A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations

RIMS-1722

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations

By

Yoshitaka WATANABE, Takehiko KINOSHITA and Mitsuhiro T. NAKAO

May 2011

R _ESEARCH I NSTITUTE FOR M ATHEMATICAL S _CIENCES

KYOTO UNIVERSITY, Kyoto, Japan

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations

Yoshitaka Watanabe

, Takehiko Kinoshita

and Mitsuhiro T. Nakao

a Research Institute for Information Technology, Kyushu University, Fukuoka 812-8581, JAPAN

bResearch Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN, Supported by GCOE ‘Fostering top leaders in mathematics’, Kyoto University

cSasebo National College of Technology, Sasebo City 857-1193, Nagasaki Prefecture, JAPAN

aEmail: [email protected]

Key words.Constructive a posteriori estimates, Galerkin method, Linear elliptic PDEs.

Abstract

This paper presents constructive a posteriori estimates of inverse operators for bound- ary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimates plays an important role in the numerical verification of the solutions for boundary value problems in nonlinear elliptic PDEs. In general, it is not easy to obtain the a priori estimates of the operator norm for inverse elliptic operators.

Even if we can obtain these estimates, they are often over estimated. Our proposed a pos- teriori estimates are based on finite-dimensional spectral norm estimates for the Galerkin approximation and expected to converge to the exact operator norm of inverse elliptic op- erators. This provides more accurate estimates, and more efficient verification results for the solutions of nonlinear problems.

1 Introduction

The main aim of this paper is to provide the positive constantC_L2,H₀¹ satisfying the operator norm:

(−∆+b·∇+c)⁻¹_L

(^L2(Ω),H₀¹(Ω))≤C_L2,H₀¹. (1)

Here, Ω⊂R^d (d =1,2,3) is a bounded polygonal or polyhedral domain, b∈L^∞(Ω)^d, c∈ L^∞(Ω). H₀¹(Ω):={

u∈H¹(Ω); u=0 on ∂Ω}

is a Hilbert space with respect to the inner product is(u,v)_H1

0(Ω):= (∇u,∇v)_L2(Ω)^d and the norm iskuk_H¹

0(Ω):= (u,u)

1 2

H₀¹(Ω). The constant C_L2,H₀¹ plays an essential role in the verification of the solutions for the boundary value prob- lems in nonlinear elliptic partial differential equations (PDEs) [8, 9] and must be numerically determined.

By definingL :=−∆+b·∇+c, the problem of obtaining the estimates of (1) is equivalent to the norm estimation of the solution ufor the following boundary value problems in linear elliptic PDEs such that

{ Lu= f, inΩ, (2a)

u=0, on∂Ω, (2b)

2 FUNCTION SPACES AND GALERKIN APPROXIMATION 2 for arbitrary f ∈L²(Ω). Here, the weak solutionu∈H₀¹(Ω)of (2a) and (2b) is defined by the following variational equation:

L(u,v) = (f,v)_L2(Ω), ∀v∈H₀¹(Ω), (3) for a bilinear formL:H₀¹(Ω)×H₀¹(Ω)→Rdefined by

L(u,v):= (∇u,∇v)_L2(Ω)^d+ ((b·∇)u,v)_L2(Ω)+ (cu,v)_L2(Ω).

If we assume the coercivity of L, then by the Lax-Milgram theorem, there exists a unique solution for (3), indicating the existence of the inverse ofL. Nakao-Hashimoto-Watanabe [6]

proposed the validated computational technique that demonstrates the existence ofL⁻¹even if the coercivity ofLis not assumed. They also derived a technique for obtaining the estimates of (1). In section 3, we introduce these results and discuss them in more detail.

However, the estimates ofL⁻¹in [6] have an unavoidable lower bound. In this study, we propose a novel technique to obtain a posteriori estimates of (1) usingL_h⁻¹that is defined by the Galerkin approximate integral operator forL⁻¹. Our new approach has no restricted lower bound; therefore, it is expected that we can obtainC_L2,H₀¹smaller than that of [6]. Moreover, we introduce a posteriori error estimates forL⁻¹andL_h⁻¹.

The contents of this paper are as follows: In section 2, we introduce the necessary function spaces and calculate the a priori error estimates for their Galerkin approximations. In section 3, we present previously reported methods of error estimation. In section 4, we propose a posteriori estimates of (1). In section 5, we propose a posteriori error estimates for L⁻¹ and L_h⁻¹. Note that in this study, the term “a posteriori error estimates” is defined as the operator norm for integral operators. This suggests that these error estimates can be calculated whenever the Galerkin approximate spaces are given. Therefore, they do not depend on f. In section 6, we compare the constants given by [6] and propose a new value ofC_L2,H₀¹ for the test problems.

2 Function spaces and Galerkin approximation

In this section, we introduce the function spaces and constructive error estimates of projections to finite dimensional subspaces. LetX(Ω):={

u∈L²(Ω); ∆u∈L²(Ω)}

be a Banach space with respect to the normkuk_X(Ω):=kuk_L²(Ω)+k∆uk_L²(Ω). We again define the linear elliptic partial differential operatorL :H₀¹(Ω)∩X(Ω)→L²(Ω)byL :=−∆+b·∇+c. The norms of Banach spaceL^∞(Ω)^d andL^∞(Ω)are defined by

kbk_L∞(Ω)^d :=ess sup

x∈Ω

√

b₁(x)²+···+b_d(x)², kck_L∞(Ω):=ess sup

x∈Ω |c(x)|. The following Theorem 2.1 is the Sobolev inequality.

Theorem 2.1 (Sobolev inequality) Let the constant p satisfy 1≤ p≤ 2^∗, where 2^∗ is the Sobolev conjugate index defined by2^∗:=_d−2^2d . Then, there exists a positive constant Cs,p>0 such that

kuk_L^p_(Ω)≤Cs,pkuk_H¹

0(Ω), ∀u∈H₀¹(Ω). (4)

2 FUNCTION SPACES AND GALERKIN APPROXIMATION 3 Let S_h(Ω) be an approximate finite dimensional subspace of H₀¹(Ω) dependent on the parameter h. For example, S_h(Ω) is considered to be a finite element subspace with the mesh size h or a set of the finite polynomial expansion with polynomial degree. Let n be a degree of freedom for S_h(Ω) and φi be the basis function of S_h(Ω). This indicates that S_h(Ω):=span_1≤i≤n{φi}.

We denote the symmetric positive definite matricesD_φ andL_φ inR^n,nby D_φ,i,j:=(

∇φj,∇φi

)

L²(Ω)^d, 1≤i,j≤n, (5)

L_φ,i,j:=( φj,φi

)

L²(Ω), 1≤i,j≤n. (6)

LetD^1/2_φ andL^1/2_φ be the Cholesky factors ofD_φ andL_φ, respectively, i.e., D_φ =D^1/2_φ D^T_φ^/2, and L_φ =L^1/2_φ L^T_φ^/2. We define theH₀¹projectionP_h¹:H₀¹(Ω)→S_h(Ω)by

(u−P_h¹u,v_h)

H₀¹(Ω)=0, ∀v_h∈S_h(Ω). (7) Therefore, the problems of the solvability of the variational equation (7) and the nonsingularity ofD_φ become equivalent. Because the matrixD_φ is positive definite, the projectionP_h¹is well defined. Similarly, we define theL²projectionP_h⁰:L²(Ω)→S_h(Ω)by

(u−P_h⁰u,v_h)

L²(Ω)=0, ∀v_h∈S_h(Ω). (8)

Now, we assume that the following estimates ofP_h¹hold.

Assumption 2.2 There exist a positive constant C(h)>0satisfying u−P_h¹u

H₀¹(Ω)≤C(h)k∆uk_L²_(Ω), ∀u∈H₀¹(Ω)∩X(Ω), (9) u−P_h¹u

L²(Ω)≤C(h)u−P_h¹u

H₀¹(Ω), ∀u∈H₀¹(Ω). (10) Assumption 2.2 is the most basic error estimates in the Galerkin method. For example, in the case of a finite element space used piecewise bilinear polynomial approximation of H₀¹(Ω), the valueC(h)is known byC(h) =_π^h. Alternatively, in the case of piecewise biquadratic poly- nomial approximation, Assumption 2.2 is satisfied byC(h) =₂^h_π. Moreover, these approxima- tions give the optimal constants (e.g., [5]). In the case ofNdegree polynomial approximation is used, Assumption 2.2 is satisfied byC(h) =O(_N^h). However, in these cases, the optimal constants are unknown (e.g., [3]).

For arbitrary f ∈L²(Ω), we define the Galerkin approximate solution u_h∈S_h(Ω) of (3) such that

(∇u_h,∇v_h)_L2(Ω)^d+ ((b·∇)u_h,v_h)_L2(Ω)+ (cu_h,v_h)_L2(Ω)= (f,v_h)_L2(Ω), ∀v_h∈S_h(Ω). (11)

3 KNOWN RESULTS 4 LetG_φ be a matrix inR^n,n, where each element is defined by

G_φ,i,j:=L(φj,φi) =(

∇φj,∇φi

)

L²+(

(b·∇)φj,φi

)

L²+( cφj,φi

)

L², 1≤i,j≤n. (12) Then, the nonsingularity of G_φ and the unique existence of the solution u_h in (11) become equivalent. Therefore, we assume the nonsingularity of G_φ. However, when applying the proposed a posteriori estimates, it is necessary to confirm the nonsingularity ofG_φ by validated computations.

Next, we define theLprojectionP_h^L:H₀¹(Ω)→S_h(Ω)by

L(u−P_h^Lu,v_h) =0, ∀v_h∈S_h(Ω). (13) From the nonsingularity of G_φ, P_h^L is well defined. If for an arbitrary f ∈L²(Ω) there exists u that is a unique solution for (3), then we denote the operator L⁻¹ :L²(Ω)→H₀¹(Ω) by u=L⁻¹f. By defining the operator L_h⁻¹:L²(Ω)→S_h(Ω), we obtain u_h, the solution of (11). Thus, we obtainL_h⁻¹=P_h^LL⁻¹from the definition ofP_h^L.

3 Known results

In this section, we introduce the result for the invertibility condition of the operatorL and its previously determined estimates. We define the following constants:

C1:=kbk_L∞(Ω)^d+C_s,2kck_L∞(Ω), K1(h):=C(h) (

Cs,2kdivbk_L∞(Ω)+C₁ )

, C2:=kbk_L∞(Ω)^d+C(h)kck_L∞(Ω), K2(h):=√

dCs,2kbk_L∞(Ω)^d+C(h)C_s,2kck_L∞(Ω), M_φ¹¹(h):=D_φ^T/2G⁻_φ¹D^1/2_φ

2,

wherek·k₂is the matrix two-norm i.e., the maximum singular value.

Theorem 3.1 ([6, Theorem 2.1 & Corollary 1]) Let K(h)>0be defined by K(h):=

{ K₁(h), if b∈W^1,∞(Ω)^d, K₂(h), if b∈L^∞(Ω)^d. Letκφ >0satisfy

κφ :=C(h)(

C₁M_φ¹¹(h)K(h) +C₂)<1. (15) Then, under Assumption 2.2, the operatorL is invertible.

We denote the symmetric positive definite matrixRinR^2,2by R:= 1

(1−κφ)²



 M_φ¹¹(h)² (

C₁²C(h)²+(

1−C₂C(h))₂)

symmetry M_φ¹¹(h)

(

C₁C(h) +(

1−C₂C(h))

M_φ¹¹(h)K(h) )

1+M_φ¹¹(h)²K(h)²



. We can obtain the estimates ofL⁻¹usedR.

4 A POSTERIORI ESTIMATES FOR INVERSE LINEAR ELLIPTIC OPERATORS 5 Theorem 3.2 ([6, Theorem 2.3]) By using the same assumptions as those in Theorem 3.1, we obtain the following estimates,

L⁻¹

L(

L²(Ω),H₀¹(Ω))≤C_s,2kRk₂¹². (16) Even ifbhas sufficient regularity, the estimates (16) is expected to converge toCs,2max{M_φ¹¹,1} ash→0. As a result, this a posteriori method over estimates the operator norm and fails to converge to its exact operator norm. Further discussion of the error in the previously reported a posteriori estimates forL⁻¹andL_h⁻¹are discussed in [7]. Next, we will improve this esti- mation method (16), and propose the new a posteriori estimates ofL⁻¹that converges to the exact operator norm.

Theorem 3.3 ([7, Theorem 6]) By using the same assumptions as those in Theorem 3.1, we obtain the following error estimates:

L⁻¹−L_h⁻¹

L(

L²(Ω),H₀¹(Ω))≤C(h)1+C_s,2M_φ¹¹(h)C₁ 1−κφ

√ 1+(

M_φ¹¹(h)K(h))2

, (17)

L⁻¹−L_h⁻¹

L(

L²(Ω),L²(Ω))≤C(h)1+C_s,2M_φ¹¹(h)C₁ 1−κφ

(

C(h) +C_s,2M_φ¹¹(h)K(h) )

. (18)

The proof of Theorem 3.3 can be obtained by using the proof of Theorem 3.2. Therefore, if the estimates of (16) can be improved, then the error estimates of Theorem 3.3 can also be improved. In Section 6, we use numerical examples to describe the results of improving these error estimates.

Remark 3.4 (Aubin-Nitsche trick) In the case of b∈W^1,∞(Ω)^d, the convergence order of (18)is O(h²). Because we can apply the L²error estimates by applying the Aubin-Nitsche trick, the convergence order of K(h)(

=K₁(h))

is O(h). On the other hand, in the case of b∈L^∞(Ω)^d and b6∈W^1,∞(Ω)^d, the convergence order of (18)is O(h). Because the solution for the dual problem of(2a)and(2b)does not have sufficient regularity, we cannot apply the Aubin-Nitsche trick. Therefore, K(h)(

=K₂(h))

does not have the order of h. Thus, when the dual problem becomes singular, it is difficult to obtain the L² error estimates whose convergence order is O(h²). To address this difficulty, we have previously proposed a technique for obtaining L² error estimates by using validated computations in [4]. When this technique is used, it is expected that K₂(h)will have the order h.

4 A posteriori estimates for inverse linear elliptic operators

In this section, we improve the previously reported estimates of (16) by proposing the new a posteriori estimates of L⁻¹, which converges to the exact operator norm. To this end, let M_φ⁰⁰(h),M_φ¹⁰(h), andM_φ⁰¹(h)be the positive constants defined by

M_φ⁰⁰(h):=L_φ^T/2G⁻_φ¹L^1/2_φ

2, M_φ¹⁰(h):=D_φ^T/2G⁻_φ¹L^1/2_φ

2, M_φ⁰¹(h):=L_φ^T/2G⁻_φ¹D^1/2_φ

2, respectively. The following lemma consists of the constantsM_φ⁰⁰ andM_φ¹⁰.

4 A POSTERIORI ESTIMATES FOR INVERSE LINEAR ELLIPTIC OPERATORS 6 Lemma 4.1 The operator norm ofL_h⁻¹satisfies the following equalities

L_h⁻¹

L(

L²(Ω),L²(Ω))=M_φ⁰⁰(h), (19) L_h⁻¹

L(

L²(Ω),H₀¹(Ω))=M_φ¹⁰(h). (20) Proof. —— Note that we only discuss the proof of (20). The proof of (19) is omitted because it is almost the same. For arbitrary f ∈L²(Ω), letu_h:=L_h⁻¹f ∈S_h(Ω). The values fromu_h toP_h⁰f are the elements ofS_h(Ω), and can be expressed by the linear combination of the basis ofS_h(Ω). This indicates thatα^{:= (}α1,···,αn)^T andβ ^{:= (}β1,···,βn)^T ∈Rⁿexists such that

u_h(x) =

∑

αiφi(x), P_h⁰f(x) =

∑

βiφi(x).

The equation (11) is rewritten usingα ^andβ ^{to give}

G_φα ^=Lφβ^{, (21)}

where the matricesG_φ andL_φ are defined by (12) and (6), respectively. BecauseL_φ andD_φ are symmetric positive definite matrices, they can be factorized by the Cholesky decomposition.

From (21), we have

ku_hk²_H¹

0(Ω)=α^TDφα ^=(D_φ^T/2α^)T(DT_φ^/2α⁾ ku_hk_H¹

0(Ω)=D^T_φ^/2α

2

=(

D^T_φ^/2G⁻_φ¹L^1/2_φ )(

L^T_φ^/2β⁾

2

≤D^T_φ^/2G⁻_φ¹L^1/2_φ

2

L^T_φ^/2β

2 (22)

=D^T_φ^/2G⁻_φ¹L^1/2_φ

2

P_h⁰f

L²(Ω)

≤D^T_φ^/2G⁻_φ¹L^1/2_φ

2kfk_L²_(Ω). (23)

Therefore, we obtain L_h⁻¹

L(

L²(Ω),H₀¹(Ω))= sup

L²(Ω)3f6=0

L_h⁻¹f

H₀¹(Ω)

kfk_L²_(Ω) ≤D_φ^T/2G⁻¹_φ L^1/2_φ

2. (24)

Next, we consider the existence of f₀∈L²(Ω)that satisfies the equalities of (22) and (23).

LetB_φ :=D^T_φ^/2G⁻¹_φ L^1/2_φ ,λ ^>0 be a maximum eigenvalue ofB^T_φB_φ, andγ6=0 be an eigenvector associated toλ. Note thatλ ^satisfies√λ ⁼D_φ^T/2G⁻_φ¹L^1/2_φ

2. Because L^T_φ^/2 is nonsingular, we denoteβ0:=(

L^T_φ^/2)₋₁

γ^{. Let f}0∈S_h(Ω)be defined by f₀:=∑ⁿi=1β0,iφi. Then, f₀satisfies

4 A POSTERIORI ESTIMATES FOR INVERSE LINEAR ELLIPTIC OPERATORS 7 the equalities (22) and (23). Practically, we obtain the equality of (22) by

B_φL^T_φ^/2β0²

2=γ^TBT_φ^Bφγ

=λkγk²₂ B_φL^T_φ^/2β0

2=D^T_φ^/2G⁻_φ¹L^1/2_φ

2kγk₂.

Furthermore,P_h⁰f₀= f₀is clear from f₀∈S_h(Ω). Therefore, we have the equality of (23). As a result, (24) satisfies the equality.

From Lemma 4.1, we can expect that accurate estimates of L⁻¹ can be obtained using M_φ¹⁰(h). Practically, we have the following theorem.

Theorem 4.2 Letκˆφ >0satisfy

κˆ_φ ^:=C(h)C2

(1+M_φ¹⁰(h)C₁)

<1. (25)

Then under the same assumptions in as those in Theorem 3.1, we have the following estimates L⁻¹

L(

L²(Ω),H₀¹(Ω))≤

√

M_φ¹⁰(h)²+C(h)²(

1+M_φ¹⁰(h)C₁)2

1−κˆ_φ ^{. (26)}

Proof. —— By assuming (15), we find that the bounded linear operator L⁻¹ :L²(Ω) → H₀¹(Ω)∩X(Ω)exists. For arbitrary f ∈L²(Ω), letu:=L⁻¹f ∈H₀¹(Ω)∩X(Ω). By using the definition ofu,usatisfies the following integral equation

u= (−∆)⁻¹(

−(b·∇)u−cu+f) ,

where(−∆)⁻¹:L²(Ω)→H₀¹(Ω)∩X(Ω)denotes the solution operator of the Poisson equation with homogeneous Dirichlet boundary conditions. We can decompose the finite and infinite dimensional parts using the projectionP_h¹such that

{P_h¹u=P_h(−∆)⁻¹(

−(b·∇)u−cu+f)

, (27a)

(I−P_h¹)u= (I−P_h¹)(−∆)⁻¹(

−(b·∇)u−cu+f)

. (27b)

In short, we denoteu_⊥:=u−P_h¹u. From (27a), for arbitraryv_h∈S_h(Ω), we obtain (∇P_h¹u,∇v_h)

L²(Ω)^d =(

∇P_h¹(−∆)⁻¹(

−(b·∇)u−cu+f) ,∇v_h)

L²(Ω)^d

= (−(b·∇)u−cu+f,v_h)_L2(Ω)

L(P_h¹u,v_h) = (−(b·∇)u_⊥−cu_⊥+f,v_h)_L2(Ω)

=( P_h⁰(

−(b·∇)u_⊥−cu_⊥+ f) ,v_h)

L²(Ω). (28)

4 A POSTERIORI ESTIMATES FOR INVERSE LINEAR ELLIPTIC OPERATORS 8 BecauseP_h¹uand P_h⁰(

−(b·∇)u_⊥−cu_⊥+f)

are the elements of S_h(Ω), they are expressible by the linear combination of the basis of S_h(Ω). This indicates that α ^{:= (}α1,···,αn)^T and β ^{:= (}β1,···,βn)^T ∈Rⁿexists such that

P_h¹u=

∑

αiφi, P_h⁰(

−(b·∇)u_⊥−cu_⊥+f)

=

∑

βiφi. (28) is rewritten usingα ^andβ ^{to give}

G_φα ^=Lφβ^{. (29)}

From (29), we have P_h¹u²

H₀¹(Ω)=α^TDφα

= (

D_φ^T/2α^)T(DT_φ^/2G_φ^−1L1/2_φ ^)(LT_φ^/2β⁾

≤P_h¹u

H₀¹(Ω)D_φ^T/2G⁻¹_φ L^1/2_φ

2

P_h⁰(

−(b·∇)u_⊥−cu_⊥+f)

L²(Ω). By using Assumption 2.2 and the fact thatP_h⁰isL²projection, we have

P_h¹u

H₀¹(Ω)≤M_φ¹⁰(h)k−(b·∇)u_⊥−cu_⊥+fk_L²_(Ω)

≤M_φ¹⁰(h)

(kbk_L∞(Ω)^dk∇u_⊥k_L²_(Ω)^d+kck_L∞(Ω)ku_⊥k_L²_(Ω)+kfk_L²_(Ω))

≤M_φ¹⁰(h)C₂k∇u_⊥k_L²_(Ω)^d+M_φ¹⁰(h)kfk_L²_(Ω). (30) Next, by calculating theH₀¹norm of (27b) from Assumption 2.2, we obtain

ku_⊥k_H¹

0(Ω)≤C(h)k−(b·∇)u−cu+fk_L²_(Ω)

≤C(h)

(kbk_L∞(Ω)^dk∇uk_L²(Ω)^d+kck_L∞(Ω)kuk_L²(Ω)+kfk_L²(Ω)

)

≤C(h)(

kbk_L∞(∇P_h¹u

L²+k∇u_⊥k_L²)

+kck_L∞(P_h¹u

L²+ku_⊥k_L²)

+kfk_L²)

≤C(h)C₁∇P_h¹u

L²(Ω)^d+C(h)C₂k∇u_⊥k_L²_(Ω)^d+C(h)kfk_L²_(Ω). (31) From (31) and (30), we obtain

ku_⊥k_H¹

0 ≤C(h)C₁ (

M_φ¹⁰(h)C₂ku_⊥k_H¹

0 +M_φ¹⁰(h)kfk_L²)

+C(h)C₂ku_⊥k_H¹

0 +C(h)kfk_L². By using Assumption (25), we obtain

ku_⊥k_H0¹_(Ω)≤C(h)1+M_φ¹⁰(h)C₁

1−κˆφ kfk_L²(Ω). (32) From (30) and (32), we have

P_h¹u

H₀¹(Ω)≤M_φ¹⁰(h)

1−κˆφ kfk_L²_(Ω). (33)

4 A POSTERIORI ESTIMATES FOR INVERSE LINEAR ELLIPTIC OPERATORS 9 Finally, from (33), (32), and the fact thatP_h¹isH₀¹projection, we have

kuk²_H¹

0(Ω)=P_h¹u²

H₀¹(Ω)+ku_⊥k²_H¹

0(Ω)

≤ M_φ¹⁰(h)²

(1−κˆφ)²kfk²_L²_(Ω)+C(h)²

(1+M_φ¹⁰(h)C₁)2

(1−κˆφ)² kfk²_L²_(Ω)

= M_φ¹⁰(h)²+C(h)²(

1+M_φ¹⁰(h)C₁)2

(1−κˆφ)² kfk²_L²_(Ω) L⁻¹f

H₀¹(Ω)≤

√

M_φ¹⁰(h)²+C(h)²(

1+M_φ¹⁰(h)C₁)₂

1−κˆφ kfk_L²_(Ω), Therefore, this proof is completed.

TheL²estimates are obtained by providing a proof similar to that of Theorem 4.2.

Theorem 4.3 By using the same assumptions as those in Theorem 4.2, we obtain the following estimates

L⁻¹

L(

L²(Ω),L²(Ω))≤ M_φ⁰⁰(h) +C(h)²(

1+M_φ¹⁰(h)C₁)

1−κˆ_φ ^{. (34)}

Proof. —— For arbitrary f ∈L²(Ω), letu:=L⁻¹f ∈H₀¹(Ω)∩X(Ω). From (29), we obtain P_h¹u²

L²(Ω)= (

L_φ^T/2α^)T(LT_φ^/2G_φ^−1L1/2_φ ^)(LT_φ^/2β⁾

≤P_h¹u

L²(Ω)L^T_φ^/2G⁻_φ¹L^1/2_φ

2

P_h⁰(

−(b·∇)u_⊥−cu_⊥+f)

L²(Ω). By using Assumption 2.2 and (32), we obtain

P_h¹u

L²(Ω)≤M_φ⁰⁰(h)C₂k∇u_⊥k_L²_(Ω)^d+M_φ⁰⁰(h)kfk_L²_(Ω)

≤M_φ⁰⁰(h)C₂C(h)1+M_φ¹⁰(h)C₁

1−κˆ_φ ^{kfkL2(Ω)+Mφ00(h)kfkL2(Ω)}

=M_φ⁰⁰(h)

1−κˆφ kfk_L²(Ω). (35)

Similarly, for the estimates ofku_⊥k_L²_(Ω), by using Assumption 2.2 and (32), we obtain ku_⊥k_L²_(Ω)≤C(h)ku_⊥k_H¹

0(Ω)≤C(h)²1+M_φ¹⁰(h)C₁

1−κˆφ kfk_L²_(Ω). (36) From (35) and (36), we obtain

kuk_L²_(Ω)≤P_h¹u

L²(Ω)+ku_⊥k_L²_(Ω)

≤ M_φ⁰⁰(h)

1−κˆφ kfk_L²_(Ω)+C(h)²1+M_φ¹⁰(h)C₁

1−κˆφ kfk_L²_(Ω),

4 A POSTERIORI ESTIMATES FOR INVERSE LINEAR ELLIPTIC OPERATORS 10 Therefore, this proof is completed.

To obtain theL^pestimates, the following theorem is necessary.

Theorem 4.4 (Gagliardo-Nirenberg) Let the constants p and q satisfy1≤p≤q^∗≤∞. Then, for arbitrary0≤θ ≤1, there exists the positive constant C_g,r,p,q>0such that

kuk_L^r_(Ω)≤Cg,r,p,qkuk^θ_L^p_(Ω)kuk_W^1−θ1,q(Ω), ∀u∈W^1,q(Ω), (37) where ¹_r = ^θ_p+^1−θ_q∗ .

It is known that the optimal constants ofC_g,r,p,qin Theorem 4.4 become the minimum eigen- value of the certain nonlinear elliptic boundary value problems (e.g., [1]). Moreover, we can obtain the upper bounds ofCg,r,p,qby Sobolev constants. For example, if we can calculate the Sobolev constants forC_s,2∗>0 in (4), then for arbitrary 2≤p≤2^∗, we obtain

kuk_L^p(Ω)≤ kuk^1−d

(1 2−¹_p) L²(Ω) kuk^d

(1 2−¹_p) L^2∗(Ω)

≤C^d

(1 2−¹_p)

s,2^∗ kuk^1−d

(1 2−¹_p) L²(Ω) kuk^d

(1 2−¹_p) H₀¹(Ω) .

Therefore, we obtainC_g,p,2,2≤C^d

(1 2−¹_p) s,2^∗ .

Finally, in this section, we present theL^pestimates.

Corollary 4.5 Assume that the following two inequalities are provided:

L⁻¹

L(

L²(Ω),L²(Ω))≤C_L2,L²

L⁻¹

L(

L²(Ω),H₀¹(Ω))≤C_L2,H₀¹

then, for arbitrary2≤p≤2^∗, we obtain L⁻¹

L(

L²(Ω),L^p(Ω))≤C_g,p,2,2C^1−d

(1 2−¹_p) L²,L² C^d

(1 2−¹_p)

L²,H₀¹ . (38)

Proof. —— For arbitrary f ∈ L²(Ω), let u :=L⁻¹f ∈H₀¹(Ω)∩X(Ω). From Gagliardo- Nirenberg inequality and assumptions, we have

kuk_L^p_(Ω)≤Cg,p,2,2kuk^1−d

(1 2−¹_p) L²(Ω) kuk^d

(1 2−¹_p) H₀¹(Ω)

≤C_g,p,2,2C^1−d

(1 2−¹_p)

L²,L² kfk^1−d

(1 2−¹_p) L²(Ω) C^d

(1 2−¹_p) L²,H₀¹ kfk^d

(1 2−¹_p) L²(Ω) . Therefore, this proof is completed.