Approximation of the biharmonic problem using P1 finite elements

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Approximation of the biharmonic problem using P1 finite elements

Robert Eymard, Raphaèle Herbin, Mohamed Rhoudaf

To cite this version:

Robert Eymard, Raphaèle Herbin, Mohamed Rhoudaf. Approximation of the biharmonic problem

using P1 finite elements. Journal of Numerical Mathematics, De Gruyter, 2011, pp.Volume 19, Issue

1, Pages 1-26. �hal-00825673�

Approximation of the biharmonic problem using P1 finite elements

R. Eymard, R.Herbin and M. Rhoudaf May 24, 2013

Abstract

We study in this paper a P1 finite element approximation of the solution inH0²(Ω) of a biharmonic problem. Since the P1 finite element method only leads to an approximate solution in H₀¹(Ω), a discrete Laplace operator is used in the numerical scheme. The convergence of the method is shown, for the general case of a solution with H₀²(Ω) regularity, thanks to compactness results and to the use of a particular interpolation of regular functions with compact supports. An error estimate is proved in the case where the solution is inC⁴(Ω). The order of this error estimate is equal to 1 if the solution has a compact support, and only 1/5 otherwise. Numerical results show that these orders are not sharp in particular situations.

1 Introduction

This paper deals with the approximation of the following problem, called the biharmonic problem, which arises in various frameworks of fluid or solid mechanics:

findusuch that ∆(∆u)(x) =f(x)−divg(x) + ∆`(x), forx∈Ω,

u(x) = 0 and∇u(x)·n∂Ω(x) = 0, forx∈∂Ω. (1) In this paper, Problem (1) is considered in the following weak sense:

findu∈H₀²(Ω) such that

∀v∈H₀²(Ω), Z

Ω

∆u(x)∆v(x)dx= Z

Ω

(f(x)v(x) +g(x)· ∇v(x) +`(x)∆v(x))dx, (2) where H₀²(Ω) denotes the closure in H²(Ω) of the set C_c^∞(Ω) of infinitely continuously differentiable functions with compact support in Ω, and where

d∈N\ {0}denotes the space dimension,

Ω is an open polygonal bounded and connected subset ofR^d, with Lipschitz-continuous boundary∂Ω,

(3) and

f ∈L²(Ω), `∈L²(Ω) and g∈(L²(Ω))^d. (4) Numerous discretization methods for Problem (2) have been proposed in the recent past. The most classical is probably the conforming finite element method; the finite element space must then be a finite dimensional subspace of the Sobolev spaceH²(Ω). Hence elementary basis functions are sought such that the reconstructed global basis functions on Ω belong toC¹(Ω). On Cartesian meshes, such basis functions are found by generalizing the one-dimensionalP³Hermite finite element to the multi-dimensional frame- work. This task becomes much more difficult on more general meshes and involves rather sophisticated finite elements such as the Argyris finite element on triangles in 2D, which unfortunately requires 21 de- grees of freedom [8]. Hence non-conforming FEMs have also been widely studied: see e.g. [8, Section 49],

[9], and references therein, and [3, 4] for more recent works. Discontinuous Galerkin methods have also been recently developed and analysed [12, 13, 14, 11]; error estimates have been derived for polynomials of degree greater or equal to two or three. Other methods which have been developed for fourth order problems include mixed methods [6, 12] (see also references therein), [15], and compact finite difference methods [7, 2, 1]. All the above methods are high order methods, and therefore, rather computationally expensive and may not be so easy to implement. Recently, a cheaper low order method based on the discretization of the Laplace operator by a cell centred finite volume scheme was proposed [10].

The idea developed in the present paper is to use the discretization of the Laplace operator, which is naturally provided by the continuous piecewise linear finite element method (see Section 2). Such a method leads to a nonconforming method in H²(Ω), since it only provides an approximate solution in H₀¹(Ω). This discrete Laplace operator is then used in a discrete bilinear form, which is applied to the elements of the P1 finite element space which vanish at the boundary. Note that the condition

∇u·n_∂Ω = 0 at the boundary of the domain is satisfied by the limit of a sequence of approximate solutions thanks to the definition of the discrete Laplace operator (see Lemma 3.4 in Section 3). Then the convergence has to be proved, using a suitable interpolation of regular test functions. In the general case, the discrete Laplace operator applied to the standard interpolation of a regular function is not consistent with the continuous Laplace operator applied to this function, although it leads to a second order discrete operator (see Lemma 3.5). Therefore a non standard interpolation of regular functions in the P1 finite element space has to be derived (see Lemma 3.6). Error estimates are derived in the case where the continuous solution has some regularity (Section 4). An order 1 is shown in the case where the solution has a compact support in Ω (Theorem 4.2), but only 1/5 for a general regular solution (Theorem 4.3). It is worth noticing that the order of these estimates is lower than that which is numerically observed in various situations (Section 5).

A short conclusion of ongoing research is finally drawn in Section 6.

2 Definition of the scheme

Let Ω be an open polyhedral domain inR^d, withd∈N^?. We consider a conforming simplicial meshT of Ω (in the standard sense provided for example in [8]). We denote byh_T the maximum of the diameters hS of allS∈ T.

LetV be the finite set of the vertices ot the mesh; the set of the vertices of a simplexS ∈ T is denoted byVS, and the subset of allS∈ T such thatz ∈ VS is denoted byTz. We denote byVext the set of all z∈ V such thatz∈∂Ω, and we denoteVint=V \ Vext the set of the interior vertices.

Let (Kz)z∈V be a family of disjoint open connected subsets of Ω such that:

1. for allz∈ V,z∈KzandKz⊂S

S∈TzS, 2. S

z∈VKz= Ω.

We denote by θ_T the minimum of (θ_S)_S∈T,(θ_z)_z∈V, where θ_S is the ratio between the radius of the largest Euclidean ball contained inS and diam(S), and

θ_z= |Kz|

|S

S∈T_zS|, ∀z∈ V.

For any z ∈ V, we denote by Vz the set of ally∈ V such that there existsS ∈ Tz with y∈ VS (which means thatVz=S

S∈TzVS). In the following, we will also use the notationT for denoting the whole set of discrete geometric definitions.

Remark 1 The results of this paper hold under the general requirements on Kz given above. In the numerical examples given in this paper, we use the following definition of Kz. For all S ∈ T and all z∈ VS, we denote byKS,zthe subset ofS of all points whose barycentric coordinate related toz is larger than that related to any z⁰ ∈ VS with z⁰ 6=z (see figure 1). We then denote for all z ∈ V by Kz the union of allKS,z, for allS∈ Tz.

z S

KS,z

∂Ω K_z

Figure 1: Definition ofKz

For any z∈ V, letξz∈H¹(Ω) be the piecewise affine basis function of the P1 finite element, such that ξz(z) = 1 and ξz(z⁰) = 0 for all z⁰ ∈ V \ {z}. We then denote byV_T the vector space spanned by all functionsξz,z ∈ V. We remark that the propertyv=P

z∈Vv(z)ξzholds for anyv∈V_T, and we define the natural P1 interpolationITϕ∈VT of any continuous functionϕby

I_Tϕ(z) =ϕ(z), ∀z∈ V, ∀ϕ∈C⁰(Ω). (5) For anyu∈VT and anyS∈ T, we denote by∇Suthe constant value of∇uinS, and we define

Tzy=− Z

Ω

∇ξz(x)· ∇ξy(x)dx=−X

S∈T

|S|∇Sξz· ∇Sξy, ∀z,y∈ V. (6) Using the propertyP

y∈VTzy= 0 sinceP

y∈Vξy(x) = 1 for allx∈Ω, let ∆z : V_T →R, for allz∈ V, be the linear form defined by

∆zu= 1

|Kz| X

y∈V

u(y)Tzy = 1

|Kz| X

y∈Vz

Tzy(u(y)−u(z)), ∀u∈VT, ∀z∈ V. (7) We then define ∆_T : V_T →L²(Ω) by

∆Tu(x) =X

z∈V

∆zu1Kz(x), for a.e. x∈Ω, ∀u∈VT. (8) We define the discrete space

V_T,0={u∈V_T, u= 0 on∂Ω}. (9)

Then the scheme for the approximation of Problem (2) consists in finding u∈V_T,0; ∀v∈V_T,0,

Z

Ω

∆_Tu(x)∆_Tv(x)dx= Z

Ω

(f(x)v(x) +g(x)· ∇v(x) +`(x)∆_Tv(x))dx. (10) An important relation for the mathematical analysis is

∀u, v∈V_T, Z

Ω

∇u(x)· ∇v(x)dx=−X

z∈V

|Kz|v(z)∆zu=− Z

Ω

P_Tv(x)∆_Tu(x)dx, (11) where we define the piecewise constant reconstruction of the elements ofV_T by

P_Tv(x) =X

z∈V

v(z)1_Kz(x), for a.e. x∈Ω, ∀v∈V_T. (12)

3 Convergence analysis

Lemma 3.1 (Piecewise reconstruction) Let us assume Hypotheses (3). Let T be a conforming simplicial mesh ofΩ. We then have the following inequality:

kv−PTvkL²(Ω)≤hTk∇vk_L2(Ω)^d,∀v∈VT. (13) Proof. LetS ∈ T, z ∈ VS and x∈ S. We have v(x)−v(z) =∇Sv·(x−z), denoting by ∇Sv the constant gradient ofv inS. This leads to|v(x)−v(z)| ≤ |∇Sv|h_T. Therefore we get

Z

Ω

(v(x)−P_Tv(x))²dx=X

z∈V

X

S∈T

Z

Kz∩S

(v(x)−v(z))²dx≤h²_T Z

Ω

|∇v(x)|²dx,

which gives (13).

Lemma 3.2

Let us assume Hypotheses(3). LetT be a conforming simplicial mesh ofΩ. Then the following inequalities hold:

k∇wk_L2(Ω)^d≤2 diam(Ω)k∆TwkL²(Ω), ∀w∈V_T,0, (14) and

kwk_L2(Ω)≤2 diam(Ω)²k∆_Twk_L2(Ω), ∀w∈V_T,0. (15) Proof. Setting v=win (11), we get

Z

Ω

|∇w(x)|²=− Z

Ω

PTw(x)∆Tw(x)dx.

Hence, by using the Cauchy-Schwarz inequality, we have

k∇wk²_L2(Ω)^d≤ kP_Twk_L2(Ω)k∆_Twk_L2(Ω). (16) The Poincar´e inequality [5], which holds since VT,0⊂H₀¹(Ω),reads

kwkL²(Ω)≤diam(Ω)k∇wk_L2(Ω)^d. By using (13) andh_T ≤diam(Ω), we have

kP_Twk_L2(Ω)≤ kP_Tw−wk_L2(Ω)+kwk_L2(Ω)≤(h_T + diam(Ω))k∇wk_L2(Ω)^d

≤2 diam(Ω)k∇wk_L2(Ω)^d.

Gathering the above results, we deduce (14) and (15) from (16).

Lemma 3.3 (Existence, uniqueness and estimate on the solution of (10)) Let us assume Hy- potheses (3) and (4). Let T be a conforming simplicial mesh of Ω. Then, for any u∈ V_T,0 satisfying (10), the following holds:

kuk_L2(Ω)≤4 diam(Ω)⁴kfk_L2(Ω)+ 4 diam(Ω)³kgk_L2(Ω)^d+ 2 diam(Ω)²k`k<sub>L</sub>2(Ω), (17) k∇uk<sub>L</sub>2(Ω)<sup>d</sup>≤4 diam(Ω)<sup>3</sup>kfk<sub>L</sub>2(Ω)+ 4 diam(Ω)<sup>2</sup>kgk<sub>L</sub>2(Ω)<sup>d</sup>+ 2 diam(Ω)k`k_L2(Ω), (18) and

k∆_Tuk_L2(Ω)≤2 diam(Ω)²kfk_L2(Ω)+ 2 diam(Ω)kgk_L2(Ω)^d+k`k_L2(Ω). (19) As a consequence, there exists one and only oneu∈V_T,0 such that (10)holds.

Proof. Letube given such that (10) holds. Let us take v=uin (10). We get Z

Ω

(∆_Tu(x))²dx= Z

Ω

(f(x)u(x) +g(x)· ∇u(x) +`(x)∆_Tu(x))dx, which leads to

k∆Tuk²_L2(Ω)≤ kukL²(Ω)kfkL²(Ω)+k∇uk_L2(Ω)^dkgk_L2(Ω)^d+k`kL²(Ω))k∆TukL²(Ω), Thanks to Lemma 3.2, the previous inequality provides

k∆TukL²(Ω)≤2 diam(Ω)²kfkL²(Ω)+ 2 diam(Ω)kgk_L2(Ω)^d+k`kL²(Ω), which is (19). We then deduce (18) and (17), using again Lemma 3.2.

On the other hand, we remark that (10) is equivalent to a square linear system. Settingf = 0 ,g= 0 and

`= 0, we get from (17) that u= 0, showing the invertibility of the matrix of the system. This implies the existence and uniqueness of the discrete solution.

Lemma 3.4 (Compactness of a sequence of approximate solutions) Let us assume Hypotheses (3). Let (Tm)m∈N be a sequence of conforming simplicial discretizations of Ω such that hT_m tends to 0 as m → ∞. Assume that there exists θ > 0 with θ < θTm for all m ∈ N. Let (um)m∈N be a sequence of functions such that u_m ∈V_T⁰

m,0 for allm ∈N. For simplicity, we shall denote the discrete operator ∆_Tm by ∆_m. Assume that the sequence(∆_mu_m)_m∈N is bounded inL²(Ω) byC ≥0; then there exists a subsequence of (T_m)_m∈N, again denoted (T_m)_m∈N, and u∈H₀²(Ω), such that the corresponding subsequence (u_m)_m∈N satisfies:

1. u_m→uin L²(Ω), 2. ∇um→ ∇uin L²(Ω)^d,

3. ∆mum*∆uweakly inL²(Ω), asm→ ∞.

Proof. Since the sequence (∆mum)_m∈Nis bounded inL²(Ω), we may extract a subsequence of (Tm, um)_m∈N, again denoted (Tm, um)_m∈N, such that (∆mum)_m∈Nconverges weakly inL²(Ω) to somew∈L²(Ω). From Lemma 3.2, we get that

k∇umk_L2(Ω)≤C, ∀m∈N,

whereC∈R+ only depends on Ω andC. Therefore, applying Rellich’s theorem, we get the existence of someu∈H₀¹(Ω) and of a subsequence of (T_m, u_m)_m∈N, again denoted (T_m, u_m)_m∈N, such that

∇u_m*∇u weakly in L²(Ω)^d, and

u_m→u strongly in L²(Ω),

asm→ ∞. Let us prove thatu∈H₀²(Ω). Letu(resp. um) denote the prolongement ofu(resp. um) by 0 inR^d\Ω. Thanks tou∈H₀¹(Ω) (resp. um∈H₀¹(Ω)), we have∇u∈L²(R^d)^d (resp. ∇um∈L²(R^d)^d), with the property

∇um*∇u weakly in L²(R^d)^d. (20)

Letϕ∈C_c^∞(R^d); note that ϕdoes not necessarily vanish at the boundary of Ω. LetImϕ denoteI_Tmϕ for short. Recall thatImϕtends toϕ inH¹(Ω) as m→ ∞, thanks to the hypothesis that there exists θ >0 withθ < θT_m for allm∈N. We then define the approximation Gmϕof∇ϕby

Gmϕ(x) =

∇Imϕ(x) for a.e x∈Ω,

∇ϕ(x) for a.e x∈R^d\Ω. (21)

LetTm=R

R^d∇um(x)·Gmϕ(x)dx. Using (20), and the convergence ofGmϕ(x) to ∇ϕin L²(R^d)^d,we get

m→+∞lim Tm= Z

R^d

∇u(x)· ∇ϕ(x)dx.

On the other hand, we have Tm=

Z

R^d

∇um(x)·Gmϕ(x)dx= Z

Ω

∇um(x)· ∇Imϕ(x)dx.

Thanks to (11), we get

Tm=− Z

Ω

P_TmImϕ(x)∆mu(x)dx.

Passing to the limit m → ∞ in the above relation, since (13) shows that PT_mImϕ converges to ϕ in L²(Ω), we get thanks to strong/weak convergence properties,

Z

R^d

∇u(x)· ∇ϕ(x)dx=− Z

Ω

ϕ(x)w(x) =− Z

R^d

ϕ(x)w(x),

where we denote bywthe prolongement of wby 0 inR^d\Ω. This proves that∇u∈Hdiv(R^d) and that

∆u=wa.e. inR^d, which means that ∆u= 0 outside Ω and that ∆u=w a.e. in Ω. Sinceu∈H¹(R^d) and ∆u∈L²(R^d), a classical result of regularity shows thatu∈H²(R^d). Since∇u= 0 inR^d\Ω, we get that the trace of∇uon∂Ω is equal to 0. Henceu∈H₀²(Ω).

Let us now prove the strong convergence of ∇umto ∇u. Using the weak convergence of this sequence, it suffices to prove the convergence of k∇u_mk_L2(Ω)^d to k∇uk_L2(Ω)^d. To this aim, we write the relation obtained by settingu=v=u_min (11):

Z

Ω

|∇um(x)|²=− Z

Ω

P_Tmum(x)∆mum(x)dx, ∀m∈N.

Passing to the limit m→ ∞ in the above relation, we get, using strong/weak convergence properties in the right hand side,

m→∞lim Z

Ω

|∇um(x)|²=− Z

Ω

u(x)∆u(x)dx= Z

Ω

|∇u(x)|²,

hence concluding the proof.

In order to conclude the convergence analysis, it is natural to examine the convergence of the discrete Laplace operator, when applied toITϕ∈V_T,0, for any regular function ϕ∈C²(Ω)∩H₀²(Ω). We show in the next lemma that this operator is indeed a second order discrete operator.

Lemma 3.5 (Order of the discrete Laplace operator applied to standard interpolation) Let us assume Hypotheses (3). Let T be a conforming simplicial mesh ofΩand let θ >0 such that θ≤θ_T. Then there existsC1>0, only depending onθ, such that

|∇Sξz| ≤ C1

h_S, ∀z ∈ VS,∀S ∈ T, (22)

and

|∆zITϕ| ≤C1|ϕ|2, ∀ϕ∈C²(Ω), ∀z∈ Vint, (23) where∆z is defined by (7),I_Tϕ is defined by (5)and|ϕ|2= maxi,j=1,dk∂_ij²ϕk_L∞(Ω).

Proof. Inequality (22) results from the fact that the ratio, between the distance from any vertex ofS to the opposite face andhS, is larger than 2θ. We now consider z∈ Vint andϕ∈C²(Ω). We can write, using (7),

∆zITϕ= 1

|Kz| X

y∈Vz

(ϕ(y)−ϕ(z))Tzy.

A Taylor expansion providesϕ(x)−ϕ(z) =G·(x−z) +D(x,z)|x−z|², where|D(x,z)| ≤d²|ϕ|2and G=∇ϕ(z). Let us check that the discrete operator ∆zvanishes on the affine functionµ : x7→G·(x−z) (which is such thatµ∈V_T). Indeed, we have, using (11) and (12),

−|Kz|∆zµ =− Z

Ω

∆_Tµ(x)1_Kz(x)dx=− Z

Ω

∆_Tµ(x)P_Tξ_z(x)dx

= Z

Ω

∇µ(x)· ∇ξz(x)dx= Z

Ω

G· ∇ξz(x)dx= 0.

We therefore get

∆zI_Tϕ= 1

|K_z| X

y∈Vz

TzyD(y,z)|y−z|²=− 1

|K_z| X

S∈Tz

X

y∈VS

|S|∇Sξz· ∇SξyD(y,z)|y−z|².

Using (22) and the regularity conditionθP

S∈Tz|S| ≤ |Kz| and|y−z| ≤hS, we conclude (23).

Since the discrete Laplace operator is a second order discrete operator, the question of its strong conver- gence to the continuous Laplace operator arises. Indeed, the proof that ∆_TI_Tϕconverges to ∆ϕfor the weak topology ofL²(Ω) results from the following property, the proof of which uses Lemma 3.1:

Z

Ω

(∆_TI_Tϕ−∆ϕ)P_Tvdx= Z

Ω

∆ϕ(v−P_Tv)dx− Z

Ω

(∇I_Tϕ− ∇ϕ)· ∇vdx≤Ch_Tk∇vk_L2(Ω)^d, ∀v∈V_T,0. Nevertheless, whatever be the choice of (Kz)z∈V satisfying the hypotheses required above, it is not possible in the general case to obtain that ∆_TITϕstrongly converges to ∆ϕas h_T →0, whileθ≤θ_T. Therefore it is not possible to conclude to the convergence of the scheme by letting v = ITϕ in (10):

another interpolation is necessary, which we introduce in the following Lemma 3.6.

Lemma 3.6 (Interpolation of regular functions with compact support) Let us assume Hypothe- ses (3). Let T be a conforming simplicial discretization of Ω, and letθ >0 be given such thatθ < θ_T. Let ϕ∈C_c²(Ω) and leta= d(support(ϕ), ∂Ω).

Then there existsIe_Tϕ∈V_T,0 andC >0 only depending onΩandθ such that keITϕ−ϕkL²(Ω)≤ChT

|ϕ|2

a² , (24)

k∇eITϕ− ∇ϕk_L2(Ω)^d≤ChT

|ϕ|2

a² , (25)

and

k∆TIeTϕ−∆TϕkL²(Ω)≤ChT

|ϕ|2

a² , (26)

where|ϕ|2= max_i,j=1,dk∂²_ijϕkL^∞(Ω) and∆_Tϕis the piecewise constant function defined by

∆_zϕ= 1

|Kz| Z

Kz

∆ϕ(x)dx, ∀z∈ V, (27)

and

∆_Tϕ(x) =X

z∈V

∆zϕ 1K_z(x), for a.e. x∈Ω. (28)

Proof. Letρ∈C_c^∞(R^d,R+) be the function defined by ρ(x) = exp(−1/(1− |x|²))

R

B(0,1)exp(−1/(1− |y|²))dy, ∀x∈B(0,1),

1

Ω

a a

ψ ϕ

Figure 2: Functionsϕandψ

andρ(x) = 0 for x∈/ B(0,1). Letψ∈C_c^∞(Ω,[0,1]) be the function defined by ψ(y) =

Z

x∈Ω,d(x,∂Ω)>^a₂

4 a

d

ρ 4

a(y−x)

dx, ∀y∈Ω. (29)

Thenψ(x) = 0 for allx∈Ω such that d(x, ∂Ω)< ^a₄ andψ(x) = 1 for allx∈Ω such that d(x, ∂Ω)>^3a₄ (see Figure 2). The idea of the construction ofIe_Tϕis to consider the approximation ofϕinV_T,0obtained by the finite element method in the case where the right hand side is given by−∆ϕ; sinceIe_Tϕmust be equal to 0 on the boundary cells, we multiply this discrete solution by ψ. Then the proof mimics the identity ∆(ψv) =v∆ψ+ 2∇ψ· ∇v+ψ∆v.

We first suppose thatT is such thath_T < ^a₄. Let us definebv andev∈V_T,0such that

∀v∈V_T,0,





 Z

Ω

∇ev(x)· ∇v(x)dx=− Z

Ω

∆ϕ(x)P_Tv(x)dx, and

Z

Ω

∇bv(x)· ∇v(x)dx=− Z

Ω

∆ϕ(x)v(x)dx.

(30)

We definewe=ev−v. By subtracting the second relation to the first one in (30), and settingb v =w, wee get

Z

Ω

|∇w(x)|e ²dx=− Z

Ω

∆ϕ(x)(PTw(x)e −w(x))dx.e Applying Lemma 3.1, we obtain

Z

Ω

|∇w(x)|e ²dx≤ k∆ϕk_L2(Ω)kP_Tw(x)e −w(x)ke _L2(Ω)≤h_Tk∆ϕk_L2(Ω)k∇wke _L2(Ω)^d. We then deduce

k∇ve− ∇bvk_L2(Ω)^d≤h_Tk∆ϕkL²(Ω).

A standard interpolation result gives the existence ofCθ, only depending on Ω and θ, such that

k∇ϕ− ∇ITϕkL²(Ω)≤CθhT|ϕ|2. (31) Thanks to

Z

Ω

|∇bv(x)− ∇ϕ(x)|²dx= Z

Ω

(∇bv(x)− ∇ϕ(x))·(∇I_Tϕ(x)− ∇ϕ(x))dx, we also get

k∇bv− ∇ϕk_L2(Ω)^d≤C_θh_T|ϕ|₂. Hence, definingw=ev− I_Tϕ, we obtain

k∇wk_L2(Ω)^d≤ k∇(ev−bv)kL²(Ω)+k∇(bv−ϕ)kL²(Ω)+k∇(ϕ− ITϕ)kL²(Ω)≤(2Cθ+ 1)hT|ϕ|2. (32)

The Poincar´e inequality, which holds since w∈H₀¹(Ω),writes

kwkL²(Ω)≤diam(Ω)k∇wk_L2(Ω)^d≤diam(Ω)(2Cθ+ 1)hT|ϕ|2. (33) Let us remark that, thanks to (30),ev satisfies

∆zev= ∆zϕ, ∀z∈ Vint, (34)

using the notation given by (27) (note that this equality is not a priori satisfied forz ∈ Vext). We now observe that we have

ψ(z)∆_zev= ∆_zϕ, ∀z∈ V. (35)

Indeed, if d(z, ∂Ω)> ^3a₄, we have ψ(z) = 1 and z ∈ Vint, which implies that (34) holds. Otherwise, if d(z, ∂Ω)≤ ^3a₄ and z ∈ Vint, we have|Kz|∆zev =R

K_z∆ϕ(x)dx= 0, and if z ∈ Vext, we haveψ(z) = 0 andR

K_z∆ϕ(x)dx= 0. We now define eITϕ∈VT,0 byIeTϕ(z) =ψ(z)ev(z), for allz∈ V. From formula (7), we get

|Kz|∆zIe_Tϕ= X

y∈Vz

Tzy(eI_Tϕ(y)−Ie_Tϕ(z)) = X

y∈Vz

Tzy(ψ(y)ev(y)−ψ(z)ev(z)).

Thanks to the identityab−cd=c(b−d) +d(a−c) + (a−c)(b−d), we get, forz,y∈ V,

ψ(y)ev(y)−ψ(z)ev(z) =ψ(z)(ev(y)−ev(z)) +ev(z)(ψ(y)−ψ(z)) + (ψ(y)−ψ(z))(v(y)e −v(z)).e This implies, from formula (7), that

|K_z|∆_zIe_Tϕ=ψ(z)|K_z|∆_zev+ev(z)|K_z|∆_zI_Tψ+ X

y∈Vz

T_zy(ψ(y)−ψ(z))(ev(y)−ev(z)). (36) We remark that

(ψ(y)−ψ(z))ϕ(y) = (ψ(y)−ψ(z))ϕ(z) = 0, ∀z∈ V, ∀y∈ Vz. (37) Indeed, assumingϕ(y)6= 0 orϕ(z)6= 0, we have d(z, ∂Ω)> aor d(y, ∂Ω)> a. Since d(z,y)≤h_T ≤a/4, we get that d(z, ∂Ω) > ^3a₄ and d(y, ∂Ω) > ^3a₄. This implies ψ(z) = 1 and ψ(y) = 1. Therefore ψ(z)−ψ(y) = 0.

We then get from (37), for allz∈ V andy∈ V_z,

(ψ(y)−ψ(z))(ev(y)−ev(z)) = (ψ(y)−ψ(z))(ev(y)−ϕ(y)−ev(z) +ϕ(z)) and

ev(z) ∆zI_Tψ=ev(z) X

y∈V_z

Tzy(ψ(y)−ψ(z)) = (ev(z)−ϕ(z)) X

y∈V_z

Tzy(ψ(y)−ψ(z)) =w(z)∆zI_Tψ.

Using (35) and the two preceding relations in (36), we obtain

|K_z|

∆_zIe_Tϕ−∆_zϕ

=w(z)|Kz|∆zITψ+ X

y∈Vz

Tzy(ψ(y)−ψ(z))(w(y)−w(z)).

Taking the square of the previous relation and applying the inequality (a+b)²≤2(a²+b²) we get,

|Kz|²

∆_zIeTϕ−∆_zϕ²

≤ 2 (w(z)|Kz|∆zITψ)² +2



 X

y∈Vz

Tzy(ψ(y)−ψ(z))(w(y)−w(z))





2

.

Using the Cauchy-Schwarz inequality and dividing by|Kz|, we obtain

|Kz|

∆_zIeTϕ−∆_zϕ²

≤2 w(z)²|Kz|(∆zITψ)² + 2

|Kz| X

y∈Vz

|Tzy|(ψ(y)−ψ(z))² X

y∈Vz

|Tzy|(w(y)−w(z))².

We now use (22), which implies that

Z

S

∇ξz(x)· ∇ξy(x)dx

≤ |S|C₁²

h²_S , ∀y,z∈ VS, ∀S∈ T. Thanks to the definition (29) ofψ, we have the existence of a constantC₃ such that

k∇ψk ≤ C₃ a .

This leads, using|ψ(y)−ψ(z)| ≤ ^C_a³h_S, to the existence ofC₄, only depending onθ, such that X

y∈Vz

|Tzy|(ψ(y)−ψ(z))²≤ C4

a²|Kz|.

Applying (23) proved in Lemma 3.5, since|ψ|2≤C5/a²,

|∆zITψ| ≤ C₆ a²,

whereC6only depends onθ(this inequality also holds forz∈ Vextsince in this case ∆zI_Tψ= 0). Hence we get

X

z∈V

|Kz|

∆zIeTϕ−∆zϕ²

≤ 2C₆² a⁴

X

z∈V

|Kz|w(z)² +2C4

a² X

z∈V

X

y∈Vz

|T_zy|(w(y)−w(z))².

We now remark that, forS ∈ T such thaty,z∈ VS, we have

|Tzy|(w(y)−w(z))²≤ X

S∈T,y,z∈VS

|T_zy^S |(∇Sw·(y−z))²≤C₁² X

S∈T,y,z∈VS

|S|

h²_S|∇Sw|²h²_S,

where we denote byT_zy^S =−|S|∇Sξz· ∇Sξy. Hence, we may write X

z∈V

X

y∈Vz

|Tzy|(w(y)−w(z))²≤2d(d+ 1)C₁² 2

X

S∈T

|S| |∇Sw|²

remarking that each edge of a simplex occurs two times in the above summation, and that any simplex has ^d(d+1)₂ edges. Gathering the above results, we thus obtain

X

z∈V

|Kz|

∆zIeTϕ−∆zϕ²

≤ 2C₆²

a⁴kwk²_L2(Ω)+ 2d(d+ 1)C4C₁²

a² k∇wk²_L2(Ω)^d.

Using (33) and (32) provides (26). We then remark that, for allv∈V_T,0, we have Z

Ω

(∆_zIe_Tϕ(x)−∆_zϕ(x))P_Tv(x)dx=− Z

Ω

∇Ie_Tϕ(x)· ∇v(x)dx+ Z

Ω

∇ev(x)· ∇v(x)dx,

thanks to both (11) and (30). Hence takingv=Ie_Tϕ−evprovides (25), as well as (24) using the Poincar´e inequality.

The proof of (26) in the casehT ≥ ^a₄ is obtained by definingIeTϕ= 0, and usingk∆TϕkL²(Ω)≤ |ϕ|2|Ω|^1/2, hT/a≥1/4 and 1/a≥1/diam(Ω). Then (25) and (24) follow in that case.

Theorem 3.7 (Convergence of the scheme) Let us assume Hypotheses (3)and(4). Letu∈H₀²(Ω) be the solution of Problem (2); letT be a conforming simplicial mesh of Ωandu_T ∈V_T,0 be the solution of (10). Then, ash_T tends to0 withθ≤θ_T, for a fixed value ofθ >0:

1. u_T converges inL²(Ω) tou, 2. ∇u_T converges inL²(Ω)^d to∇u, 3. ∆_Tu_T converges inL²(Ω) to∆u.

Proof. Let (Tm)_m∈N be a sequence of conforming simplicial meshes of Ω such that h_Tm tends to 0 as m → ∞ and θ < θ_Tm for all m ∈N. Let u_m ∈ V_T⁰

m,0, for all m∈ N, be the solution of (10). Thanks to Lemmas 3.3 and 3.4, we get the existence of a subsequence of (T_m)_m∈N, again denoted (T_m)_m∈N, and of u ∈ H₀²(Ω) such that the conclusion of Lemma 3.4 holds. Let ϕ ∈ C_c^∞(Ω) be given. We take, in (10) with T =Tm, v =Ie_Tmϕ defined by Lemma 3.6. Passing to the limit as m → ∞ in the resulting equation (thanks to the weak/strong convergence properties provided by Lemmas 3.4 and 3.6) and using the density ofC_c^∞(Ω) in H₀²(Ω), we get that uis the solution of Problem (2). By a classical uniqueness argument, we get that the whole sequence converges. Settingv=umin (10), we get thatk∆_Tmumk²_L2(Ω)

converges toR

Ω(f(x)u(x) +g(x)· ∇u(x) +`(x)∆u(x)) dx=R

Ω(∆u(x))²dxasm→ ∞. Together with the weak convergence of ∆_Tmumto ∆uasm→ ∞, this provides the convergence inL²(Ω) of ∆_Tmumto

∆u.

4 Error estimates

Let us first prove a technical lemma used in the following error estimate results.

Lemma 4.1 (An inequality for regular continuous solutions.)

Under Hypotheses (3), let u ∈ C⁴(Ω) be given and let f = ∆(∆u). Let T be a conforming simplicial mesh of Ω and let θ >0 be such that θ < θ_T. Letu_T ∈V_T,0 be the solution of (10) in the case where f = ∆(∆u), g = 0 and ` = 0. Then there exists C > 0, only depending on Ω and θ, such that the following inequality holds:

k∆u−∆_Tu_TkL²(Ω)≤Ch_Tkuk∞,4+ 2 k∆Tu−∆_TvkL²(Ω), ∀v∈V_T,0, (38) where, for allk∈N,kuk_∞,k denotes the maximum value of the absolute value ofuand of its derivatives until orderk and∆_T is defined by (28).

Proof. Letv, w∈V_T,0be given. We have Z

Ω

(∆_Tv(x)−∆_Tu_T(x))∆_Tw(x) dx= Z

Ω

(∆_Tv(x)−∆_Tu(x))∆_Tw(x) dx +

Z

Ω

(∆_Tu(x)−P_T(I_T∆u)(x))∆_Tw(x) dx +

Z

Ω

(PT(IT∆u)(x)−∆TuT(x))∆Tw(x) dx.

Using (10),f = ∆(∆u) andw∈H₀¹(Ω), we may write Z

Ω

∆TuT(x)∆Tw(x) dx= Z

Ω

∆(∆u)(x)w(x) dx=− Z

Ω

∇(∆u)(x)· ∇w(x) dx.

We have, from (11), Z

Ω

P_T(IT∆u)(x)∆_Tw(x) dx=− Z

Ω

∇(IT∆u)(x)· ∇w(x) dx.

Hence we get Z

Ω

(∆_Tv(x)−∆_Tu_T(x))∆_Tw(x) dx= Z

Ω

(∆_Tv(x)−∆_Tu(x))∆_Tw(x) dx +

Z

Ω

(∆_Tu(x)−P_T(I_T∆u)(x))∆_Tw(x) dx +

Z

Ω

∇(∆u− I_T∆u)(x)· ∇w(x) dx.

Using (14) and the Cauchy-Schwarz inequality, we obtain Z

Ω

(∆_Tv(x)−∆_Tu_T(x))∆_Tw(x) dx

≤

k∆Tv−∆TukL²(Ω)+k∆Tu−PT(IT∆u)kL²(Ω)+ 2diam(Ω)k∇(∆u− IT∆u)k_L2(Ω)^d

k∆TwkL²(Ω).

Takingw=v−uT in the above inequality, we get

k∆Tv−∆TuTkL²(Ω)≤ k∆Tv−∆TukL²(Ω)+k∆Tu−PT(IT∆u)kL²(Ω)

+2 diam(Ω)k∇(∆u− IT∆u)k_L2(Ω)^d. We now write

k∆u−∆_Tu_Tk_L2(Ω)≤ k∆u−∆_Tuk_L2(Ω)+k∆_Tu−∆_Tvk_L2(Ω)+k∆_Tv−∆_Tu_Tk_L2(Ω). Thanks to the two above inequalities, we get

k∆u−∆_Tu_Tk_L2(Ω)≤ 2k∆_Tu−∆_Tvk_L2(Ω)+k∆u−∆_Tuk_L2(Ω)+k∆_Tu−P_T(I_T∆u)k_L2(Ω)

+2 diam(Ω)k∇(∆u− I_T∆u)k_L2(Ω)^d. Thanks to the regularity of ∆u, we have

k∆u−∆_Tuk²_L2(Ω) = X

z∈V

Z

K_z

1

|Kz| Z

K_z

(∆u(y)−∆u(x))dx ²

dy

≤ |Ω|4 h²_T4kuk²_∞,3.

(39)

We may also write

k∆_Tu−P_T(I_T∆u)k²_L2(Ω) = X

z∈V

|Kz| 1

|Kz| Z

Kz

(∆u(x)−∆u(z))dx ²

≤ |Ω|h²_T4kuk²_∞,3.

Applying standard results on the interpolation error of the regular function ∆u in V_T, we have the existence ofCP1, only depending onθand Ω, such that

k∇(∆u− I_T∆u)k_L2(Ω)^d≤C_P1h_Tkuk_∞,4. Therefore the proof of (38) follows.

We can then state the following result.

Theorem 4.2 (Error estimate in the case where u∈C_c⁴(Ω))

Let us assume Hypotheses (3)and (4), letu∈C_c⁴(Ω)be given and letf = ∆(∆u). LetT be a conforming simplicial mesh ofΩand letθ >0 be such thatθ < θT. LetuT ∈VT,0 be the solution of (10)in the case wheref = ∆(∆u),g= 0 and`= 0. Then there exists C >0, only depending onΩ, θandusuch that

ku_T −uk_L2(Ω)≤Ch_T, (40)

k∇u_T − ∇uk_L2(Ω)^d≤Ch_T, (41) and

k∆_Tu_T −∆uk_L2(Ω)≤Ch_T. (42)

Proof. We apply Lemma 4.1 withv=eI_Tu. Thanks to Lemma 3.6, we get the existence ofC >0, only depending on Ω, θandu(by its derivatives, and the distance of the support ofuto the boundary of the domain) such that (42) holds. Then, writing

k∇uT − ∇uk_L2(Ω)^d ≤ k∇uT − ∇eITuk_L2(Ω)^d+k∇eITu− ∇uk_L2(Ω)^d, we can apply Lemma 3.2. We get

k∇uT − ∇uk_L2(Ω)^d ≤ 2 diam(Ω)k∆Tu_T −∆_TIeTuk_L2(Ω)^d+k∇eITu− ∇uk_L2(Ω)^d

≤ 2 diam(Ω)(k∆TuT −∆uk_L2(Ω)^d+k∆u−∆TIeTuk_L2(Ω)^d) +k∇eI_Tu− ∇uk_L2(Ω)^d.

Using (42) and Lemma 3.6 provide (41). Then (40) results from the Poincar´e inequality.

Let us now state the result, without assuming that the solution has a compact support.

Theorem 4.3 (Error estimate in the case where u∈C⁴(Ω)∩H₀²(Ω))

Let us assume Hypotheses (3) and (4), let u∈ C⁴(Ω)∩H₀²(Ω) be given and let f = ∆(∆u). Let T be a conforming simplicial mesh of Ω and letθ >0 be such that θ < θ_T. Letu_T ∈V_T,0 be the solution of (10)in the case where f = ∆(∆u),g = 0 and `= 0. Then there exists C >0, only depending onΩ, θ andusuch that

kuT −uk_L2(Ω)≤ChT

1

5, (43)

k∇u_T − ∇uk_L(Ω)^d ≤Ch_T¹⁵, (44) and

k∆_Tu_T −∆uk_L2(Ω)≤Ch_T¹⁵. (45) Proof.

For a givena >0, we define the functionψa by (29), and the functionua by u_a(x) =u(x)ψ_a(x) ∀x∈Ω.

We remark that, for anyi, j= 1, . . . , d,

∂_ij²u_a(x) =ψ_a(x)∂_ij²u(x) +∂_iψ_a(x)∂_ju(x) +∂_jψ_a(x)∂_iu(x) +u(x)∂_ij²ψ_a(x). (46) Thanks to a Taylor expansion of u and ∇u from any pointy ∈ ∂Ω such that |x−y| = d(x, ∂Ω), we get the existence ofCu >0,only depending on usuch that, for all x ∈Ω, |∇u(x)| ≤ Cud(x, ∂Ω) and

|u(x)| ≤Cud(x, ∂Ω)². Thanks to|∂iψa(x)| ≤C/aand|∂_ij²ψa(x)| ≤C/a², we get from (46) the existence ofC_u⁰, only depending onu, such that

|ua|2≤C_u⁰, ∀a∈[0,diam(Ω)]. (47)

Hence we have the existence ofC_u⁰⁰, only depending onu,

|∆ua(x)−∆u(x)| ≤C_u⁰⁰, ∀x∈Ω such that d(x, ∂Ω)≤a.

Since ∆u_a(x) = ∆u(x) for all x∈Ω such that d(x, ∂Ω)≥a, we get

k∆u_a(x)−∆u(x)k²_L2(Ω)≤ |{x∈Ω,d(x, ∂Ω)≤a}|(C_u⁰⁰)². Thanks to Hypotheses (3), there exists someC_Ω>0 such that

|{x∈Ω,d(x, ∂Ω)≤a}| ≤CΩa, ∀a∈[0,diam(Ω)].

On the other hand, since the distance between the support of ua ∈C_c^∞(Ω) and∂Ω is greater thana/4, Lemma 3.6 gives the existence ofC7>0, only depending on Ω andθ, such that

k∆TIeTua−∆Tuak_L2(Ω)≤C7

hT|ua|2

a² ≤C7

hTC_u⁰ a² . We have

k∆TeITua−∆ukL²(Ω)≤ k∆TIeTua−∆TuakL²(Ω)+k∆Tua−∆TukL²(Ω)+k∆Tu−∆ukL²(Ω). Thanks to the Cauchy-Schwarz inequality, we derive

k∆Tua−∆Tuk²_L2(Ω) =X

z∈V

|Kz| 1

|Kz| Z

K_z

(∆ua(x)−∆u(x))dx 2

≤X

z∈V

Z

Kz

(∆ua(x)−∆u(x))²dx=k∆ua−∆uk²_L2(Ω). Since (39) provides

k∆_Tu−∆uk_L2(Ω)≤4|Ω|^1/2h_Tkuk_∞,3, we then get the existence ofC8>0 independent on the mesh, such that

k∆TIeTu_a−∆ukL²(Ω)≤C₇h_TC_u⁰

a² + (C_Ωa)^1/2 (C_u⁰⁰) +C₈h_T.

Choosinga0=h^2/5_T leads to the existence ofC9>0, only depending onu, Ω andθ, such that k∆_TIe_Tua₀−∆uk_L2(Ω)≤C9h^1/5_T .

Applying Lemma 4.1 with v = IeTu_a0, we conclude the proof of the theorem, following the proof of Theorem 4.2 for the derivation of (44) and (43).

5 Numerical results

Let us introduce the following error norms for the solution, its gradient and its Laplacian:

E₀= X

z∈V

|K_z|(u(z)−u_T(z))²

!1/2

/kuk_L2(Ω),

E1= X

S∈T

|S| |∇Su_T − ∇u(xS)|²

!1/2

/k∇uk_L2(Ω),

denoting byxS the center of gravity ofS, and

E2= X

z∈V

|Kz|(∆zuT −∆u(z))²

!^1/2

/k∆uk_L2(Ω).

We are going to study these error norms for a one-dimensional and a two-dimensional examples.

5.1 One-dimensional case

We approximate the solution

u(x) = (x(1−x))²

24 , ∀x∈[0,1], of the problem

u⁽⁴⁾(x) = 1, x∈[0,1], u(0) =u(1) =u⁰(0) =u⁰(1) = 0,

using Scheme (10), with different 1D meshes with N interior points. In the following figure and table, we show the numerical results obtained in the case where the mesh is uniform, i.e. the pointsz ∈ V are located at the abscissaei/N, fori= 0, . . . , N.

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

N E₀ order E₁ order E₂ order

5 0.366 - 0.246 - 8.94E-2 -

10 9.16E-2 '2 6.24E-2 '2 2.24E-2 '2 20 2.29E-2 '2 1.57E-2 '2 5.59E-3 '2 40 5.73E-3 '2 3.92E-3 '2 1.40E-3 '2 80 1.43E-3 '2 9.80E-4 '2 3.49E-4 '2 160 3.58E-4 '2 2.45E-4 '2 8.73E-5 '2 320 8.95E-5 '2 6.13E-5 '2 2.18E-5 '2 640 2.25E-5 '2 1.54E-5 '2 5.50E-6 '2

An order 2 is numerically obtained for the solution, its gradient and its discrete Laplacian, which is much more than the theoretical order proved in this case (1/5). In the following figure and table, we show the numerical results obtained when the interior points z ∈ V are located at the abscissae (i+α_i)/N, for i= 1, . . . , N−1, whereα_i is a random value between−0.3 and 0.3.

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

N E₀ order E₁ order E₂ order

5 0.416 - 0.283 - 0.111 -

10 9.66E-2 '2 6.64E-2 '2 2.49E-2 '2 20 2.40E-2 '2 1.70E-2 '2 6.24E-3 '2 40 6.45E-3 '2 4.67E-3 '2 1.62E-3 '2 80 1.64E-3 '2 1.19E-3 '2 4.15E-4 '2 160 4.42E-4 '2 3.31E-4 '2 1.05E-5 '2 320 1.03E-5 '2 7.56E-5 '2 2.53E-5 '2 640 2.70E-6 '2 2.02E-5 '2 6.50E-6 '2

Again, an order 2 is numerically obtained for the solution, its gradient and its discrete Laplacian, which shows the robustness of the scheme in this less regular case.

5.2 Two-dimensional cases

We consider Scheme (10) for the approximation of the 2D problem, where Ω = (0,1)² and where the continuous solution is given by:

u(x1, x2) = (1−cos(2πx1))(1−cos(2πx2)), ∀(x1, x2)∈[0,1]²,

which satisfies (2) for the ad hocdata f = ∆(∆u), g = 0,` = 0, Ω =]0,1[²; hence we choosef as the function defined by:

f(x1, x2) = ∆(∆u)(x1, x2) = (2π)⁴ 4 cos(2πx1) cos(2πx2)−(cos(2πx1) + cos(2πx2))