. Henceforth, we denote

κ_s,p,h=M_s,p,h

m_s,p,h, κ_t,p,h= M_t,p,h

m_t,p,h, κ_p,h=√

κ_s,p,hκ_t,p,h. (17)

It is possible to estimate m_s,p,h and M_s,p,h (resp. m_t,p,h and M_t,p,h) whenW_h (resp. V_h) is a finite element space and the global shape functions are such that their support is restricted to a number of mesh cells that is uniformly bounded with respect to the meshsize. For instance, if the mesh family{Th}h>0 is quasi-uniform, κ_s,p,handκ_t,p,h are uniformly bounded with respect toh; see Appendix A.

3. Bounds on κ

(A)

The goal of this section is to derive upper and lower bounds for the ^p-condition number of the stiffness matrixA.

3.1. Main results

Introduce the following notation:

α_p,h= inf

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h) whL^p(Ω)vh_Lp(Ω)

, (18)

ω_p,h= sup

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h)

wh_L^p_(Ω)vh_Lp(Ω)· (19)

A first result is the following:

Theorem 3.1. Under the above assumptions,

∀h, κ⁻²_p,hω_p,h

α_p,h ≤κ_p(A)≤κ²_p,hω_p,h

α_p,h· (20)

Proof. (1) Upper bound onAp. Consider W ∈R^N. Then, owing to definition (19) and using the notation C_Vhv_h=V andC_Whw_h=W,

AWp= sup

V∈R^N

(AW,V)_N Vp

= sup

V∈R^N

a_h(w_h, v_h) wh_L^p_(Ω)vh_Lp(Ω)

whL^p(Ω)

Wp

vh_Lp(Ω)

Vp Wp

≤ω_p,hw_hL^p(Ω)

Wp

sup

V∈R^N

vh_Lp(Ω)

Vp Wp. Using inequalities (15)–(16) yieldsAWp≤ω_p,hM_s,p,hM_t,p,hWp. That is to say,

Ap≤ω_p,hM_s,p,hM_t,p,h.

(2) Upper bound onA⁻¹p. Using again (15)–(16) together with definition (18) yields α_p,hm_s,p,hWp≤α_p,hw_hL^p(Ω)≤ sup

vh∈Vh

a_h(w_h, v_h) vh_Lp(Ω)

= sup

V∈R^N

(AW,V)N

vh_Lp(Ω) ≤ AWp sup

V∈R^N

Vp

vh_Lp(Ω) ≤m⁻¹_t,p,hAWp.

Hence, settingZ=AW, we inferα_p,hm_s,p,hA⁻¹Zp≤m⁻¹_t,p,hZp. SinceZ is arbitrary, this means A⁻¹p≤ 1

α_p,hm⁻¹_s,p,hm⁻¹_t,p,h. The upper bound in (20) is a direct consequence of the above estimates.

(3) Lower bound onA⁻¹p. SinceW_h is finite-dimensional, there isw_h= 0 inW_h such that α_p,h= sup

vh∈Vh

a_h(w_h, v_h) w_hL^p_(Ω)v_hLp(Ω)·

As a result, setting W=C_Whw_h andV=C_Vhv_hyields AWp= sup

V∈R^N

(AW,V)_N Vp

= sup

vh∈Vh

a_h(w_h, v_h) wh_L^p_(Ω)vh_Lp(Ω)

vh_Lp(Ω)

Vp

whL^p(Ω)

Wp Wp

≤α_p,hM_t,p,hM_s,p,hWp. Hence,

1

α_p,hM_t,p,h⁻¹ M_s,p,h⁻¹ ≤ A⁻¹p.

(4) Lower bound onAp. SinceW_h is finite-dimensional, there isw_h= 0 inW_h such that ω_p,h= sup

vh∈Vh

a_h(w_h, v_h) w_hL^p(Ω)v_hLp(Ω)·

This implies

m_s,p,hWp≤ wh_L^p_(Ω)= 1 ω_p,h sup

vh∈Vh

a_h(w_h, v_h) vh_Lp(Ω)

= 1

ω_p,h sup

vh∈Vh

(AW,V)N

Vp

Vp

v_hLp(Ω) ≤ 1

ω_p,hm⁻¹_t,p,hAWp≤ 1

ω_p,hm⁻¹_t,p,hApWp. SinceW = 0 this yields

ω_p,hm_s,p,hm_t,p,h≤ Ap.

The lower bound in (20) easily follows from the above estimates.

To account for a possible polynomial dependence of α_p,h and ω_p,h onh, we make the following additional technical hypotheses:

∃γ,

⎧⎨

⎩

0< c^α_inf= lim inf

h→0 α_p,hh^−γ <+∞, 0< c^α_sup= lim sup

h→0 α_p,hh^−γ <+∞, (21)

∃δ,

⎧⎨

⎩

0< c^ω_inf = lim inf

h→0 ω_p,hh^δ <+∞, 0< c^ω_sup= lim sup

h→0 ω_p,hh^δ <+∞. (22)

As a consequence of Theorem 3.1, we deduce the following:

Theorem 3.2. Under the assumptions (21)–(22), the following holds true: for all ∈ ]0,1[, there is h such that for all h≤h,

(1− ) c^ω_inf

c^α_supκ⁻²_p,hh^−γ−δ≤κ_p(A)≤(1 + )c^ω_sup

c^α_inf κ²_p,hh^−γ−δ. (23) Proof. Let ∈]0,1[.

(1) There is h such that for all h ≤ h, (1− ₃)c^α_infh^γ ≤ α_p,h and ω_p,h ≤ 1 +₃

c^ω_suph^−δ. Then, apply Theorem 3.1 to deduce the upper bound.

(2) Owing to the definition ofc^α_sup, there is h such that for all 0< h≤h, there isw_h∈W_h satisfying

sup

vh∈Vh

a_h(w_h, v_h)

w_hL^p(Ω)v_hLp(Ω) ≤ 1 +2

c^α_suph^γ.

Then, proceed as in step (3) of the proof of Theorem 3.1 to derive the lower bound A⁻¹p≥(1 +

2)⁻¹(c^α_sup)⁻¹M_s,p,h⁻¹ M_t,p,h⁻¹ h^−γ.

(3) The definition ofc^ω_inf implies the existence ofhsuch that for all 0< h≤h, there is w_h∈W_hsatisfying (1−

2)c^ω_infh^−δ≤ sup

vh∈Vh

a_h(w_h, v_h) w_hL^p(Ω)v_hLp(Ω)·

Then, proceed as in step (4) of the proof of Theorem 3.1 to infer Ap ≥ (1−₂)c^ω_infm_s,p,hm_t,p,hh^−δ. The

lower bound onκ_p(A) then results from the above estimates.

Remark 3.3. Observe that in (20) and (23) the lower bound is multiplied byκ⁻²_p,h and the upper bound is multiplied by κ²_p,h; hence, these estimates are sharp only if κ_p,h = √

κ_s,p,hκ_t,p,h is uniformly bounded with respect toh. It is shown in the appendix that this holds true when the global shape functions{ϕ1, . . . ϕ_N}and {ψ₁, . . . ψ_N}spanningV_h andW_hrespectively have localized supports. The definition ofα_p,handω_p,hmust be modified if the bases ofV_h andW_h are hierarchical.

3.2. Estimates based on natural stability norms

α_h= inf

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h) whWhvhVh

, (24)

ω_h= sup

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h) w_hWhv_hVh

· (25)

In general, one may expect that the norms of W_h andV_h are selected so thatα_h is uniformly bounded from below away from zero and ω_h is uniformly bounded. Hence, boundingκ_p(A) in terms ofα_h andω_hmay yield valuable information.

For this purpose, we make the following technical assumptions:

∃csP >0, ∀wh∈W_h, c_sPwh_L^p_(Ω)≤ whWh, (26)

∃c_tP >0, ∀v_h∈V_h, c_tPv_hLp(Ω)≤ v_hVh, (27)

∃s >0,∃csI, ∀wh∈W_h, whWh ≤c_sIh^−swh_L^p_(Ω), (28)

∃t >0,∃ctI, ∀vh∈V_h, vhVh ≤c_tIh^−tvh_Lp(Ω). (29) Estimates (26) and (27) are Poincar´e-like inequalities expressing the fact that the norms equipping W_h and V_h control the L^p-norm and the L^p-norm, respectively. In other words, the injections W_h ⊂ [L^p(Ω)]ⁿ and V_h⊂[L^p(Ω)]ⁿ are uniformly continuous. Furthermore, (28) and (29) are inverse inequalities. When the mesh family{Th}h>0is quasi-uniform, the constantssandtcan be interpreted as the order of the differential operator used to define the norms inW_h andV_h, respectively.

As a consequence of Theorem 3.1, we deduce the following:

Corollary 3.4. Under the assumptions(26)–(29), the following bound holds:

∀h, κ_p(A)≤κ_s,p,hc_sI

c_sP κ_t,p,hc_tI c_tP

ω_h

α_hh^−s−t. (30)

Proof. Let us estimateα_p,h andω_p,h. (1) It is clear that

α_h= inf

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h) whWhvhVh

≤ 1

c_sPc_tP inf

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h) wh_L^p_(Ω)vh_Lp(Ω)· Hence α_p,h≥c_sPc_tPα_h.

(2) Moreover, ω_p,h= sup

wh∈Wh

sup

vh∈Vh

a_h(w_h, v_h)

wh_L^p_(Ω)vh_Lp(Ω) ≤ω_h sup

wh∈Wh

whWh

wh_L^p_(Ω) sup

vh∈Vh

vhVh

vh_Lp(Ω) ≤ω_hc_sIc_tIh^−s−t. Hence,ω_p,h≤c_sIc_tIω_hh^−s−t.

(3) Conclude using Theorem 3.1.

Remark 3.5. It may happen that (30) is not sharp; see Section 4.3 and (68).

In addition to (26)–(29), we assume the following:

∃µ,

⎧⎪

⎪⎨

⎪⎪

⎩

0< d^α_inf= lim inf

h→0

α_p,h

α_h h^µ<+∞, 0< d^α_sup= lim sup

h→0

α_p,h

α_h h^µ<+∞, (31)

∃ν,

⎧⎪

⎪⎨

⎪⎪

⎩

0< d^ω_inf= lim inf

h→0

ω_p,h

ω_h h^s+t−ν<+∞, 0< d^ω_sup= lim sup

h→0

ω_p,h

ω_h h^s+t−ν <+∞. (32)

The constantsµandν are meant to measure the possible default to optimality of Corollary 3.4. Proceeding as in the proof of Theorem 3.2, it is clear that the following holds true:

Corollary 3.6. Under the assumptions (26)–(29) and (31)–(32), the following holds true: for all ∈ ]0,1[, there ish such that for all h≤h,

(1− ) d^ω_inf

d^α_supκ⁻²_p,hω_h

α_hh^{−s−t+µ+ν} ≤κ_p(A)≤(1 + )d^ω_sup

d^α_inf κ²_p,hω_h

α_hh^{−s−t+µ+ν}. (33)

4. Applications

This section presents various applications of the theoretical results derived in Section 3 to finite element approximations of PDE’s posed on a bounded domain Ω inR^d. For the sake of simplicity, we assume that Ω is a polyhedron. Let{Th}_h>0be a shape-regular family of meshes of Ω.

4.1. Elliptic PDE’s in variational form

Consider the Laplacian with homogeneous Dirichlet boundary conditions. Set W =H₀¹(Ω),V =H⁻¹(Ω), and A : W w −→ −∆w ∈ V. Clearly A : W → V is an isomorphism. Introduce the bilinear form a(w₁, w₂) =

Ω∇w₁·∇w₂,∀(w₁, w₂)∈W×W.

LetW_hbe a finite-dimensional space based on the meshTh. We assume thatW_h⊂W,i.e., the approximation isH¹-conforming. We assume thatW_his such that there isc, independent ofh, such that the following global inverse inequality holds:

∀wh∈W_h, ∇wh_L²_(Ω)≤c h⁻¹wh_L²_(Ω). (34) This hypothesis holds whenever W_h is a finite element space constructed using a quasi-uniform mesh family;

see,e.g., [4, 5, 8, 10].

Consider the approximate problem:

Seek u_h∈W_h such that

a(u_h, v_h) = (f, v_h)_L2(Ω), ∀vh∈W_h, (35) for some data f ∈L²(Ω). LetA be the stiffness matrix associated with (35). The main result concerning the Euclidean condition number ofAis the following:

Theorem 4.1. If the mesh family {Th}_h>0 is quasi-uniform and (34) holds, there are 0 < c₁ ≤ c₂, both independent of h, such that

c₁κ⁻²_2,hh⁻²≤κ₂(A)≤c₂κ²_2,hh⁻². (36)

Proof. (1) Forw_h∈W_h\ {0}, defineR(w_h) = ^∇wh

2L2(Ω)

wh²_L2(Ω) . Then α_2,h= inf

wh∈Wh

R(w_h), ω_2,h= sup

wh∈Wh

R(w_h). (37)

(2) Let ˜z_hbe given by Lemma A.5 withZ =H₀¹(Ω),Z_h=W_hequipped with theH¹-seminorm, andL=L²(Ω).

SinceR(˜z_h)≤c, we inferα_2,h≤R(˜z_h)≤c uniformly inh. Moreover, the Poincar´e inequality inH₀¹(Ω) implies that α_2,his uniformly bounded from below away from zero.

(3) Lettingw_h in (37) be one of the global shape functions inW_h, it is clear thatω_2,h≥ch⁻². Moreover, owing to the inverse inequality (34),ω_2,h≤ch⁻².

(4) To conclude, use Theorem 3.1 (or Thm. 3.2 withγ= 0 andδ= 2).

Remark 4.2. Observe that owing to the quasi-uniformity of{Th}h>0 and Lemma A.1, the constant κ_2,h =

√κ_s,2,hκ_t,2,his bounded from below and from above uniformly with respect tohwhen the global shape functions have localized supports.

Remark 4.3. The Euclidean condition numberκ₂(A) can also be estimated using Corollary 3.4. One readily verifies that α_h can be bounded from below uniformly with respect to h, ω_h can be bounded from above uniformly with respect toh, and thats=t= 1. Hence, (30) yieldsκ₂(A)≤ch⁻²,i.e., the estimate is sharp.

One also verifies thatµ=ν= 0 in (31)–(32), confirming the optimality of Corollary 3.4.

Remark 4.4. Estimate (36) extends to more general second-order elliptic operators,e.g., advection-diffusion- reaction equations.

4.2. Elliptic PDE’s in mixed form

In this section we investigate the following non-standard Galerkin technique introduced in [6] to approximate the Laplacian in mixed form. Let H(div; Ω) = {v ∈ [L²(Ω)]^d;∇·v ∈ L²(Ω)}, W = H(div; Ω)×H₀¹(Ω), and V = [L²(Ω)]^d×L²(Ω). Introduce the operator

A:W (u, p) −→ (u+∇p,∇·u)∈V. (38)

One readily verifies thatA:W →V is an isomorphism. For (w, v)∈W×V, define the bilinear form

a((u, p),(v, q)) = (u, v)_L2(Ω)+ (∇p, v)_L²_(Ω)+ (∇·u, q)_L²_(Ω). (39) By analogy with Darcy’s equations,uis termed the velocity andpthe pressure.

The non-standard Galerkin approximation consists of seeking the discrete velocity in the Raviart–Thomas finite element space of lowest order and the discrete pressure in the Crouzeix–Raviart finite element space.

Denote byF_h,F_h^∂, andF_hⁱ the set of faces, boundary faces, and interior faces of the mesh, respectively. Define X_h={uh;∀K∈ Th, u_h|K∈RT₀;∀F ∈ F_hⁱ,

F[[u_h·n]] = 0}, (40) Y_h={ph;∀K∈ Th, p_h|K∈P₁;∀F ∈ Fh,

F[[p_h]] = 0}, (41)

where RT₀ = [P₀]^d⊕xP₀, [[u_h·n]] denotes the jump of the normal component of u_h across interfaces, and [[p_h]]

the jump ofp_h across interfaces (with the convention that a zero outer value is taken wheneverF∈ F_h^∂). Test functions for both the velocity and the pressure are taken to be piecewise constants. Introducing the spaces W_h=X_h×Y_h and

V_h={(vh, q_h);∀K∈ Th, v_h|K ∈[P0]^d andq_h|K∈P0}, (42) and defining the bilinear forma_h∈ L(Wh×V_h;R) such that

a_h((u_h, p_h),(v_h, q_h)) = (u_h, v_h)_L2(Ω)+ (∇·u_h, q_h)_L2(Ω)+

K∈Th

(∇p_h, v_h)_L2(K), (43)

the discrete problem is formulated as follows:

Seek (u_h, p_h)∈W_hsuch that

a_h((u_h, p_h),(v_h, q_h)) = (f, q_h)_L2(Ω), ∀(v_h, q_h)∈V_h, (44) for some dataf ∈L²(Ω). Note that the approximation setting is conforming on the velocity and non-conforming on the pressure. Moreover, it is readily checked that the total number of unknowns in (44) equals the total number of equations. Indeed, the former is the number of faces plus the number of interior faces, the latter is equal to (d+ 1) times the number of elements, and these two quantities are equal owing to the Euler relations.

EquipW_h with the norm

(u_h, p_h)²_Wh =u_h²_L2(Ω)+∇·u_h²_L2(Ω)+p_h²_L2(Ω)+

K∈Th

∇p_h²_L2(K), (45)

and equipV_hwith the norm(vh, q_h)²_Vh =vh²_L2(Ω)+qh²_L2(Ω). In the framework of the BNB Theorem, the main stability result for (44) is the following:

Lemma 4.5. There are c >0 andh₀ such that for allh≤h₀,

(uh,pinfh)∈Wh

sup

(vh,qh)∈Vh

a_h((u_h, p_h),(v_h, q_h)) (uh, p_h)Wh(vh, q_h)Vh

≥c. (46)

Proof. Since this is a non-classical result, the proof is briefly sketched; see [6] and [8] for further details.

(1) Let (u_h, p_h)∈W_h. Denote byu_h the function whose restriction to each elementK∈ Th is the mean value ofu_h. Use a similar notation forp_h. Denote by∇hp_hthe function whose restriction to each elementK∈ Th is

∇p_h|K. Setv_h =u_h+∇hp_h andq_h= 2p_h+∇·u_h. Note that the pair (v_h, q_h) is inV_h since the gradient and the divergence terms are piecewise constant owing to the present choice of discrete spaces. Hence,

a_h((u_h, p_h),(v_h, q_h)) = (u_h, u_h)_L2(Ω)+∇·uh²_L2(Ω)+

K∈Th

∇ph²_L2(K)

+ 2(∇·u_h, p_h)_L2(Ω)+

K∈Th

(u_h,∇p_h)_L2(K)+ (∇p_h, u_h)_L2(K)

=uh²_L2(Ω)+∇·uh²_L2(Ω)+

K∈Th

∇ph²_L2(K), since (∇·uh, p_h)_L2(Ω)+

K∈Th(u_h,∇ph)_L2(K)= 0.

(2) For u_h ∈ X_h, one readily verifies that ∀K ∈ Th, ∀x∈ K, u_h(x) = u_h+_d¹(x−g_K)∇·uh where g_K is the barycenter ofK. This implies that there isc, independent ofh, such that

∀uh∈X_h, uh_L²_(K)≤ uh_L²_(K)+c h_K∇·uh_L²_(K). Hence,

a_h((u_h, p_h),(v_h, q_h))≥cuh²_L2(Ω)+ (1−ch²)∇·uh²_L2(Ω)+

K∈Th

∇ph²_L2(K). Ifhis small enough, (1−ch²) is bounded from below by ¹₂.

(3) Use the extended Poincar´e inequality (see,e.g., [7, 8] for a proof)

∀p_h∈Y_h,

K∈Th

∇p_h²_L2(K)≥cp_h²_L2(Ω),