Unification of distance inequalities for linear variational problems

Abstract

In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Céa’s or Babuška’s for coercive and non-coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang’s inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.

Appendix

The purpose of this Appendix is to illustrate the use of the theory developed in this work.

Certainly the most common practical application of the study of non-coercive variational problems arises in the framework of the numerical solution of differential equations. The particular case we consider here is far from new, and the estimates we derive for it are also well-known. The point deserving the reader’s attention is that we set the problem in a special functional framework, using the tools developed in Sects. 3, 4, 5 and 6. As a consequence we provide a different interpretation of the classical piecewise linear finite element method first introduced by Courant (1943) to solve second-order elliptic equations, which can be helpful to users.

Actually the application we consider below can be viewed as a non-standard finite volume method (see e.g. Leveque 2002 or Eymard et al. 2000). This is because the unknown function is represented like in the (piecewise linear) finite element method, but the test functions are replaced by characteristic functions of control volumes. More specifically the problem to solve is either the homogeneous Poisson equation or a second-order ordinary differential equation in a bounded domain $\Omega$ of $\mathfrak {R}^N$ for $N=1,2,3$, with boundary $\Gamma$, namely:

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u + \sigma u = f &{} \text{ in } \Omega \\ u = 0 &{} \text{ on } \Gamma , \end{array} \right. \end{aligned}$$

(33)

where $f$ is a given function in $C^0(\bar{\Omega })$, and $\sigma$ is a function satisfying $0 < \sigma _m \le \sigma (x) \le \sigma _M$ $\forall x \in \bar{\Omega }$ for $N=1$ and $\sigma =0$ for $N=2,3$. For the sake of simplicity we assume that $\Omega$ is an interval when $N=1$, a polygon when $N=2$ and a polyhedron when $N=3$.

The existence and uniqueness of $u$ were established a long time ago, and we know that $u$ lies in the Sobolev space $H^2(\Omega )$ (cf. Adams 1975), if $\Omega$ is convex (Grisvard 1985).

The (almost) standard equivalent variational form of problem (33) is

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \text{ Find } u \in X &{} \text{ such } \text{ that }\\ \bar{a}(u,v) = L(v) &{} \forall v \in Y, \end{array} \right. \end{aligned}$$

(34)

where $Y=H^1_0(\Omega )$, that is, the subspace of Sobolev space $H^1(\Omega )$ consisting of functions that vanish a.e. on $\Gamma$. $X$ is the subspace of $Y$ consisting of functions whose laplacian belongs to $L^2(\Omega )$, $\bar{a}(u,v):= \int _{\Omega } {\mathbf {grad}} \; u \cdot {\mathbf {grad}} \; v \;\mathrm{d}{\mathbf {x}}$ and $L(v):= \int _{\Omega } fv \; \mathrm{d}{\mathbf {x}}$.

Now referring to classical books on the finite element method such as Braess (1997), Ciarlet (1978) or Ern and Guermond (2004), among many others, let ${\mathcal P}=\{{\mathcal T}_h\}_h$ be a quasi-uniform family of partitions of $\Omega$ into disjoint intervals, triangles or tetrahedra, according to the value of $N$, satisfying certain compatibility conditions such as $\displaystyle \cup _{K \in {\mathcal T}_h} \bar{K} = \bar{\Omega }$, $\forall {\mathcal T}_h \in {\mathcal P}$. We refer to any of the above references for the other compatibility conditions. The usual interpretation of the subscript $h$ is a reference length characterizing the partition ${\mathcal T}_h$, most commonly the maximum edge length of all its elements. We denote by $G_K$ the centroid of an element $K \in {\mathcal T}_h$.

Now for a given $h$, let $X_h$ be the subspace of $Y$ consisting of continuous functions, whose restriction to each element in ${\mathcal T}_h$ is a polynomial of degree less than or equal to one. Let $P_i$, $i=1,2,\ldots ,I_h$, be a node—i.e. a vertex—of the partition ${\mathcal T}_h$ in the interior of $\Omega$, and $\Pi _i$ be a control volume associated with $P_i$: the union of all the elements in ${\mathcal T}_h$ whose closure contains $P_i$. We denote by $\Gamma _i$ the boundary of $\Pi _i$ and further introduce the space $Y_h$ spanned by the characteristic functions $\chi _i$ of the $\Pi _i$’s. We denote by $v_i$ the coefficients in the expansion of a function $v \in Y_h$ with respect to the $\chi _i$’s, i.e. $v=\displaystyle \sum \nolimits _{i=1}^{I_h} v_i \chi _i$.

We may equip $X_h$ with the same norm as we equip both $X$ and $Y$, namely,

$$\begin{aligned} \parallel v \parallel _X = \parallel v \parallel _Y := \displaystyle \left\{ \int _{\Omega } \left[ v^2 + \displaystyle \sum _{j=1}^N (\partial v/\partial x_j)^2 \right] \mathrm{d}{\mathbf {x}} \right\} ^{1/2} \end{aligned}$$

for which $Y$ is a Hilbert space (cf. Lions and Magenes 1968), and of course so is $X_h$, but not $X$. On the other hand we must define a discrete analog $\parallel v \parallel _{Y_h}$ of this norm for $v \in Y_h$, more specifically, $\parallel v \parallel _{Y_h} = \displaystyle \left[ \int _{\Omega } v^2 \; {\mathbf {x}} + \int _{\Omega } |{\mathbf {grad}}_h v|^2 \; {\mathbf {x}} \right] ^{1/2}$, where ${\mathbf {grad}}_h v$ is a discrete analog of the gradient operator for discontinuous functions on the interfaces of the $\Pi _i$’s. For practical purposes it is possible to avoid the exact definition of such an operator, as seen hereafter. Nevertheless this definition will be given for $N=1$, the only case we address in detail.

As one can easily check, $Y_h \cap Y = \{ 0_Y \}$. Hence the norm of an element $t=v+v_h$ in the direct sum $T$ of $Y$ and $Y_h$, for $v \in Y$ and $v_h \in Y_h$, is given by $\parallel t \parallel _T = [\parallel v \parallel _Y^2 + \parallel v_h \parallel _{Y_h}^2]^{1/2}$. On the other hand, noting that $X \cap X_h=\{0_X\}$, the direct sum $Z=X+X_h$ is simply normed by $\parallel \cdot \parallel _X$.

Using the identity $\int _{\Pi _i} \Delta u \; d{\mathbf {x}} \equiv \oint _{\Gamma _i} \partial u_{|\Pi _i}/\partial n_i dS$ for $u \in X$, where $\partial \cdot /\partial n_i$ denotes the outer normal derivative over $\Gamma _i$, we consider the following problem to approximate (33):

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \text{ Find } u_h \in X_h &{} \text{ such } \text{ that }\\ a_h(u_h,v) = L_h(v) &{} \forall v \in Y_h, \end{array} \right. \end{aligned}$$

(35)

where $a_h \in {\mathcal L}_{2c}(X_h \times Y_h)$ and $L_h \in Y^{'}$ are defined by

• $a_h(u,v):= \displaystyle \sum _{i=1}^{I_h} v_i \left[ - \displaystyle \frac{N+1}{N} \oint _{\Gamma _i} \displaystyle \frac{\partial u_{|\Pi _i}}{\partial n_i} \; \mathrm{d}S + \int _{\Pi _i} \sigma u \; \mathrm{d}{\mathbf {x}} \right]$;

• $L_h(v)=\int _{\Omega } f_h v \; \mathrm{d}{\mathbf {x}}$ .

$f_h$ being defined by $f_h({\mathbf {x}})=f(G_K)$ for every ${\mathbf {x}} \in K$, $\forall K \in {\mathcal T}_h$.

The fact that (35) has a unique solution for $N=2$ or $N=3$ is a consequence of its equivalence with another well-posed problem, corresponding to the classical approximation of (33) by the piecewise linear finite element method, namely,

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \text{ Find } \bar{u}_h \in X_h &{} \text{ such } \text{ that }\\ \bar{a}(\bar{u}_h,v) = L_h(v) &{} \forall v \in X_h, \end{array} \right. \end{aligned}$$

(36)

Indeed it is possible to prove that the $I_h \times I_h$ matrix corresponding to problem (35) is exactly the matrix corresponding to problem (36) multiplied by $N+1$. This assertion can be verified by geometric arguments, which we refrain from developing here, since they require a series of definitions and calculations, that would divert attention from the essence of our purposes. On the other hand the right-hand side vector of (35) equals the one corresponding to (36) multiplied by $N+1$, and thus this factor accounts for the only difference between both problems for $N>1$.

There is a large amount of work devoted to the relationship between finite volume and finite element schemes for two- and three-dimensional boundary value problems. Far from being exhaustive, we refer to Idelsohn and Oñate (1994), Knabner and Angermann (2003) and Ye (2001) for details on this issue. Actually the one-dimensional case provides an ideal framework for illustrating Theorem 6.2, and hence Theorems 6.1 and 5.1 too. For this reason in the rest of this section we address this case in detail, assuming first of all that $\sigma$ is constant. Afterwards we briefly extend the study to the case of a variable $\sigma$.

First we adjust the $x$-coordinate in such a way that $\Omega =(0,L)$, and label the elements in ${\mathcal T}_h$ as $K_i,\; i=1,2,\ldots ,I_h+1$, where $K_i=(x_{i-1},x_i)$, with $0=x_0 < x_1 < \ldots < x_{I_h} < x_{I_h+1} = L$. We further set $h_i:=x_i-x_{i-1}$, $\forall i \in \{1,2,\ldots ,I_h+1\}$.

Next setting $v_0=v_{I_h+1}=0$, we define a norm in $Y_h$ as follows:

$$\begin{aligned} \parallel v \parallel _{Y_h}^2 := \int _0^L v^2 \;\mathrm{d}x + \displaystyle \sum _{i=1}^{I_h+1} \left[ \frac{v_{i}-v_{i-1}}{h_i} \right] ^2 h_i. \end{aligned}$$

Clearly $T$ equipped with the underlying norm is a Hilbert space, and moreover we have

$$\begin{aligned} a_h(u,v) \le M_h \parallel u \parallel _{X} \parallel v \parallel _{Y_h} \quad \forall u \in X, \quad \forall v \in Y_h, \end{aligned}$$

(37)

for a suitable constant $M_h$ that we will specify later on, after having extended $a_h$ to $X$.

For the sake of simplicity we consider the particular case where $x_i-x_{i-1}=h$, $\forall i {\,\in \,} \{1,2,\ldots ,I_h+1\}$, with $h = L/(I_h+1)$. Since $\tilde{a}(u,v) {\le } \max [1,\sigma ]\parallel {u} \parallel _X \parallel {v} \parallel _Y \quad \forall u \in X, \quad \forall v \in Y$, it is clear that the bilinear form $a \in {\mathcal L}_{2c}(Z \times T)$ defined by $a(z,t)=\bar{a}(z,v)+a_h(z,v_h)$, where $z=u+u_h$ with $u \in X$ and $u_h \in X_h$, and $t=v+v_h$ with $v \in Y$ and $v_h \in Y_h$, satisfies

$$\begin{aligned} a(z,t) \le M \parallel z \parallel _{X} \parallel t \parallel _{T} \quad \forall z \in Z, \quad \forall t \in T, \end{aligned}$$

(38)

for a constant $M$ given as a function of $\sigma$ and $M_h$. To determine $M_h$ first we extend $a_h$ to $X$ as:

$$\begin{aligned} a_h(u,v)&= - 2\displaystyle \sum _{i{=}1}^{I_h} h^{{-}1} \left[ \int _{x_{i{-}1}}^{x_i} u^{'}(x)\;\mathrm{d}x - \int _{x_i}^{x_{i{+}1}} u^{'}(x) \mathrm{d}x \right] v_i \nonumber \\&+\, \sigma \int _0^L uv \; \mathrm{d}x, \quad \forall u \in X, \quad \forall v \in Y_h. \end{aligned}$$

(39)

Let us consider the case where $u \in X_h$. Noting that $u^{'}$ is constant in both $K_i$ and $K_{i+1}$, setting $u_i=u(x_i)$ we have,

$$\begin{aligned} a_h(u,v)= 2 \displaystyle \sum _{i=1}^{I_h} \left[ \frac{u_{i}-u_{i+1}}{h}+ \frac{u_i-u_{i-1}}{h} \right] v_i + \sigma \int _0^L u v \; \mathrm{d}x. \end{aligned}$$

(40)

Applying a simple manipulation in the first term of the summation in (40), we easily derive

$$\begin{aligned} a_h(u,v)= 2 \displaystyle \sum _{i=1}^{I_h+1} \frac{(u_{i}-u_{i-1})(v_{i}-v_{i-1})}{h} + \sigma \int _0^L u v \; \mathrm{d}x. \end{aligned}$$

(41)

Now using the Cauchy–Schwarz inequality this gives,

$$\begin{aligned} a_h(u,v)&\le 2 \displaystyle \left[ \sum _{i=1}^{I_h+1} \displaystyle \frac{(u_{i}-u_{i-1})^2}{h} \right] ^{1/2} \displaystyle \left[ \sum _{i=1}^{I_h+1} \displaystyle \frac{(v_{i}-v_{i-1})^2}{h} \right] ^{1/2}\nonumber \\&+\, \sigma \displaystyle \left[ \int _0^L u^2 \; \mathrm{d}x \right] ^{1/2} \displaystyle \left[ \int _0^L v^2 \; \mathrm{d}x \right] ^{1/2}, \end{aligned}$$

(42)

which finally yields $\forall u \in X_h$ and $\forall v \in Y_h$,

$$\begin{aligned} a_h(u,v) \le \max [2,\sigma ] \parallel u \parallel _X \parallel v \parallel _{Y_h}. \end{aligned}$$

(43)

Next taking $u \in X$, similar manipulations produce:

$$\begin{aligned} a_h(u,v)= 2 \displaystyle \sum _{i=1}^{I_h+1} \int ^{x_i}_{x_{i-1}} \displaystyle \frac{u^{'}(v_i - v_{i-1})}{h} \mathrm{d}x + \sigma \int _0^L u v \; \mathrm{d}x. \end{aligned}$$

(44)

After application of the Cauchy–Schwarz inequality to (44) we obtain:

$$\begin{aligned} a_h(u,v)&\le 2 \displaystyle \left[ \sum _{i=1}^{I_h+1} \int _{x_{i-1}}^{x_i} |u^{'}|^2 \right] ^{1/2} \displaystyle \left[ \sum _{i=1}^{I_h+1} \left( \displaystyle \frac{v_{i}-v_{i-1}}{h}\right) ^2 h \right] ^{1/2}\nonumber \\&+\, \sigma \displaystyle \left[ \int _0^L u^2 \; \mathrm{d}x \right] ^{1/2} \displaystyle \left[ \int _0^L v^2 \; \mathrm{d}x \right] ^{1/2}, \end{aligned}$$

(45)

which gives (43) $\forall v \in Y_h$ and $\forall u \in X$. It follows that $M_h=\max [2,\sigma ]$.

The next step is to prove that $a$ is weakly coercive on $X_h \times Y_h$, and to exhibit the underlying constant $\alpha _h >0$. For this purpose we proceed as follows:

Let $u$ be given in $X_h$ and $v \in Y_h$ be defined by $v_i=u_i$, $i=1,2,\ldots ,I_h$. Recalling (41) and noticing that $v_{|K_i}=v_i+v_{i-1}$ for $i=1,2,\ldots ,I_h+1$ and $\int _{K_i} u \; \mathrm{d}x= [u_i+u_{i-1}]h/2$, for $i=1,2,\ldots ,I_h+1$, we have

$$\begin{aligned} a_h(u,v)= 2\displaystyle \sum _{i=1}^{I_h+1} \left[ \left( \frac{u_{i}-u_{i-1}}{h} \right) ^2 + \sigma \displaystyle \frac{(u_i + u_{i-1})^2}{2} \right] \; h. \end{aligned}$$

(46)

Straightforward calculations lead to

$$\begin{aligned} \int _0^L [u^{'}]^2 \; \mathrm{d}x = \displaystyle \sum _{i=1}^{I_h+1} \left( \frac{u_i-u_{i-1}}{h}\right) ^2 h. \end{aligned}$$

(47)

On the other hand by the Friedrichs–Poincaré inequality (cf. Lions and Magenes 1968) we have

$$\begin{aligned} \int _{0}^L u^2 \; \mathrm{d}x \le 4 L^2 \int _{0}^L |u^{'}|^2 \; \mathrm{d}x. \end{aligned}$$

(48)

Plugging (47) and (48) into (46) we easily obtain,

$$\begin{aligned} a_h(u,v) \ge 2(1+4 L^2)^{-1} \parallel u \parallel _X^2. \end{aligned}$$

(49)

On the other hand we have, $\int _{K_i} v^2 \; \mathrm{d}x = (u_{i-1}+u_i)^2 h \le [u_{i-1}^2+(u_{i-1}+u_i)^2+u_i^2]h=6 \int _{K_i} u^2 \;\mathrm{d}x$. Summing from $i=1$ through $i=I_h+1$ and recalling both (47) and the definition of $\parallel \cdot \parallel _{Y_h}$, we obtain,

$$\begin{aligned} \parallel v \parallel _{Y_h} \le \sqrt{6} \parallel u \parallel _X. \end{aligned}$$

(50)

Finally combining (49) and (50) we immediately conclude that $a_h$ is weakly coercive on $X_h \times Y_h$ with constant $\alpha _h = 2(1+4 L^2)^{-1}/\sqrt{6}$.

Theorem 6.2 (for the case $\tilde{a} = a$) can now be applied, leading to

$$\begin{aligned} \parallel u - u_h \parallel _X \le \displaystyle \frac{1}{\alpha _h} \left[ M d_X(u,X_h) + \displaystyle \sup _{v \in Y_h \setminus \{0_Y\}} \frac{a(u,v)-L(v)}{\parallel v \parallel _{Y_h}} \right] + \displaystyle \frac{1}{\alpha } \displaystyle \sup _{v \in Y_h \setminus \{0_Y\}} \frac{L_h(v)-L(v)}{\parallel v \parallel _{Y_h}}, \end{aligned}$$

(51)

where $\alpha$ is the constant of weak coercivity of $a$ on $X_h \times Y_h$, that is $\alpha =\alpha _h$.

From standard approximation results (cf. Strang 1972), there exists a constant $C_I$ independent of $h$ such that

$$\begin{aligned} d_X(u,X_h) \le C_I h \parallel u^{''} \parallel , \end{aligned}$$

(52)

where $\parallel \cdot \parallel$ stands for the norm of $L^2(\Omega )$, that is, $\parallel [\cdot ] \parallel := \displaystyle \left\{ \int _{\Omega } [\cdot ]^2 \; \mathrm{d}x \right\} ^2.$

Using the Cauchy–Schwarz inequality, we have

$$\begin{aligned} |L_h(v)-L(v)|&= \displaystyle \left| \displaystyle \sum _{i=1}^{I_h+1} \int _{K_i} [f-f_h](x) [v_i+v_{i-1}] \; \mathrm{d}x \right| \\&\le \, \displaystyle \left[ \int _0^L [ f - f_h ]^2(x) \; \mathrm{d}x \right] ^{1/2} \displaystyle \left[ \int _0^L v^2(x) \; \mathrm{d}x \right] ^{1/2}. \end{aligned}$$

Now assuming that $f \in H^1(\Omega )$, from the same standard approximations results (see e.g. Ciarlet 1978), there exists a constant $C_F$ independent of $h$ such that $\displaystyle \left\{ \int _0^L [ f - f_h ]^2(x) \; \mathrm{d}x \right\} ^{1/2}$ $\le C_F h \parallel f^{'} \parallel$. It follows that

$$\begin{aligned} \displaystyle \sup _{v \in Y_h \setminus \{0_Y\}} \frac{L_h(v)-L(v)}{\parallel v \parallel _{Y_h}} \le C_F h \parallel f^{'} \parallel . \end{aligned}$$

(53)

We still have to estimate the sup term with numerator equal to $a(u,v)-L(v)=a_h(u,v)-\int _0^L fv \; \mathrm{d}x$, for $v \in Y_h$. In this aim we first note that

$$\begin{aligned} \int _0^L [\sigma u-f]v \; \mathrm{d}x=\int ^L_0 u^{''} v\; \mathrm{d}x = \displaystyle \sum _{i=1}^{I_h} \int _{x_{i-1}}^{x_{i+1}} u^{''}v_i \mathrm{d}x. \end{aligned}$$

(54)

The assumption $f \in H^1(\Omega )$ allows us to legitimately assert that $u \in H^3(\Omega )$. Hence we may use the following identities, which can be proved by straightforward calculations:

$$\begin{aligned} \left\{ \begin{array}{l} \int _{x_{i-1}}^{x_i} u^{'}(x) \mathrm{d}x = \int _{x_{i-1}}^{x_i} [u^{'}(x_i)+(x-x_i)u^{''}(x_i)+\int _x^{x_i} u^{'''}(s)(s-x)\mathrm{d}s]\mathrm{d}x \\ \int ^{x_{i+1}}_{x_i} u^{'}(x) \mathrm{d}x = \int ^{x_{i+1}}_{x_i} [u^{'}(x_i)+(x-x_i)u^{''}(x_i)-\int ^x_{x_i} u^{'''}(s)(s-x)\mathrm{d}s]\mathrm{d}x. \end{array} \right. \end{aligned}$$

(55)

Combining this with (54) we easily obtain,

$$\begin{aligned}&a_h(u,v)-L(v) = \displaystyle \sum _{i=1}^{I_h} \left\{ \int _{x_{i-1}}^{x_{i+1}} u^{''}(x) \mathrm{d}x - 2 h u^{''}(x_i) \right. \nonumber \\&\quad \left. + 2 h^{-1} \left[ \int _{x_{i-1}}^{x_i} \int _x^{x_i} u^{'''}(s)(s-x)\mathrm{d}s \; \mathrm{d}x +\int ^{x_{i+1}}_{x_i}\int ^x_{x_i} u^{'''}(s)(s-x)\mathrm{d}s \; \mathrm{d}x\right] \right\} v_i. \end{aligned}$$

(56)

But since $\int _{x_{i-1}}^{x_{i+1}} u^{''}(x)\mathrm{d}x= 2hu^{''}(x_i)+\int _{x_{i-1}}^{x_{i+1}} \int _{x_i}^x u^{'''}(s)\mathrm{d}s\; \mathrm{d}x$, it follows from (56) that

$$\begin{aligned} a_h(u,v)-L(v)&\le \displaystyle \sum _{i=1}^{I_h} \left\{ \int _{x_{i-1}}^{x_{i+1}} \int _{x_{i-1}}^{x_{i+1}} | u^{'''}(s)|| v_i | \mathrm{d}s\; \mathrm{d}x \right. \nonumber \\&\left. +2 \displaystyle \left[ \int _{x_{i-1}}^{x_i} \int _{x_{i-1}}^{x_i} |u^{'''}(s)| | v_i |\mathrm{d}s \; \mathrm{d}x + \int ^{x_{i+1}}_{x_i} \int ^{x_{i+1}}_{x_i}| u^{'''}(s)| | v_i | \mathrm{d}s \;\mathrm{d}x \right] \right\} . \end{aligned}$$

(57)

Now we adjust the range of $i$ in the summation of the second term in brackets from $i=2$ up to $i=I_h+1$, taking into account that $v_0=v_{I_h+1}=0$. Combining this with the obvious relation $|v_i|+|v_{i-1}| \le |v_i+v_{i-1}|+|v_i-v_{i-1}|L/h$, the Cauchy–Schwarz inequality leads to

$$\begin{aligned} a_h(u,v)-L(v) \le C_A h \parallel u^{'''} \parallel \parallel v \parallel _{Y_h}. \end{aligned}$$

(58)

for a suitable constant $C_A$ independent of $h$.

For $f \in H^1(\Omega )$, collecting (51), (52), (53) and (58), we have thus proven the error estimate:

$$\begin{aligned} \parallel u - u_h \parallel _X \le C h [ \parallel u^{''} \parallel + \parallel u^{'''} \parallel + \parallel f^{'} \parallel ], \text{ with } C=\max [MC_I,C_F,C_A]/\alpha . \end{aligned}$$

(59)

To complete this example let us briefly examine the case where $\sigma$ is non-constant. Here, in principle it is necessary to work with a polynomial approximation $\sigma _h$ of $\sigma$. More precisely, assuming that $\sigma \in C^0(\bar{\Omega })$ we set $\sigma _h(x) = \sigma (G_{K_i})$ $\forall x \in K_i$, for $i=1,2,\ldots ,I_h+1$, and define the corresponding bilinear form $\tilde{a}_h \in {\mathcal L}_{2c}(X_h \times Y_h)$ by

$$\begin{aligned} \tilde{a}_h(u,v) := 2 \displaystyle \sum _{i=1}^{I_h} [u^{'}(x_{i+1}^{-})-u^{'}(x_{i-1}^{+})]v_i+ \int _0^L \sigma _h uv \; \mathrm{d}x. \end{aligned}$$

(60)

An extension of $\tilde{a}_h$ to $X \times Y_h$ is defined in the same manner as the one of $a_h$. We also define $\bar{a}_h \in {\mathcal L}([X+X_h],Y)$ by replacing $\sigma$ with $\sigma _h$ in the expression of $\bar{a}$.

Accordingly we modify the approximate problem (35) into,

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \text{ Find } \tilde{u}_h \in X_h &{} \text{ such } \text{ that }\\ \tilde{a}_h(\tilde{u}_h,v) = L_h(v) &{} \forall v \in Y_h. \end{array} \right. \end{aligned}$$

(61)

We must also introduce a perturbed bilinear form $\tilde{a} \in {\mathcal L}_{2c}(Z \times T)$ given by $\tilde{a}(z,t) = \bar{a}_h(z,v)+\tilde{a}_h(z,v_h), \text{ where } z=u+u_h \text{ and } t=v+v_h, \text{ with } u \in X, \; u_h \in X_h, \; v \in Y \text{ and } v_h \in Y_h.$

It is not difficult to see that all the bounds that hold for $a_h$ similarly apply to $\tilde{a}_h$, if we replace the constant $\sigma$ by $\sigma _M$ in the upper bounds. In particular this yields constants $\tilde{M}_h$ and $\tilde{\alpha }_h$ that play the same role for $\tilde{a}_h$ as $M_h$ and $\alpha _h$ do for $a_h$. Keeping the same definition of $a \in {\mathcal L}_{2c}(Z \times T)$ as before, we now denote the theoretical solution of (35) by $u_h^{*}$.

In fact the only issue really new here is that now we have to estimate another sup term in (32), namely,

$$\begin{aligned} \displaystyle \sup _{v \in Y_h \setminus \{0_T\}} \displaystyle \frac{[\tilde{a}-a](u_h^{*},v)}{\parallel v \parallel _{Y_h}} = \displaystyle \sup _{v \in Y_h \setminus \{0_T\}} \displaystyle \frac{\int _0^L (\sigma _h-\sigma ) u^{*}_h v \; \mathrm{d}x}{\parallel v \parallel _{Y_h}} \le \parallel \sigma _h - \sigma \parallel _{\infty } \parallel u^{*}_h \parallel , \end{aligned}$$

(62)

where $\parallel g \parallel _{\infty }$ represents the standard maximum norm of a bounded function g in $\bar{\Omega }$, that is $\parallel g \parallel _{\infty } = \max _{x \in \bar{\Omega }} |g(x)|$. Assuming that $\sigma \in C^1(\bar{\Omega })$ again by standard approximation results (see e.g. Quarteroni et al. 2010) we have $\parallel \sigma _h - \sigma \parallel _{\infty } \le C_S h \parallel \sigma ^{'} \parallel _{\infty }$, where $C_S$ is a constant independent of $h$.

Adapting the analysis leading to (59) to the case of a variable $\sigma$, it can be shown that an estimate entirely analogous to (59) holds for $u^{*}_h$. More precisely, for a suitable constant $C^{*}$ depending on $L$ and $\sigma _M$ but not on $h$ we have,

$$\begin{aligned} \parallel u - u^{*}_h \parallel _X \le C^{*} h [ \parallel u^{''} \parallel + \parallel u^{'''} \parallel + \parallel f^{'} \parallel ]. \end{aligned}$$

(63)

From (63) it follows that $\parallel u_h^{*} \parallel \le \parallel u \parallel + C^{*} h [ \parallel u^{''} \parallel + \parallel u^{'''} \parallel + \parallel f^{'} \parallel ]$. On the other hand $\forall v \in X$, $\parallel v^{'} \parallel ^2 = -\int _0^L vv^{''}\;\mathrm{d}x \le \parallel v \parallel \parallel v^{''} \parallel$. Thus using again (48) we obtain $\parallel u \parallel \le 4 L^2 \parallel u^{''} \parallel$. This finally leads to estimate (64), where $\tilde{C}$ is a suitable constant depending on $L,\sigma _M,\parallel \sigma ^{'} \parallel _{\infty }$ but not on $h$:

$$\begin{aligned} \parallel u - \tilde{u}_h \parallel _X \le \tilde{C} h [ \parallel u^{''} \parallel + \parallel u^{'''} \parallel +\parallel f^{'} \parallel ]. \end{aligned}$$

(64)

In short we can assert that, provided the data $f$ and $\sigma$ are sufficiently smooth, the approximate solution $\tilde{u}_h$ converges linearly to $u$ in the natural norm $\parallel \cdot \parallel _X$, as $h$ goes to zero.

Remark 3

The analysis carried out above was deliberately long. This is because we wanted to work in a very broad hilbertian framework, thereby showing how to handle methods that are not usually considered in a variational setting, such as the finite volume method. Indeed this leads to different exact and approximate bilinear forms and right-hand side functionals, spaces not included in each other in the exact and the approximate problem, among other differences. Of course we could have simply compared the linear system corresponding to the approximate problem, to the one of the classical piecewise linear finite element methods. Akin to the multi-dimensional case, we would have found that the latter is identical to the former, except for the numerical quadrature formulae employed to integrate non-constant terms. However we avoided this line of argument, for we are persuaded that our strategy can serve as a guide to the analysis of other problems in future work, to which such a similarity does not apply. $\square$

Remark 4

Incidentally this example provides a noticeable physical interpretation of the classical piecewise linear finite element method. Indeed, as pointed out above, the underlying non-standard finite volume scheme is roughly equivalent to such a finite element method. This means that the latter possesses the flux conservation property across control volumes, though overlapping ones, i.e. the $\Pi _i$’s. $\square$