Some Remarks on the Rigorous Estimation of Inverse Linear Elliptic Operators

Abstract

This paper presents a new numerical method to obtain the rigorous upper bounds of inverse linear elliptic operators. The invertibility of a linearized operator and its norm estimates give important informations when analyzing the nonlinear elliptic partial differential equations (PDEs). The computational costs depend on the concerned elliptic problems as well as the approximation properties of used finite element subspaces, e.g., mesh size or so. We show the proposed new estimate is effective for an intermediate mesh size.

1 Introduction

The main aim of this paper is to provide an efficient estimates of a solution of the following linear elliptic partial differential equations (PDEs) with the Dirichlet boundary condition:

for an arbitrary $f\in L^2(\varOmega )$. Here, $\varOmega \subset \mathbb {R}^d$, $(d \in \{1,2,3\})$ is a bounded polygonal or polyhedral domains, $b\in L^\infty (\varOmega )^d$, and $c\in L^\infty (\varOmega )$. As well known, many physical problems have a linearized problem of the form (1a)-(1b), e.g., the stationary Burgers equations [7].

Now let $L^2(\varOmega )$ be the set of all measurable functions from $\varOmega$ to $\mathbb {C}$ with square integrable, which is a Hilbert space with associated inner product $\left( u,v\right) _{L^2(\varOmega )}:=$$\int _\varOmega u(x)\overline{v(x)}\,dx$, where $\overline{\,\cdot \,}$ shows the complex conjugate. Let $H_0^1(\varOmega ):=\{u\in H^1(\varOmega ) u=0$$\text {on} \partial \varOmega \}$ be the usual Sobolev space with respect to the inner product $\left( u,v\right) _{H_0^1(\varOmega )}:=\left( \nabla u,\nabla v\right) _{L^2(\varOmega )^d}$. Let $L:H_0^1(\varOmega ) \times H_0^1(\varOmega ) \rightarrow \mathbb {C}$ be a bilinear form defined by

$$\begin{aligned} L(u,v):=\left( \nabla u,\nabla v\right) _{L^2(\varOmega )^d} + \left( (b\cdot \nabla )u,v\right) _{L^2(\varOmega )} + \left( cu,v\right) _{L^2(\varOmega )}, \quad \forall u, v \in H_0^1(\varOmega ). \end{aligned}$$

We define the weak solution $u \in H_0^1(\varOmega )$ of (1a)-(1b) by a solution of the following variational equation:

$$\begin{aligned} L(u,v) = \left( f,v\right) _{L^2(\varOmega )}, \quad \forall v\in H_0^1(\varOmega ). \end{aligned}$$

(2)

If we assume the coercivity of L, then, by the Lax-Milgram theorem, there exists a unique solution for (2). Moreover, it can be proved that this weak solution is a solution of (1a)-(1b) by a regularity argument (see e.g., [1]). This fact means that the linear elliptic operator $\mathcal {L}:=-\triangle +b\cdot \nabla +c$ has the inverse operator.

On the other hand, Plum [10], Oishi [9], Nakao-Hashimoto-Watanabe [7] and Kinoshita-Watanabe-Nakao [5] proposed a computational technique to verify the existence of $\mathcal {L}^{-1}$ even though the coercivity of L is not assumed. In this paper, we also do not assume the coercivity to L at all. Moreover, we try to find the quantitative value of $C_{L^2,H_0^1}$ satisfying

$$\begin{aligned} \left\| \mathcal {L}^{-1}\right\| _{\mathcal {L}\left( L^2(\varOmega ),H_0^1(\varOmega )\right) } \le C_{L^2,H_0^1}. \end{aligned}$$

(3)

The constant $C_{L^2,H_0^1}$ plays an essential role in the numerical verification of solutions for the boundary value problems for nonlinear elliptic PDEs [9, 10] and it is desirable to compute $C_{L^2,H_0^1}$ as small as possible. Particularly, the constant $C_{L^2,H_0^1}$ proposed by Watanabe-Kinoshita-Nakao [13] is expected to converge to exact norm $\left\| \mathcal {L}^{-1}\right\| _{\mathcal {L}\left( L^2(\varOmega ),H_0^1(\varOmega )\right) }$ as the discretization parameter $h \rightarrow 0$ on the suitable assumptions. Therefore, in the asymptotic sense, the estimates of (3) by [13] would give better bounds than the results in [7]. Indeed, many numerical examples show this situation. However, in order to get successful calculation of $C_{L^2,H_0^1}$ in [13], we often need smaller mesh size h than [7]. In other words, we could verify the existence of $\mathcal {L}^{-1}$ by the method in [7] with smaller computational costs than [13].

In this paper, we present a new method to compute the constant $C_{L^2,H_0^1}$ in (3) based on the perturbation theory of linear operator with technique in [7]. The verification condition of the existence of $\mathcal {L}^{-1}$ by the proposed method is essentially same as in [7]. But as shown in the numerical results, the proposed $C_{L^2,H_0^1}$ is often better.

The contents of this paper are as follows: In Sect. 2, we define the necessary notations and function spaces. In Sect. 3, we introduce previous results of the invertibility of $\mathcal {L}$ and its a posteriori estimates. In Sect. 4, we propose a new verification condition for the invertibility of $\mathcal {L}$ and its a posteriori estimates. In Sect. 5, we show several verification results for the proposed procedures.

2 Notations

Let $\mathcal {X}$ and $\mathcal {Y}$ be the Banach spaces. We represent the space of the bounded linear operators from $\mathcal {X}$ to $\mathcal {Y}$ by $\mathcal {L}(\mathcal {X},\mathcal {Y})$. Especially, $\mathcal {L}(\mathcal {X})$ denotes $\mathcal {L}(\mathcal {X},\mathcal {X})$. Let $\mathcal {L}_C(\mathcal {X},\mathcal {Y}) \subset \mathcal {L}(\mathcal {X},\mathcal {Y})$ be the space of the compact operators from $\mathcal {X}$ to $\mathcal {Y}$. Moreover, $\mathcal {L}_F(\mathcal {X},\mathcal {Y}) \subset \mathcal {L}(\mathcal {X},\mathcal {Y})$ denotes the set of the bounded Fredholm operators from $\mathcal {X}$ to $\mathcal {Y}$. For any linear operator $\mathcal {A}:\mathcal {X} \rightarrow \mathcal {Y}$, $D(\mathcal {A})$, $R(\mathcal {A})$, and $N(\mathcal {A})$ denote the domain, range, and kernel of $\mathcal {A}$, respectively. We define the norm of $D(\mathcal {A})$ by $\left\| u\right\| _{D(\mathcal {A})}:=\left\| u\right\| _{\mathcal {X}}+\left\| \mathcal {A}u\right\| _{\mathcal {Y}}$, which is called graph norm. As well known, if $\mathcal {A}$ is a closed operator then $D(\mathcal {A})$ becomes a Banach space with respect to $\left\| \,\cdot \,\right\| _{D(\mathcal {A})}$.

Let $-\triangle :D(-\triangle ) \subset L^2(\varOmega ) \rightarrow L^2(\varOmega )$ be a Laplace operator, where the domain $D(-\triangle )$ is defined by

$$\begin{aligned} D(-\triangle )={\left\{ {u \in H_0^1(\varOmega )}~;~{-\triangle u \in L^2(\varOmega )}\right\} }. \end{aligned}$$

Then, $-\triangle$ is a closed operator from $L^2(\varOmega )$ to $L^2(\varOmega )$. We define the differential operator $B \in \mathcal {L}\bigl (H_0^1(\varOmega ),L^2(\varOmega )\bigr )$ by $B:=b\cdot \nabla +c$. The differential operators are treated as the closed operators in many cases. However, it is more convenient to treat the differential operators as the bounded operators in our verification method. Let $I_e:D(-\triangle ) \hookrightarrow H_0^1(\varOmega )$ be an embedding operator. Then, $I_e \in \mathcal {L}_C\bigl (D(-\triangle ),H_0^1(\varOmega )\bigr )$ is satisfied by the Rellich compactness theorem because $\varOmega$ is in a class of the bounded domain with the Lipschitz continuous boundary. Moreover, $BI_e \in \mathcal {L}_C\bigl (D(-\triangle ),L^2(\varOmega )\bigr )$ is satisfied because composition operator of the bounded operator and compact operator is a compact operator. The bounded operator $\mathcal {L} \in \mathcal {L}\bigl (D(-\triangle ),L^2(\varOmega )\bigr )$ is represented by $\mathcal {L}:=-\triangle + BI_e = -\triangle + b\cdot \nabla I_e + cI_e$. Especially, the domain of $\mathcal {L}$ is defined by $D(\mathcal {L})=D(-\triangle )$. Then, $\mathcal {L} \in \mathcal {L}_F\bigl (D(-\triangle ),L^2(\varOmega )\bigr )$ and $\mathrm {ind\,}(\mathcal {L})=0$ by [5].

The norms of Banach space $L^\infty (\varOmega )^d$ and $L^\infty (\varOmega )$ are defined by

respectively.Let $C_{s,2}$ be a positive constant satisfying $\left\| u\right\| _{L^2(\varOmega )} \le C_{s,2}\left\| u\right\| _{H_0^1(\varOmega )}$ for all $u\in H_0^1(\varOmega )$, which is called the Poincaré constant.

Let $S_h(\varOmega )$ be an approximate finite dimensional subspace of $H_0^1(\varOmega )$ dependent on the parameter h. For example, $S_h(\varOmega )$ is considered to be a finite element subspace with the mesh size h or a set of polynomials less than a fixed degree. Let n be a degree of freedom for $S_h(\varOmega )$ and $\{\phi _i\}_{i=1}^n$ be the basis functions of $S_h(\varOmega )$. Namely, .

We denote the self-adjoint positive definite (SPD) matrices $D_\phi$ and $L_\phi$ in $\mathbb {C}^{n \times n}$ by

$$\begin{aligned} D_{\phi ,i,j} := \left( \nabla \phi _j,\nabla \phi _i\right) _{L^2(\varOmega )^d}, \quad L_{\phi ,i,j} := \left( \phi _j,\phi _i\right) _{L^2(\varOmega )}, \quad \forall i,j \in \{1,\ldots ,n\}. \end{aligned}$$

Since $D_\phi$ and $L_\phi$ are SPD, these have the Cholesky factorization. Let $D_\phi ^{1/2}$ and $L_\phi ^{1/2}$ be the Cholesky factors of $D_\phi$ and $L_\phi$, respectively, i.e.,

$$\begin{aligned} D_\phi =D_\phi ^{1/2}D_\phi ^{H/2}, \quad \text {and} \quad L_\phi =L_\phi ^{1/2}L_\phi ^{H/2} \end{aligned}$$

where $D_\phi ^{H/2}$ shows the conjugate matrix of $D_\phi ^{1/2}$. We define the $H_0^1$ projection $P_h^1:H_0^1(\varOmega ) \rightarrow S_h(\varOmega )$ by

$$\begin{aligned} \left( u-P_h^1u,v_h\right) _{H_0^1(\varOmega )}=0, \quad \forall v_h\in S_h(\varOmega ). \end{aligned}$$

(4)

Therefore, the problems of the solvability of the variational Eq. (4) and the nonsingularity of $D_\phi$ are equivalent. Because the matrix $D_\phi$ is positive definite, the projection $P_h^1$ is well defined. Now, we assume that the following error estimates of $P_h^1$ hold throughout this paper.

Assumption 1

There exists a positive constant $C(h)>0$ satisfying

$$\begin{aligned} \left\| u-P_h^1u\right\| _{H_0^1(\varOmega )}&\le C(h)\left\| \triangle u\right\| _{L^2(\varOmega )}, \quad \forall u \in D(-\triangle ), \end{aligned}$$

(5)

$$\begin{aligned} \left\| u-P_h^1u\right\| _{L^2(\varOmega )}&\le C(h)\left\| u-P_h^1u\right\| _{H_0^1(\varOmega )}, \quad \forall u \in H_0^1(\varOmega ). \end{aligned}$$

(6)

Assumption 1 is the most basic error estimates in the Galerkin method. For example, in the case of the one dimensional bounded interval as $\varOmega$, if $S_h(\varOmega )$ is a finite element space using piecewise linear polynomials, the value C(h) is known by $C(h)=\frac{h}{\pi }$. Alternatively, in the case of piecewise quadratic polynomials, Assumption 1 is satisfied by $C(h)=\frac{h}{2\pi }$. Moreover, these approximations give the optimal constants (e.g., [6]). In case that N degree polynomials are used, Assumption 1 is satisfied by $C(h)=O(\frac{h}{N})$. However, in these cases, the optimal constants are unknown (e.g., [3]). In case of the two or three dimensional bounded rectangular or rectangular cuboid domain as $\varOmega$, if $S_h(\varOmega )$ is a finite element space using the tensor product of one dimensional piecewise polynomial spaces, C(h) is attained same constants in one dimensional case (e.g., [6]). In case of the two dimensional bounded polygonal domain as $\varOmega$, if $S_h(\varOmega )$ is the P1 finite element space with triangular mesh, Assumption 1 is satisfied. The details of C(h) are shown in e.g., [2].

Let $G_\phi$ be a matrix in $\mathbb {R}^{n \times n}$, where each elements are defined by

$$\begin{aligned} G_{\phi ,i,j} := L(\phi _j,\phi _i)=\left( \nabla \phi _j,\nabla \phi _i\right) _{L^2} + \left( (b\cdot \nabla )\phi _j,\phi _i\right) _{L^2} + \left( c\phi _j,\phi _i\right) _{L^2}, \quad \forall i,j \in \{1,\ldots ,n\}. \end{aligned}$$

We assume that $G_\phi$ is nonsingular throughout this paper. Applying the proposed verification method, it is necessary to confirm the nonsingularity of $G_\phi$ by validated computations.

3 Previous Results

In this section, we introduce the results for the invertibility condition of the operator $\mathcal {L}$ and its a posteriori estimates. We define the following constants:

$$\begin{aligned} C_1&:= \left\| b\right\| _{L^\infty (\varOmega )^d}+C_{s,2}\left\| c\right\| _{L^\infty (\varOmega )},&C_2&:= \left\| b\right\| _{L^\infty (\varOmega )^d}+C(h)\left\| c\right\| _{L^\infty (\varOmega )}, \\ M_\phi ^{11}(h)&:= \left\| D_\phi ^{H/2}G_\phi ^{-1}D_\phi ^{1/2}\right\| _{2},&M_\phi ^{10}(h)&:= \left\| D_\phi ^{H/2}G_\phi ^{-1}L_\phi ^{1/2}\right\| _{2} \end{aligned}$$

where $\left\| \,\cdot \,\right\| _{2}$ is the matrix two-norm, i.e., the maximum singular value.

Theorem 1

([7, Theorem 2.1 & Corollary 1 & Theorem 2.3] & [8]). Let $\tilde{K}(h)>0$ be defined by

$$\begin{aligned} {\tilde{K}(h):=} \left\{ \begin{array}{ll} C(h)\left( C_{s,2}\left\| \mathrm {div}\,b\right\| _{L^\infty (\varOmega )}+C_1 \right) &{} if\, b \in W^{1,\infty }(\varOmega )^d, \\ &{}\\ C_{s,2}C_2 &{} if\, b \in L^\infty (\varOmega )^d \backslash W^{1,\infty }(\varOmega )^d. \end{array}\right. \end{aligned}$$

And let $\tilde{\kappa }_\phi >0$ be a constant satisfying

$$\begin{aligned} \tilde{\kappa }_\phi :=C(h)\bigl (C_1M_\phi ^{11}(h)\tilde{K}(h)+C_2) < 1. \end{aligned}$$

(7)

Then, there exists $\mathcal {L}^{-1} \in \mathcal {L}\bigl (L^2(\varOmega ),D(-\triangle )\bigr )$ and $C_{L^2,H_0^1}$ in (3) can be taken as

$$\begin{aligned} C_{L^2,H_0^1} = \frac{C_{s,2}}{1-\tilde{\kappa }_\phi } \left\| \begin{pmatrix} M_\phi ^{11}(h)\bigl (1-C_2C(h)\bigr ) &{} M_\phi ^{11}(h) \tilde{K}(h) \\ M_\phi ^{11}(h)C_1C(h) &{} 1 \end{pmatrix}\right\| _{2}. \end{aligned}$$

(8)

If b has sufficient regularity, from the fact that $\tilde{K}(h)=O\bigl (C(h)\bigr )$, the $C_{L^2,H_0^1}$ defined (8) converges to:

$$\begin{aligned} C_{L^2,H_0^1} \rightarrow C_{s,2}\left\| \begin{pmatrix} M_\phi ^{11}(0) &{} 0 \\ 0 &{} 1 \end{pmatrix}\right\| _{2} = C_{s,2} \max \left\{ M_\phi ^{11}(0),1\right\} \end{aligned}$$

(9)

as $h \rightarrow 0$, where $M_\phi ^{11}(0) := \lim _{h \rightarrow 0} M_\phi ^{11}(h)$. This a posteriori estimates fails to converge to its exact operator norm. On the other hand, Watanabe-Kinoshita-Nakao proposed another a posteriori estimates in [5, 13] as follows.

Theorem 2

([13, Theorem 4.2] & [5, Theorem 4.3]). Assume that $\hat{\kappa }_\phi >0$ satisfy

$$\begin{aligned} \hat{\kappa }_\phi :=C(h)C_2\bigl (1+M_\phi ^{10}(h)C_1\bigr )<1. \end{aligned}$$

(10)

Then, there exists $\mathcal {L}^{-1} \in \mathcal {L}\bigl (L^2(\varOmega ),D(-\triangle )\bigr )$ and $C_{L^2,H_0^1}$ in (3) can be taken as

$$\begin{aligned} C_{L^2,H_0^1} = \frac{\sqrt{M_\phi ^{10}(h)^2+C(h)^2\bigl (1+M_\phi ^{10}(h)C_1\bigr )^2}}{1-\hat{\kappa }_\phi }. \end{aligned}$$

(11)

The right hand side of (11) is expected to converge to the exact operator norm as $h \rightarrow 0$. Therefore, we expect that (11) would give better estimates than (8). In fact, we can prove $M_\phi ^{10}(h) \le C_{s,2}M_\phi ^{11}(h)$ for arbitrary $h > 0$. However, in the actual verification process, we often meet the situation such that the criterion (10) is harder than (7) for a fixed h. Therefore, Theorem 1 should be effective for the problem that h cannot be taken so small. We now try to derive $C_{L^2,H_0^1}$ smaller than (8) in Theorem 1 with the same criterion (7).

Note that, in order to obtain the values of $M_\phi ^{11}(h)$ and $M_\phi ^{10}(h)$, it is necessary to solve numerically some corresponding generalized matrix eigenvalue problems. If it succeeded in the verification of the finite upper bound of $M_\phi ^{11}(h)$ or $M_\phi ^{10}(h)$, it means that $G_\phi$ is nonsingular. Rump proposed an efficient method for solving this eigenvalue problem with result verification in [12].

4 Main Theorem

We describe a main theorem of this paper as Theorem 3 in this section. Before describing it, we need to get several lemmas as below.

Lemma 1

Let $b\in L^{\infty }(\varOmega )^d$ and $c\in L^\infty (\varOmega )$. Then, we obtain the following estimates:

$$\begin{aligned} \left\| P_h^1u\right\| _{H_0^1(\varOmega )}&\le M_\phi ^{11}(h)\left\| P_h^1(-\triangle )^{-1}\bigl ((b\cdot \nabla + c)(u-P_h^1u) - \mathcal {L}u\bigr )\right\| _{H_0^1(\varOmega )} \end{aligned}$$

(12)

for all $u \in D(-\triangle )$.

Proof

For an arbitrary $u \in D(-\triangle )$, let $u_\perp := u - P_h^1u$ and $f:=\mathcal {L}u=-\triangle u + b\cdot \nabla u + cu \in L^2(\varOmega )$. Then, u satisfies (2). We take a test function v as $v=v_h \in S_h(\varOmega ) \subset H_0^1(\varOmega )$ in (2), from the definition of $H_0^1$-projection, we have

$$\begin{aligned} \left( \nabla u,\nabla v_h\right) _{L^2(\varOmega )^d} + \left( b\cdot \nabla u + cu,v_h\right) _{L^2(\varOmega )}&= \left( f,v_h\right) _{L^2(\varOmega )} \nonumber \\ L(P_h^1u,v_h)&= \left( -b\cdot \nabla u_\perp - cu_\perp + f,v_h\right) _{L^2(\varOmega )}. \end{aligned}$$

(13)

We set $\psi :=(-\triangle )^{-1}\bigl (-b\cdot \nabla u_\perp - cu_\perp + f\bigr ) \in D(-\triangle )$. In (13), from the definition of $H_0^1$-projection, we obtain

$$\begin{aligned} L(P_h^1u,v_h)&= \left( -b\cdot \nabla u_\perp - cu_\perp + f,v_h\right) _{L^2(\varOmega )}, \quad \forall v_h \in S_h(\varOmega ), \nonumber \\&= \left( -\triangle (-\triangle )^{-1}\bigl (-b\cdot \nabla u_\perp - cu_\perp + f\bigr ),v_h\right) _{L^2(\varOmega )} \nonumber \\&= \left( \nabla \psi ,\nabla v_h\right) _{L^2(\varOmega )^d} \nonumber \\&= \left( \nabla P_h^1\psi ,\nabla v_h\right) _{L^2(\varOmega )^d}. \end{aligned}$$

(14)

Since $P_h^1u$ and $P_h^1\psi$ are elements of $S_h(\varOmega )$, they are represented as linear combinations of the basis of $S_h(\varOmega )$. Namely, there exist $\alpha =(\alpha _1,\ldots ,\alpha _n)^T,~\gamma =(\gamma _1,\ldots ,\gamma _n)^T \in \mathbb {C}^n$ such that

$$\begin{aligned} P_h^1u=\sum _{i=1}^n \alpha _i\phi _i, \quad P_h^1\psi =\sum _{i=1}^n \gamma _i\phi _i. \end{aligned}$$

Then, (14) is rewritten using $\alpha$ and $\gamma$ to have

$$\begin{aligned} G_\phi \alpha = D_\phi \gamma . \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \left\| P_h^1u\right\| _{H_0^1(\varOmega )}^2 =\alpha ^HD_\phi \alpha&=\left( D_\phi ^{H/2}\alpha \right) ^H\left( D_\phi ^{H/2}G_\phi ^{-1}D_\phi ^{1/2}\right) \left( D_\phi ^{H/2}\gamma \right) \\&\le \left\| P_h^1u\right\| _{H_0^1(\varOmega )}\left\| D_\phi ^{H/2}G_\phi ^{-1}D_\phi ^{1/2}\right\| _{2}\left\| P_h^1\psi \right\| _{H_0^1(\varOmega )}, \end{aligned}$$

which proves the lemma.

Let $L_{\mathrm {div}}^{\infty }(\varOmega )^d := {\left\{ {u \in L^\infty (\varOmega )^d}~;~{\mathrm {div\,}u \in L^\infty (\varOmega )}\right\} }$. The right hand side of (12) can be estimated by the following lemma.

Lemma 2

Let $b\in L_{\mathrm {div}}^{\infty }(\varOmega )^d$ and $c\in L^\infty (\varOmega )$. Then, we obtain the following estimates:

$$\begin{aligned} \left\| P_h^1(-\triangle )^{-1}(b\cdot \nabla + c)(u-P_h^1u)\right\| _{H_0^1(\varOmega )}&\le K_1(h)\left\| u-P_h^1u\right\| _{H_0^1(\varOmega )} \end{aligned}$$

(15)

for all $u\in H_0^1(\varOmega )$, where $K_1(h):=C(h)\left( C_{s,2}\left\| \mathrm {div\,}b\right\| _{L^\infty (\varOmega )}+C_1\right)$.

Proof

For an arbitrary $u\in H_0^1(\varOmega )$, let $u_\perp :=u-P_h^1u \in H_0^1(\varOmega )$ and $\psi :=(-\triangle )^{-1}$$(b\cdot \nabla +c)u_\perp \in D(-\triangle )$. Then, we have

$$\begin{aligned} \left\| P_h^1\psi \right\| _{H_0^1(\varOmega )}^2&=\left( \nabla \psi ,\nabla P_h^1\psi \right) _{L^2(\varOmega )^d} \\&=\left( -\triangle \psi ,P_h^1\psi \right) _{L^2(\varOmega )} \\&=\left( (b\cdot \nabla )u_\perp ,P_h^1\psi \right) _{L^2(\varOmega )}+\left( cu_\perp ,P_h^1\psi \right) _{L^2(\varOmega )} \\&=-\left( u_\perp ,\mathrm {div\,}(\overline{b}P_h^1\psi )\right) _{L^2(\varOmega )}+\left( u_\perp ,\overline{c}P_h^1\psi \right) _{L^2(\varOmega )} \\&\le \left( \left\| \mathrm {div\,}(\overline{b}P_h^1\psi )\right\| _{L^2(\varOmega )}+\left\| \overline{c}P_h^1\psi \right\| _{L^2(\varOmega )}\right) \left\| u_\perp \right\| _{L^2(\varOmega )} \\&\le \left( \left\| P_h^1\psi \mathrm {div\,}\overline{b}\right\| _{L^2(\varOmega )}+\left\| (\overline{b}\cdot \nabla )P_h^1\psi \right\| _{L^2(\varOmega )}+\left\| \overline{c}P_h^1\psi \right\| _{L^2(\varOmega )}\right) \left\| u_\perp \right\| _{L^2(\varOmega )} \\&\le \left( \left\| \mathrm {div\,}b\right\| _{L^\infty }\left\| P_h^1\psi \right\| _{L^2}+\left\| b\right\| _{L^\infty }\left\| \nabla P_h^1\psi \right\| _{L^2}+\left\| c\right\| _{L^\infty }\left\| P_h^1\psi \right\| _{L^2}\right) \left\| u_\perp \right\| _{L^2} \\&\le \left( C_{s,2}\left\| \mathrm {div\,}b\right\| _{L^\infty }+\left\| b\right\| _{L^\infty }+C_{s,2}\left\| c\right\| _{L^\infty }\right) \left\| \nabla P_h^1\psi \right\| _{L^2}\left\| u_\perp \right\| _{L^2}. \end{aligned}$$

Applying (6), we obtain (15).

Even if the regularity of b is only $L^\infty (\varOmega )^d$, there exists the following lemma by [4].

Lemma 3

([4, Theorem 3.3]). Let $b\in L^{\infty }(\varOmega )^d$, $c\in L^\infty (\varOmega )$ and let $W_h(\varOmega )$ be a finite element space of $H(\mathrm {div},\varOmega ) := \{ \phi \in L^2(\varOmega )^d; \; \mathrm {div} \phi \in L^2(\varOmega )\}$. For an arbitrary $\psi _h \in S_h$, let $(w_h,v_h)\in W_h(\varOmega )\times S_h(\varOmega )$ be the solution of the following problem:

And define $\sigma _0(h)$ and $\sigma _1(h)$ as follows

$$\begin{aligned} \sigma _0(h) := \sup _{S_h\ni \psi _h\not =0}\frac{\left\| w_h+\nabla v_h-b\psi _h\right\| _{L^2(\varOmega )^d}}{\left\| \nabla \psi _h\right\| _{L^2(\varOmega )^d}}~,\quad \sigma _1(h) := \sup _{S_h\ni \psi _h\not =0}\frac{\left\| \mathrm {div\,}w_h\right\| _{L^2(\varOmega )}}{\left\| \nabla \psi _h\right\| _{L^2(\varOmega )^d}}. \end{aligned}$$

Then, we have

$$\begin{aligned} \left\| P_h^1(-\triangle )^{-1}(b\cdot \nabla + c)(u-P_h^1u)\right\| _{H_0^1(\varOmega )}&\le K_0(h)\left\| u-P_h^1u\right\| _{H_0^1(\varOmega )} \end{aligned}$$

for all $u\in H_0^1(\varOmega )$, where $K_0(h) :=\sigma _0(h)+C(h)\sigma _1(h)+C(h)C_{s,2}\left\| c\right\| _{L^\infty (\varOmega )}$.

Now, let K(h) be a positive constant defined by:

$$\begin{aligned} K(h) := {\left\{ \begin{array}{ll} K_1(h) &{}\mathrm {if~~}b \in L_{\mathrm {div}}^{\infty }(\varOmega )^d, \\ \min \bigl \{C_{s,2}C_2,~K_0(h)\bigr \} &{}\mathrm {if~~}b \in L^{\infty }(\varOmega )^d \backslash L_{\mathrm {div}}^{\infty }(\varOmega )^d. \end{array}\right. } \end{aligned}$$

(16)

From Lemmas 2 and 3, (12) is estimated by

$$\begin{aligned} \left\| P_h^1u\right\| _{H_0^1(\varOmega )}&\le M_\phi ^{11}(h)\left\| P_h^1(-\triangle )^{-1}\bigl ((b\cdot \nabla + c)(u-P_h^1u) - \mathcal {L}u\bigr )\right\| _{H_0^1(\varOmega )} \nonumber \\&\le M_\phi ^{11}(h)K(h)\left\| u-P_h^1u\right\| _{H_0^1(\varOmega )} + M_\phi ^{11}(h)\left\| P_h^1(-\triangle )^{-1}\mathcal {L}u\right\| _{H_0^1(\varOmega )} \nonumber \\&\le M_\phi ^{11}(h)K(h)\left\| u-P_h^1u\right\| _{H_0^1(\varOmega )} + M_\phi ^{11}(h)C_{s,2}\left\| \mathcal {L}u\right\| _{L^2(\varOmega )} \end{aligned}$$

(17)

Lemma 4

Let $b\in L^{\infty }(\varOmega )^d$ and $c\in L^\infty (\varOmega )$. Then, we obtain the following estimates:

$$\begin{aligned} \left\| u-P_h^1u\right\| _{H_0^1(\varOmega )}&\le C(h)\left( C_1\left\| P_h^1u\right\| _{H_0^1(\varOmega )} + C_2\left\| u-P_h^1u\right\| _{H_0^1(\varOmega )} + \left\| \mathcal {L}u\right\| _{L^2(\varOmega )}\right) \end{aligned}$$

(18)

for all $u \in D(-\triangle )$.

Proof

For an arbitrary $u \in D(-\triangle )$, let $u_\perp :=u-P_h^1u$. From the Poincaré inequality and (6), we have

$$\begin{aligned} \left\| \triangle u\right\| _{L^2}&= \left\| \mathcal {L}u - b\cdot \nabla u - cu\right\| _{L^2(\varOmega )} \\&\le \left\| \mathcal {L}u\right\| _{L^2(\varOmega )} + \left\| b\right\| _{L^\infty (\varOmega )^d}\left\| \nabla u\right\| _{L^2(\varOmega )^d} + \left\| c\right\| _{L^\infty (\varOmega )}\left\| u\right\| _{L^2(\varOmega )} \\&\le \left\| \mathcal {L}u\right\| _{L^2} + \left\| b\right\| _{L^\infty }\Bigl (\left\| P_h^1u\right\| _{H_0^1}+\left\| u_\perp \right\| _{H_0^1}\Bigr ) + \left\| c\right\| _{L^\infty }\Bigl (\left\| P_h^1u\right\| _{L^2}+\left\| u_\perp \right\| _{L^2}\Bigr ) \\&\le \left\| \mathcal {L}u\right\| _{L^2} + \Bigl (\left\| b\right\| _{L^\infty } + C_{s,2}\left\| c\right\| _{L^\infty }\Bigr )\left\| P_h^1u\right\| _{H_0^1} + \Bigl (\left\| b\right\| _{L^\infty } + C(h)\left\| c\right\| _{L^\infty }\Bigr )\left\| u_\perp \right\| _{H_0^1}. \end{aligned}$$

Therefore, from (5), we obtain

$$\begin{aligned} \left\| \nabla u_\perp \right\| _{L^2(\varOmega )^d}&\le C(h)\left\| \triangle u\right\| _{L^2(\varOmega )} \\&\le C(h)\left( C_1\left\| P_h^1u\right\| _{H_0^1(\varOmega )} + C_2\left\| u_\perp \right\| _{H_0^1(\varOmega )} + \left\| \mathcal {L}u\right\| _{L^2(\varOmega )}\right) . \end{aligned}$$

By the effective use of the above lemmas, we propose the following estimates based on the Fredholm theory.

Theorem 3

Let $K(h)>0$ be defined by (16). And let $\kappa _\phi >0$ be a constant satisfying

$$\begin{aligned} \kappa _\phi :=C(h)\bigl (C_1M_\phi ^{11}(h)K(h)+C_2) < 1. \end{aligned}$$

(19)

Then, there exists $\mathcal {L}^{-1} \in \mathcal {L}\bigl (L^2(\varOmega ),D(-\triangle )\bigr )$ and $C_{L^2,H_0^1}$ in (3) can be taken as

$$\begin{aligned} C_{L^2,H_0^1} = \frac{\sqrt{M_\phi ^{11}(h)^2\Bigl (C_{s,2} + C(h)\left( K(h)-C_{s,2}C_2\right) \Bigr )^2 + C(h)^2\left( 1+C_{s,2}M_\phi ^{11}(h)C_1\right) ^2}}{1-\kappa _\phi }. \end{aligned}$$

(20)

Proof

For an arbitrary $u \in D(-\triangle )$, we set $u_\perp :=u-P_h^1u \in H_0^1(\varOmega )$. From (17) and (18), we obtain

$$\begin{aligned} \begin{pmatrix} 1 &{} -K(h)M_\phi ^{11}(h) \\ -C(h)C_1 &{} 1-C(h)C_2 \end{pmatrix} \begin{pmatrix} \left\| P_h^1u\right\| _{H_0^1(\varOmega )} \\ \left\| u_\perp \right\| _{H_0^1(\varOmega )} \end{pmatrix}&\le \begin{pmatrix} C_{s,2}M_\phi ^{11}(h) \\ C(h) \end{pmatrix}\left\| \mathcal {L}u\right\| _{L^2(\varOmega )} \end{aligned}$$

where the inequality is meant componentwise. From the assumption (19),

$$\begin{aligned} \det \begin{pmatrix} 1 &{} -K(h)M_\phi ^{11}(h) \\ -C(h)C_1 &{} 1-C(h)C_2 \end{pmatrix} = 1 - \kappa _\phi > 0 \end{aligned}$$

is satisfied. Therefore, the solution of this simultaneous inequalities can be written as

$$\begin{aligned} \begin{pmatrix} \left\| P_h^1u\right\| _{H_0^1(\varOmega )} \\ \left\| u_\perp \right\| _{H_0^1(\varOmega )} \end{pmatrix}&\le \frac{1}{1-\kappa _\phi } \begin{pmatrix} 1-C(h)C_2 &{} K(h)M_\phi ^{11}(h) \\ C(h)C_1 &{} 1 \end{pmatrix} \begin{pmatrix} C_{s,2}M_\phi ^{11}(h) \\ C(h) \end{pmatrix}\left\| \mathcal {L}u\right\| _{L^2(\varOmega )} \\&= \frac{1}{1-\kappa _\phi } \begin{pmatrix} C_{s,2}M_\phi ^{11}(h) + C(h)M_\phi ^{11}(h)\left( K(h)-C_{s,2}C_2\right) \\ C(h)\left( 1+C_{s,2}M_\phi ^{11}(h)C_1\right) \end{pmatrix}\left\| \mathcal {L}u\right\| _{L^2(\varOmega )}. \end{aligned}$$

Then, we have

$$\begin{aligned} \left\| u\right\| _{H_0^1(\varOmega )}^2&= \left\| P_h^1u\right\| _{H_0^1(\varOmega )}^2 + \left\| u_\perp \right\| _{H_0^1(\varOmega )}^2 \\&\le \left( \frac{C_{s,2}M_\phi ^{11}(h) + C(h)M_\phi ^{11}(h)\left( K(h)-C_{s,2}C_2\right) }{1-\kappa _\phi }\right) ^2\left\| \mathcal {L}u\right\| _{L^2(\varOmega )}^2 \\&\qquad + \left( \frac{C(h)\left( 1+C_{s,2}M_\phi ^{11}(h)C_1\right) }{1-\kappa _\phi }\right) ^2\left\| \mathcal {L}u\right\| _{L^2(\varOmega )}^2. \end{aligned}$$

Finally, the invertibility of $\mathcal {L}$ is followed by the same arguments in [5, Theorem4.3].

Remark 1

If $b \in W^{1,\infty }(\varOmega )^d$, the criterion (19) is equal to (7) because $K(h)=\tilde{K}(h)$. Therefore, the attainability of criteria (19) and (7) are essentially same. On the other hand, even if the convergence order $K(h)=O(1)$, namely, independent of smoothness of the function b, the constant $C_{L^2,H_0^1}$ of (20) converges to $C_{s,2}M_\phi ^{11}(0)$ as $h \rightarrow 0$. Comparing this result with (9), we can say that (20) is better than (8) in the asymptotic sense as $h \rightarrow 0$.

5 Numerical Results

In this section, we show some verified computation results of constants $C_{L^2,H_0^1}$ by (8), (11), and (20). Let $\mathcal {L} = -\triangle + b\cdot \nabla + c : D(-\triangle ) \rightarrow L^2(\varOmega )$ be a non-self-adjoint operator with $b := R\begin{pmatrix}-x_2+1/2\\ x_1-1/2\end{pmatrix}$, $R \in \mathbb {R}$, and $c \in \mathbb {C}$ on $\varOmega :=(0,1)\times (0,1) \subset \mathbb {R}^2$. We adopted P1 finite element space with uniform triangular meshes as $S_h(\varOmega )$. Then, discretization parameter $h>0$ is the element side length. In this case, Assumption 1 holds with $C(h)=0.493h$([2]) and $C_{s,2}=\frac{1}{\pi \sqrt{2}}$. Note that, of course our arguments above can also be applied for not only P1 element but also any finite element spaces. We use the interval arithmetic toolbox INTLAB [11] Version 7 with MATLAB 8.0.0.783 (R2012b) on Intel Core i7 3.4 GHz with Mac OSX 10.8.3.

Table 1. $R=10, \quad c=15$

In Table 1, the short line segment means that the corresponding criteria (7), (10), or (19) were not satisfied, which also implies we failed to compute the rigorous upper bounds $C_{L^2,H_0^1}$. From these results, we can say that, for sufficiently small h, the estimates (11) should be finest. On the other hand, if h is not so small, then our proposed estimates (20) is better than others. Therefore, we conclude that three kinds of methods would have their own ranges of suitable applicability depending on each problem.

6 Conclusion

We presented an alternative approach to the numerical verification method for linear ellitipc problems based on Theorem 3. It is proved that our new method gives a better results from the viewpoint in computational costs. As the future subjects, we will show that the present method can also be applied to fourth order elliptic problems or more general linear elliptic operators.