A stabilized finite element method for inverse problems subject to the convection–diffusion equation. I: diffusion-dominated regime

Abstract

The numerical approximation of an inverse problem subject to the convection–diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are of a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the approximation order, the problem’s stability and perturbations in data. The convergence order is the same for both norms, but the $H^1$-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.

1 Introduction

We consider the convection–diffusion equation

$$\begin{aligned} \mathcal {L} u := - \mu \Delta u + \beta \cdot \nabla u = f \quad \text {in } \Omega , \end{aligned}$$

(1)

where $\Omega \subset \mathbb {R}^n$ is open, bounded and connected, $\mu >0$ is the diffusion coefficient and $\beta \in [W^{1,\infty }(\Omega )]^n$ is the convective velocity field. We assume that no information is given on the boundary $\partial \Omega$ and that there exists a solution $u\in H^2(\Omega )$ satisfying (1). For an open and connected subset $\omega \subset \Omega$, define the perturbed restriction $\tilde{U}_\omega := u\vert _{\omega } + \delta$, where $\delta \in L^2(\omega )$ is an unknown function modelling measurement noise. The data assimilation (or unique continuation) problem consists in finding u given f and $\tilde{U}_\omega$. Here the coefficients $\mu$ and $\beta$, and the source term f are assumed to be known. This linear problem is ill-posed and it is closely related to the elliptic Cauchy problem, see e.g. [1]. Potential applications include for example flow problems for which full boundary data are not accessible, but where local measurements (in a subset of the domain or on a part of the boundary) can be obtained.

The aim is to design a finite element method for data assimilation with weakly consistent regularization applied to the convection–diffusion equation (1). In the present analysis we consider the regime where diffusion dominates and in the companion paper [3] we treat the one with dominating convective transport. To make this more precise we introduce the Péclet number associated to a given length scale l by

$$\begin{aligned} Pe(l) := \frac{|\beta | l}{\mu }, \end{aligned}$$

for a suitable norm $|\cdot |$ for $\beta$. If h denotes the characteristic length scale of the computation, we define the diffusive regime by $Pe(h)<1$ and the convective regime by $Pe(h)>1$. It is known that the character of the system changes drastically in the two regimes and we therefore need to apply different concepts of stability in the two cases. In the present paper we assume that the Péclet number is small and we use an approach similar to that employed for the Laplace equation in [9], for the Helmholtz equation in [4] and for the heat equation in [5], that is we combine conditional stability estimates for the physical problem with optimal numerical stability obtained using a bespoke weakly consistent stabilizing term. For high Péclet numbers on the other hand, we prove in [3] weighted estimates directly on the discrete solution, that reflect the anisotropic character of the convection–diffusion problem.

In the case of optimal control problems subject to convection–diffusion problems that are well-posed, there are several works in the literature on stabilized finite element methods. In [11] the authors considered stabilization using a Galerkin least squares approach in the Lagrangian. Symmetric stabilization in the form of local projection stabilization was proposed in [10] and using penalty on the gradient jumps in [16, 20]. The key difference between the well-posed case and the ill-posed case that we consider herein is that we can not use stability of neither the forward nor the backward equations. Crucial instead is the convergence of the weakly consistent stabilizing terms and the matching of the quantities in the discrete method and the available (best) stability of the continuous problem. Such considerations lead to results both in the case of high and low Péclet numbers, but the different stability properties in the two regimes lead to a different analysis for each case that will be considered in the two parts of this paper.

The main results of this current work are the convergence estimates with explicit dependence on the Péclet number in Theorems 1 and 2, that rely on the continuous three-ball inequalities in Lemma 2 and Corollary 2.

2 Stability estimates

We prove conditional stability estimates for the unique continuation problem subject to the convection–diffusion equation (1) in the form of three-ball inequalities, see e.g. [19] and the references therein. The novelty here is that we keep track of explicit dependence on the diffusion coefficient $\mu$ and the convective vector field $\beta$. The first such inequality is proven in Corollary 1, followed by Lemma 2 and Corollary 2, where the norms for measuring the size of the data are weakened to serve the purpose of devising a finite element method in Sect. 3.

First we prove an auxiliary logarithmic convexity inequality, which is a more explicit version of [17, Lemma 5.2].

Lemma 1

Suppose that $a, b, c \ge 0$ and $p, q > 0$ satisfy $c \le b$ and $c \le e^{p\lambda } a + e^{-q\lambda } b$ for all $\lambda > \lambda _0 \ge 0$. Then there are $C > 0$ and $\kappa \in (0,1)$ (depending only on p and q) such that

$$\begin{aligned} c \le C e^{q \lambda _0} a^\kappa b^{1-\kappa }. \end{aligned}$$

Proof

We may assume that $a, b > 0$, since $c = 0$ if $a = 0$ or $b = 0$. The minimizer $\lambda _*$ of the function $f(\lambda ) =e^{p\lambda } a + e^{-q\lambda } b$ is given by

$$\begin{aligned} \lambda _* = \frac{1}{p+q} \log \frac{qb}{pa}, \end{aligned}$$

and writing $r = q/p$, the minimum value is

$$\begin{aligned} f(\lambda _*)= & {} a\left( \frac{qb}{pa}\right) ^{p/(p+q)} + b\left( \frac{qb}{pa}\right) ^{-q/(p+q)}\\= & {} \left( r^{p/(p+q)} + r^{-q/(p+q)} \right) a^{q/(p+q)} b^{p/(p+q)}. \end{aligned}$$

This shows that if $\lambda _* > \lambda _0$ then

$$\begin{aligned} c \le C_1 a^\kappa b^{1-\kappa }, \end{aligned}$$

where $\kappa = q/(p+q)$ and $C_1 = r^{p/(p+q)} + r^{-q/(p+q)}$. On the other hand, if $\lambda _* \le \lambda _0$ then it holds that $e^{-q \lambda _0} \le e^{-q \lambda _*} = a^{q/(p+q)} (r b)^{-q/(p+q)}$, or equivalently,

$$\begin{aligned} b^{q/(p+q)} \le e^{q \lambda _0} a^{q/(p+q)} r^{-q/(p+q)}. \end{aligned}$$

Therefore

$$\begin{aligned} c \le b = b^{q/(p+q)} b^{p/(p+q)} \le e^{q \lambda _0} r^{-q/(p+q)} a^{q/(p+q)} b^{p/(p+q)}. \end{aligned}$$

That is, if $\lambda _* \le \lambda _0$ then

$$\begin{aligned} c \le C_2 e^{q \lambda _0} a^\kappa b^{1-\kappa }, \end{aligned}$$

where $C_2 = r^{-q/(p+q)}$. As $e^{q \lambda _0} \ge 1$ and $C_1 > C_2$, the claim follows by taking $C = C_1$. $\square$

The following Carleman inequality is well-known, see e.g. [17]. For the convenience of the reader we have included an elementary proof in Appendix A.

Proposition 1

Let $\rho \in C^3(\Omega )$ and $K \subset \Omega$ be a compact set that does not contain critical points of $\rho$. Let $\alpha ,\tau >0$ and $\phi = e^{\alpha \rho }$. Let $w \in C^2_0(K)$ and $v = e^{\tau \phi } w$. Then there is $C>0$ such that

$$\begin{aligned} \int _K e^{2\tau \phi } (\tau ^3 w^2 + \tau |\nabla w|^2) \,\mathrm {d}x \le C \int _K e^{2 \tau \phi } |\Delta w|^2 \,\mathrm {d}x, \end{aligned}$$

for $\alpha$ large enough and $\tau \ge \tau _0$, where $\tau _0 > 1$ depends only on $\alpha$ and $\rho$.

Using the above Carleman estimate we prove a three-ball inequality that is explicit with respect to $\mu$ and $\beta$, i.e. the constants in the inequality are independent of the Péclet number. The corresponding inequality with constant depending implicitly on the Péclet number is proven for instance in [19]. We denote by B(x, r) the open ball of radius r centred at x, and by $d(x,\partial \Omega )$ the distance from x to the boundary of $\Omega$.

Corollary 1

Let $x_0 \in \Omega$ and $0< r_1< r_2 < d(x_0,\partial \Omega )$. Define $B_j = B(x_0, r_j)$, $j=1,2$. Then there are $C > 0$ and $\kappa \in (0,1)$ such that for $\mu >0$, $\beta \in [L^\infty (\Omega )]^n$ and $u \in H^2(\Omega )$ it holds that

$$\begin{aligned} \left\| u \right\| _{H^1(B_2)} \le C e^{C \tilde{Pe}^2} \left( \left\| u \right\| _{H^1(B_1)} + \frac{1}{\mu } \left\| \mathcal {L} u \right\| _{L^2(\Omega )}\right) ^\kappa \left\| u \right\| _{H^1(\Omega )}^{1-\kappa }, \end{aligned}$$

where $\tilde{Pe} = 1+|\beta |/\mu$ and $|\beta | = \left\| \beta \right\| _{[L^\infty (\Omega )]^n}$.

Proof

Due to the density of $C^2(\Omega )$ in $H^2(\Omega )$, it is enough to consider $u \in C^2(\Omega )$. Let now $0<r_0<r_1$ and $r_2< r_3< r_4 < d(x_0,\partial \Omega )$. We choose non-positive $\rho \in C^\infty (\Omega )$ such that $\rho (x) = -d(x,x_0)$ outside $B_0$. Since $|\nabla \rho | = 1$ outside $B_0$, $\rho$ does not have critical points in $B_4 {\setminus } B_0$. Let $\chi \in C_0^\infty (B_4 {\setminus } B_0)$ satisfy $\chi = 1$ in $B_3 {\setminus } B_1$, and set $w = \chi u$. We apply Proposition 1 with $K = \overline{B_4} {\setminus }B_0$ to get

$$\begin{aligned} \mu ^2 \int _{B_4 {\setminus } B_0} (\tau ^3 |w|^2 + \tau |\nabla w|^2) e^{2\tau \phi }\, \,\mathrm {d}x \le C \int _{B_4 {\setminus } B_0} |\mu \Delta w|^2 e^{2 \tau \phi }\, \,\mathrm {d}x, \end{aligned}$$

(2)

for $\phi = e^{\alpha \rho }$, with large enough $\alpha >0$, and $\tau \ge \tau _0$ (where $\tau _0 > 1$ depends only on $\alpha$ and $\rho$). We bound from above the right-hand side by a constant times

$$\begin{aligned} \int _{B_4 {\setminus } B_0} |\mu \Delta w - \beta \cdot \nabla w|^2 e^{2 \tau \phi }\, \,\mathrm {d}x + |\beta |^2 \int _{B_4 {\setminus } B_0} |\nabla w|^2 e^{2 \tau \phi }\, \,\mathrm {d}x. \end{aligned}$$

Taking $\tau \ge 2 |\beta |^2 / \mu ^2$, the second term above is absorbed by the left-hand side of (2) to give

$$\begin{aligned} \mu ^2 \int _{B_4 {\setminus } B_0} (\tau ^3 |w|^2 + \frac{\tau }{2} |\nabla w|^2) e^{2\tau \phi }\, \,\mathrm {d}x \le C \int _{B_4 {\setminus } B_0} |\mu \Delta w - \beta \cdot \nabla w|^2 e^{2 \tau \phi }\, \,\mathrm {d}x. \end{aligned}$$

(3)

Since $\phi \le 1$ everywhere, by defining $\Phi (r) = e^{-\alpha r}$ we now bound from below the left-hand side in (3) by

$$\begin{aligned} \mu ^2 \int _{B_2 {\setminus } B_1} (\tau ^3 |w|^2 + \tau |\nabla w|^2) e^{2\tau \phi }\, \,\mathrm {d}x \ge \mu ^2 \tau e^{2\tau \Phi (r_2)} \left\| u \right\| ^2_{H^1(B_2)} - \mu ^2 \tau e^{2\tau } \left\| u \right\| ^2_{H^1(B_1)}. \end{aligned}$$

An upper bound for the right-hand side in (3) is given by

$$\begin{aligned}&C \int _{B_4} |\mu \Delta u - \beta \cdot \nabla u|^2 e^{2 \tau \phi }\, \,\mathrm {d}x + C \int _{(B_4 {\setminus } B_3) \cup B_1} |(\mu [\Delta , \chi ] - \beta \cdot \nabla \chi )u|^2 e^{2 \tau \phi }\, \,\mathrm {d}x\\&\quad \le C e^{2 \tau } \left\| \mu \Delta u - \beta \cdot \nabla u \right\| ^2_{L^2(B_4)} + C e^{2 \tau \Phi (r_3)} (\mu ^2 + |\beta |^2) \left\| u \right\| ^2_{H^1(B_4 {\setminus } B_3)} \\&\qquad + C e^{2 \tau } (\mu ^2 + |\beta |^2) \left\| u \right\| ^2_{H^1(B_1)}. \end{aligned}$$

Combining the last two inequalities we thus obtain that

$$\begin{aligned} \mu ^2 e^{2\tau \Phi (r_2)} \left\| u \right\| ^2_{H^1(B_2)}&\le C e^{2\tau } \left( (\mu ^2 + |\beta |^2) \left\| u \right\| ^2_{H^1(B_1)} + \left\| \mu \Delta u - \beta \cdot \nabla u \right\| ^2_{L^2(B_4)} \right) \\&\quad + C e^{2 \tau \Phi (r_3)} (\mu ^2 + |\beta |^2) \left\| u \right\| ^2_{H^1(B_4)}, \end{aligned}$$

for $\tau \ge \tau _0 + 2|\beta |^2/\mu ^2$. We divide by $\mu ^2$ and conclude by Lemma 1 with $p = 1 - \Phi (r_2) > 0$ and $q = \Phi (r_2) - \Phi (r_3) > 0$, followed by absorbing the $\tilde{Pe} = 1 + |\beta |/\mu$ factor into the exponential factor $e^{C\tilde{Pe}^2}$. $\square$

We now shift down the Sobolev indices in Corollary 1 by making a similar argument to that in Section 4 of [12] or Section 2.2 of [4], based on semiclassical pseudodifferential calculus.

Lemma 2

Let $x_0 \in \Omega$ and $0< r_1< r_2 < d(x_0,\partial \Omega )$. Define $B_j = B(x_0, r_j)$, $j=1,2$. Then there are $C > 0$ and $\kappa \in (0,1)$ such that for $\mu >0$, $\beta \in [L^\infty (\Omega )]^n$ and $u \in H^2(\Omega )$ it holds that

$$\begin{aligned} \left\| u \right\| _{L^2(B_2)} \le C e^{C \tilde{Pe}^2} \left( \left\| u \right\| _{L^2(B_1)} + \frac{1}{\mu } \left\| \mathcal {L} u \right\| _{H^{-1}(\Omega )} \right) ^\kappa \left\| u \right\| _{L^2(\Omega )}^{1-\kappa }, \end{aligned}$$

where $\tilde{Pe} = 1+|\beta |/\mu$ and $|\beta | = \left\| \beta \right\| _{[L^\infty (\Omega )]^n}$.

Proof

Let $\hbar > 0$ be the semiclassical parameter that satisfies $\hbar = 1/\tau$, where $\tau$ is the parameter previously introduced in Proposition 1. We will make use of the theory of semiclassical pseudodifferential operators, which we briefly recall in Appendix B for the convenience of the reader. In particular we will use semiclassical Sobolev spaces with norms given by

$$\begin{aligned} \left\| u \right\| _{H_\text {scl}^s(\mathbb {R}^n)} = \left\| J^s u \right\| _{L^2(\mathbb {R}^n)}, \end{aligned}$$

where the scale of the semiclassical Bessel potentials is defined by

$$\begin{aligned} J^s = (1-\hbar ^2 \Delta )^{s/2}, \quad s \in \mathbb {R}. \end{aligned}$$

We will also use the following commutator and pseudolocal estimates, see Appendix B. Suppose that $\eta ,\vartheta \in C_0^\infty (\mathbb {R}^n)$ and that $\eta = 1$ near ${{\,\mathrm{supp}\,}}(\vartheta )$, and let $A_\psi , B_\psi$ be two semiclassical pseudodifferential operators of orders s, m, respectively. Then for all $p,q,N \in \mathbb {R}$, there is $C>0$,

$$\begin{aligned} \left\| [A_\psi ,B_\psi ] u \right\| _{H_\text {scl}^{p}(\mathbb {R}^n)}&\le C \hbar \left\| u \right\| _{H_\text {scl}^{p+s+m-1}(\mathbb {R}^n)}, \end{aligned}$$

(4)

$$\begin{aligned} \left\| (1-\eta ) A_\psi \vartheta u \right\| _{H_\text {scl}^{p}(\mathbb {R}^n)}&\le C \hbar ^N \left\| u \right\| _{H_\text {scl}^{q}(\mathbb {R}^n)}. \end{aligned}$$

(5)

Let $0<r_j< r_{j+1} < d(x_0,\partial \Omega ),\, j=0,\ldots ,4$ and $B_j=B(x_0,r_j)$, keeping $B_1, B_2$ unchanged. Let $\tilde{r}_j \in (r_{j-1}, r_j)$ and $\tilde{B}_j=B(x_0,\tilde{r}_j),\, j=0,\ldots ,3$, where $r_{-1} = 0$. Choose $\rho \in C^\infty (\Omega )$ such that $\rho (x) = -d(x,x_0)$ outside $\tilde{B_0}$, and define $\phi = e^{\alpha \rho }$ for large enough $\alpha$. Consider $v \in C_0^\infty (B_5{\setminus } \tilde{B_0})$. As in Appendix A, by taking $\ell = \phi / \hbar$ and $\sigma = \Delta \ell + 3 \alpha \lambda \phi / \hbar$, we obtain

$$\begin{aligned} C \int _{\mathbb {R}^n} |e^{\phi / \hbar } \Delta (e^{-\phi / \hbar } v) |^2 \,\mathrm {d}x \ge \int _{\mathbb {R}^n} ({\hbar }^{-1} |\nabla v|^2 + \hbar ^{-3} v^2 - |\nabla v|^2 - \hbar ^{-2} v^2) \,\mathrm {d}x. \end{aligned}$$

Scaling this with $\mu ^2 \hbar ^4$, we insert the convective term and obtain that

$$\begin{aligned}&C \int _{\mathbb {R}^n} (\mu e^{\phi / \hbar } \hbar ^2 \Delta (e^{-\phi / \hbar } v)- e^{\phi / \hbar } \hbar ^2 \beta \cdot \nabla (e^{-\phi / \hbar } v) )^2 \,\mathrm {d}x \end{aligned}$$

can be bounded from below by

$$\begin{aligned}&\int _{\mathbb {R}^n} \hbar \mu ^2 (\hbar ^2 |\nabla v|^2 + v^2) \,\mathrm {d}x - \int _{\mathbb {R}^n} \hbar ^{2} \mu ^2 (\hbar ^2 |\nabla v|^2 + v^2) \,\mathrm {d}x\\&\quad - \int _{\mathbb {R}^n} (e^{\phi / \hbar } \hbar ^2 \beta \cdot \nabla (e^{-\phi / \hbar } v))^2 \,\mathrm {d}x. \end{aligned}$$

Since

$$\begin{aligned} e^{\phi / \hbar } \hbar ^2 \beta \cdot \nabla (e^{-\phi / \hbar } v) = -\hbar (\beta \cdot \nabla \phi ) v + \hbar ^2 \beta \cdot \nabla v, \end{aligned}$$

introducing the conjugated operator $P v = -\hbar ^2 e^{\phi / \hbar } \mathcal L (e^{-\phi / \hbar }v)$, the previous bound implies

$$\begin{aligned} C \left\| Pv \right\| _{L^2(\mathbb {R}^n)}^2&\ge \hbar \mu ^2 \left\| v \right\| _{H_\text {scl}^1(\mathbb {R}^n)}^2 - \hbar ^{2} \mu ^2 \left\| v \right\| _{H_\text {scl}^1(\mathbb {R}^n)}^2 - \hbar ^2 |\beta |^2 \left\| v \right\| _{H_\text {scl}^1(\mathbb {R}^n)}^2. \end{aligned}$$

The last two terms in the right-hand side can be absorbed by the first one when

$$\begin{aligned} \hbar \le \frac{1}{2} \quad \text { and }\quad {\hbar } \le \frac{1}{2} \frac{\mu ^2}{|\beta |^2}, \end{aligned}$$

(6)

thus obtaining

$$\begin{aligned} \sqrt{\hbar } \mu \left\| v \right\| _{H_\text {scl}^1(\mathbb {R}^n)} \le C \left\| P v \right\| _{L^2(\mathbb {R}^n)}. \end{aligned}$$

(7)

Let now $\eta , \vartheta \in C_0^\infty (B_5{\setminus } \tilde{B_0})$ and suppose that $\vartheta =1$ near $B_4{\setminus } B_0$ and $\eta = 1$ near ${{\,\mathrm{supp}\,}}(\vartheta )$. Let also $\chi \in C_0^\infty (B_4 {\setminus } B_0)$ satisfy $\chi = 1$ in $B_3 {\setminus } \tilde{B}_1$. Then there is $\hbar _0>0$ such that for $v=\chi w,\, w \in C^\infty (\Omega )$, and $\hbar <\hbar _0$,

$$\begin{aligned} \left\| v \right\| _{L^2(\mathbb {R}^n)} \le \left\| \eta J^{-1} v \right\| _{H_\text {scl}^{1}(\mathbb {R}^n)} + \left\| (1-\eta ) J^{-1} \vartheta v \right\| _{H_\text {scl}^{1}(\mathbb {R}^n)} \le C \left\| \eta J^{-1} v \right\| _{H_\text {scl}^{1}(\mathbb {R}^n)}, \end{aligned}$$

(8)

where we used (5) to absorb one term by the left-hand side. From (8) and (7) we have

$$\begin{aligned} \sqrt{\hbar } \mu \left\| v \right\| _{L^2(\mathbb {R}^n)}\le C\sqrt{\hbar } \mu \left\| \eta J^{-1} v \right\| _{H_\text {scl}^1(\mathbb {R}^n)} \le C \left\| P (\eta J^{-1} v) \right\| _{L^2(\mathbb {R}^n)}, \end{aligned}$$

(9)

and the commutator estimate (4) gives

$$\begin{aligned} \left\| [P,\eta J^{-1}] v \right\| _{L^2(\mathbb {R}^n)} \le C \hbar \mu \left\| v \right\| _{L^2(\mathbb {R}^n)} + C \hbar ^2 |\beta | \left\| v \right\| _{H^{-1}_\text {scl}(\mathbb {R}^n)}. \end{aligned}$$

Recalling the assumption (6), these terms can be absorbed by the left-hand side of (9), obtaining

$$\begin{aligned} \sqrt{\hbar } \mu \left\| v \right\| _{L^2(\mathbb {R}^n)}\le C \left\| \eta J^{-1} (Pv) \right\| _{L^2(\mathbb {R}^n)} \le C \left\| P v \right\| _{H_\text {scl}^{-1}(\mathbb {R}^n)}. \end{aligned}$$

(10)

We now combine this estimate with the technique used to prove Corollary 1. Consider $u \in C^\infty (\mathbb {R}^n)$ and set $w = e^{\phi /\hbar } u$. Take $\psi \in C_0^\infty (\Omega )$ supported in $B_1 \cup (B_5 {\setminus } \tilde{B}_3)$ with $\psi = 1$ in $(\tilde{B}_1{\setminus } B_0) \cup (B_4{\setminus } B_3)$. Recall that $\chi \in C_0^\infty (B_4 {\setminus } B_0)$ satisfies $\chi = 1$ in $B_3 {\setminus } \tilde{B}_1$. Using (4) to bound the commutator

$$\begin{aligned} \left\| [P,\chi ] w \right\| _{H_\text {scl}^{-1}(\mathbb {R}^n)} \le \left\| [P,\chi ] \psi w \right\| _{H_\text {scl}^{-1}(\mathbb {R}^n)} \le C \hbar (\mu +|\beta |) \left\| \psi w \right\| _{L^2 (\mathbb {R}^n)}, \end{aligned}$$

we obtain from (10) that

$$\begin{aligned} \sqrt{\hbar } \mu \left\| \chi w \right\| _{L^2(\mathbb {R}^n)}&\le C \left\| \chi P w \right\| _{H_\text {scl}^{-1}(\mathbb {R}^n)} + C \hbar (\mu +|\beta |) \left\| \psi w \right\| _{L^2 (\mathbb {R}^n)}. \end{aligned}$$

This leads to

$$\begin{aligned}&\sqrt{\hbar } \mu \left\| \chi e^{\phi /\hbar } u \right\| _{L^2(\mathbb {R}^n)} \le C \left\| \chi e^{\phi / \hbar } (\mu \Delta u - \beta \cdot \nabla u ) \right\| _{H^{-1}(\mathbb {R}^n)}\\&\quad + C \hbar (\mu +|\beta |) \left\| \psi e^{\phi /\hbar } u \right\| _{L^2 (\mathbb {R}^n)}, \end{aligned}$$

where we used the norm inequality $\left\| \cdot \right\| _{H_\text {scl}^{-1}(\mathbb {R}^n)} \le C \hbar ^{-2} \left\| \cdot \right\| _{H^{-1}(\mathbb {R}^n)}$. Letting $\Phi (r) = e^{-\alpha r}$ and using a similar argument as in the proof of Corollary 1, we find that

$$\begin{aligned} \mu e^{\Phi (r_2)/\hbar } \left\| u \right\| _{L^2(B_2)}&\le C e^{1/\hbar } \left( (\mu +|\beta |) \left\| u \right\| _{L^2(B_1)} + \hbar ^{-\frac{3}{2}}\left\| (\mu \Delta u - \beta \cdot \nabla u ) \right\| _{H^{-1}(\Omega )} \right) \\&\quad +\,C e^{\Phi (\tilde{r}_3)/\hbar } \hbar ^{\frac{1}{2}} (\mu +|\beta |) \left\| u \right\| _{L^2 (\Omega )}, \end{aligned}$$

when $\hbar$ satisfies (6) and is small enough. Absorbing the negative power of $\hbar$ in the exponential, we then use Lemma 1 and conclude by absorbing the $\tilde{Pe} = 1 + |\beta |/\mu$ factor into the exponential factor $e^{C\tilde{Pe}^2}$. $\square$

Making the additional coercivity assumption $\nabla \cdot \beta \le 0$, we can weaken the norms just in the right-hand side of Corollary 1 by using the stability estimate for a well-posed convection–diffusion problem with homogeneous Dirichlet boundary conditions.

Corollary 2

Let $x_0 \in \Omega$ and $0< r_1< r_2 < d(x_0,\partial \Omega )$. Define $B_j = B(x_0, r_j)$, $j=1,2$. Then there are $C > 0$ and $\kappa \in (0,1)$ such that for $\mu >0$, $\beta \in [W^{1,\infty }(\Omega )]^n$ having ${{\,\mathrm{ess\,sup}\,}}_{\Omega } \nabla \cdot \beta \le 0$, and $u \in H^2(\Omega )$ it holds that

$$\begin{aligned} \left\| u \right\| _{H^1(B_2)}&\le C e^{C \tilde{Pe}^2} \left( \left\| u \right\| _{L^2(B_1)} + \frac{1}{\mu } \left\| \mathcal {L} u \right\| _{H^{-1}(\Omega )}\right) ^\kappa \left( \left\| u \right\| _{L^2(\Omega )} + \frac{1}{\mu } \left\| \mathcal {L} u \right\| _{H^{-1}(\Omega )} \right) ^{1-\kappa }, \end{aligned}$$

where $\tilde{Pe} = 1+|\beta |/\mu$ and $|\beta | = \left\| \beta \right\| _{[L^\infty (\Omega )]^n}$.

Proof

Let the balls $B_0,B_3\subset \Omega$ such that $B_{j}\subset B_{j+1}$, for $j=0,2$. Consider the well-posed problem

$$\begin{aligned} \mathcal {L} w = \mathcal {L} u \text { in } B_3,\quad w=0 \text { on } \partial B_3. \end{aligned}$$

Since ${{\,\mathrm{ess\,sup}\,}}_{\Omega } \nabla \cdot \beta \le 0$, as a consequence of the divergence theorem we have

$$\begin{aligned} \Vert w\Vert _{H^1(B_3)} \le C \frac{1}{\mu } \Vert \mathcal {L} u\Vert _{H^{-1}(B_3)}. \end{aligned}$$

Taking $v = u - w$, we have $\mathcal {L}v = 0$ in $B_3$. The stability estimate in Corollary 1 used for $B_0,B_2,B_3$ reads as

$$\begin{aligned} \left\| v \right\| _{H^1(B_2)} \le C e^{C \tilde{Pe}^2} \left\| v \right\| ^\kappa _{H^1(B_0)} \left\| v \right\| _{H^1(B_3)}^{1-\kappa }, \end{aligned}$$

and the following estimates hold

$$\begin{aligned} \left\| u \right\| _{H^1(B_2)}&\le \left\| v \right\| _{H^1(B_2)} + \left\| w \right\| _{H^1(B_2)}\\&\le C e^{C \tilde{Pe}^2} \left( \left\| u \right\| _{H^1(B_0)} + \frac{1}{\mu } \left\| \mathcal {L} u \right\| _{H^{-1}(\Omega )} \right) ^\kappa \left( \left\| u \right\| _{H^1(B_3)} + \frac{1}{\mu } \left\| \mathcal {L} u \right\| _{H^{-1}(\Omega )} \right) ^{1-\kappa }. \end{aligned}$$

Now we choose a cutoff function $\chi \in C^{\infty }_0(B_1)$ such that $\chi = 1$ in $B_0$. Then $\chi u$ satisfies

$$\begin{aligned} \mathcal {L}(\chi u)=\chi \mathcal {L}u + [\mathcal {L}, \chi ]u,\quad \chi u = 0 \text { on } \partial B_1, \end{aligned}$$

and we obtain

$$\begin{aligned} \left\| u \right\| _{H^1(B_0)} \le \left\| \chi u \right\| _{H^1(B_1)}&\le C \frac{1}{\mu } \left( \left\| [\mathcal {L},\chi ]u \right\| _{H^{-1}(B_1)} + \left\| \chi \mathcal {L}u \right\| _{H^{-1}(B_1)} \right) \\&\le C \frac{1}{\mu } \left( (\mu + |\beta |)\left\| u \right\| _{L^2(B_1)} + \left\| \mathcal {L}u \right\| _{H^{-1}(\Omega )} \right) \end{aligned}$$

The same argument for $B_3 \subset \Omega$ gives

$$\begin{aligned} \left\| u \right\| _{H^1(B_3)} \le C \frac{1}{\mu } \left( (\mu + |\beta |)\left\| u \right\| _{L^2(\Omega )} + \left\| \mathcal {L}u \right\| _{H^{-1}(\Omega )} \right) , \end{aligned}$$

thus leading to the conclusion after absorbing the $\tilde{Pe} = 1 + |\beta |/\mu$ factor into the exponential factor $e^{C\tilde{Pe}^2}$. $\square$

Remark 1

In the geometric setting of this section one can be more precise about the Hölder exponent $\kappa$ in the conditional stability estimates. For this we recall some known results for second-order elliptic equations: we refer to [1, Theorem 2.1] for the Laplace equation, and for the case including lower-order terms to [19, Theorem 3]. Let u be a homogeneous solution of (1) with $f\equiv 0$. For a constant $C_{st}$ depending implicitly on the coefficients $\mu$ and $\beta$, the following three-ball inequality holds

$$\begin{aligned} \Vert u\Vert _{L^2(B_2)} \le C_{st} \Vert u\Vert _{L^2(B_1)}^{\kappa } \Vert u\Vert _{L^2(B_3)}^{1-\kappa }, \end{aligned}$$

where $B_j$ are concentric balls in $\Omega$ with increasing radii $r_j$. The constant $C_{st}$ does not depend on the radii $r_1,\,r_2$, but it does depend on $r_3$. The exponent $\kappa \in (0,1)$ is given by

$$\begin{aligned} \kappa = \frac{\log \frac{r_3}{r_2}}{C_3\log \frac{r_2}{r_1} + \log \frac{r_3}{r_2}}, \end{aligned}$$

where $C_3>0$ is a constant depending on $r_3$.

3 Finite element method

Let $V_h$ denote the space of piecewise affine finite element functions defined on a conforming computational mesh $\mathcal {T}_h = \{ K\}$. $\mathcal {T}_h$ consists of shape regular triangular elements K with diameter $h_K$ and is quasi-uniform. We define the global mesh size by $h = \max _{K\in \mathcal {T}_h} h_K$. The interior faces of the triangulation will be denoted by $\mathcal {F}_i$, the jump of a quantity across a face F by $\llbracket \cdot \rrbracket _F$, and the outward unit normal by n.

Let $\beta \in [W^{1,\infty }(\Omega )]^n$ and adopt the shorthand notation $|\beta | := \Vert \beta \Vert _{[L^\infty (\Omega )]^n}$. As already stated in Sect. 1, we consider the diffusion-dominated regime given by the low Péclet number

$$\begin{aligned} Pe(h) := \frac{|\beta | h}{\mu } < 1. \end{aligned}$$

(11)

We will denote by C a generic positive constant independent of the mesh size and the Péclet number. Let $\pi _h:L^2(\Omega ) \mapsto V_h$ denote the standard $L^2$-projection on $V_h$, which for $k=1,2$ and $m=0,k-1$ satisfies

$$\begin{aligned} \left\| \pi _h u \right\| _{H^m(\Omega )}&\le C \left\| u \right\| _{H^m(\Omega )},\quad u\in H^m(\Omega ), \\ \left\| u-\pi _h u \right\| _{H^m(\Omega )}&\le C h^{k-m} \left\| u \right\| _{H^k(\Omega )},\quad u\in H^k(\Omega ) . \end{aligned}$$

We introduce the standard inner products with the induced norms

$$\begin{aligned} (v_h,w_h)_\Omega&:= \int _{\Omega } v_h w_h \,\mathrm {d}x,\\ \left\langle v_h,w_h\right\rangle _{\partial \Omega }&:= \int _{\partial \Omega } v_h w_h \,\mathrm {d}s, \end{aligned}$$

and the following bilinear forms

$$\begin{aligned} a_h(v_h,w_h)&:= (\beta \cdot \nabla v_h, w_h )_\Omega + ( \mu \nabla v_h , \nabla w_h )_{\Omega }- \left\langle \mu \nabla v_h \cdot n , w_h \right\rangle _{\partial \Omega },\\ s_\Omega (v_h,w_h)&:= \gamma \sum _{F \in \mathcal {F}_i} \int _{F}h (\mu + |\beta |h) \llbracket \nabla v_h \cdot n \rrbracket _F \llbracket \nabla w_h \cdot n \rrbracket _F \,\mathrm {d}s,\\ s_{\omega }(v_h,w_h)&:= ((\mu + |\beta | h) v_h, w_h)_{ \omega },\\ s(v_h,w_h)&:= s_\Omega (v_h,w_h) + s_{\omega }(v_h,w_h),\\ \end{aligned}$$

and

$$\begin{aligned} s_*(v_h,w_h) :=\gamma _* \left( \left\langle (\mu h^{-1} + |\beta |) v_h,w_h\right\rangle _{\partial \Omega }+ (\mu \nabla v_h, \nabla w_h )_\Omega + s_\Omega (v_h,w_h)\right) . \end{aligned}$$

The terms $s_\Omega$ and $s_*$ are stabilizing terms, while the term $s_\omega$ is aimed for data assimilation. After scaling with the coefficients in the above forms, Lemma 2 in [2] writes as

$$\begin{aligned} \Vert (\mu ^{\frac{1}{2}} h + |\beta |^{\frac{1}{2}} h^{\frac{3}{2}}) v_h\Vert _{H^1(\Omega )} \le C \gamma ^{-\frac{1}{2}} s(v_h,v_h)^{\frac{1}{2}}, \quad \forall v_h \in V_h, \end{aligned}$$

(12)

and Lemma 2 in [5] gives the jump inequality

$$\begin{aligned} s_\Omega (\pi _h u,\pi _h u) \le C \gamma (\mu + |\beta |h) h^2 |u|^2_{H^2(\Omega )},\quad \forall u\in H^2(\Omega ). \end{aligned}$$

(13)

The parameters $\gamma$ and $\gamma _*$ in $s_\Omega$ and $s_*$, respectively, are fixed at the implementation level and, to alleviate notation, our analysis covers the choice $\gamma =1=\gamma _*$.

We can then use the general framework in [8] to write the finite element method for unique continuation subject to (1) as follows. Consider a discrete Lagrange multiplier $z_h\in V_h$ and aim to find the saddle points of the functional

$$\begin{aligned} L_h(u_h,z_h)&:= \frac{1}{2} s_\omega (u_h - \tilde{U}_\omega , u_h - \tilde{U}_\omega ) + a_h(u_h,z_h) - (f,z_h)_\Omega \\&\quad +\frac{1}{2} s_\Omega (u_h,u_h) - \frac{1}{2} s_*(z_h,z_h), \end{aligned}$$

where we recall that $\tilde{U}_\omega = u\vert _{\omega } + \delta$ and $u\in H^2(\Omega )$ is a solution to (1). The Euler–Lagrange equations for $L_h$ lead to the following discrete problem: find $(u_h,z_h) \in [V_h]^2$ such that

$$\begin{aligned} \left\{ \begin{array}{l} a_h(u_h,w_h) - s_*(z_h,w_h) = (f,w_h)_\Omega \\ a_h(v_h,z_h) + s(u_h,v_h) =s_{\omega }(\tilde{U}_\omega ,v_h) \end{array} \right. ,\quad \forall (v_h,w_h) \in [V_h]^2, \end{aligned}$$

(14)

We observe that by the ill-posed character of the problem, only the stabilization operators $s_\Omega$ and $s_*$ provide some stability to the discrete system, and the corresponding system matrix is expected to be ill-conditioned. To quantify this effect we first prove an upper bound on the condition number.

Proposition 2

The finite element formulation (14) has a unique solution $(u_h,z_h) \in [V_h]^2$ and the Euclidean condition number $\mathcal {K}_2$ of the system matrix satisfies

$$\begin{aligned} \mathcal {K}_2 \le C h^{-4}. \end{aligned}$$

Proof

We write (14) as the linear system $A[(u_h,z_h),(v_h,w_h)] = (f,w_h)_\Omega + s_{\omega }(\tilde{U}_\omega ,v_h)$, for all $(v_h,w_h) \in [V_h]^2$, where

$$\begin{aligned} A[(u_h,z_h),(v_h,w_h)] := a_h(u_h,w_h) - s_*(z_h,w_h) + a_h(v_h,z_h) + s(u_h,v_h). \end{aligned}$$

Since $A[(u_h,z_h),(u_h,-z_h)] = s(u_h,u_h) + s_*(z_h,z_h)$, using (12) the following inf-sup condition holds

$$\begin{aligned} \Psi _h:= \inf _{(u_h,z_h)\in [V_h]^2} \sup _{(v_h,w_h)\in [V_h]^2} \frac{A[(u_h,z_h),(v_h,w_h)]}{\Vert (u_h,z_h)\Vert _{L^2(\Omega )}\Vert (v_h,w_h)\Vert _{L^2(\Omega )}} \ge C \mu (1+Pe(h)) h^2. \end{aligned}$$

This provides the existence of a unique solution for the linear system. We use [14, Theorem 3.1] to estimate the condition number by

$$\begin{aligned} \mathcal {K}_2 \le C \frac{\Upsilon _h}{\Psi _h}, \end{aligned}$$

(15)

where

$$\begin{aligned} \Upsilon _h:= \sup _{(u_h,z_h)\in [V_h]^2} \sup _{(v_h,w_h)\in [V_h]^2} \frac{A[(u_h,z_h),(v_h,w_h)]}{\Vert (u_h,z_h)\Vert _{L^2(\Omega )}\Vert (v_h,w_h)\Vert _{L^2(\Omega )}}. \end{aligned}$$

We recall the following discrete inverse inequality, see for instance [13, Lemma 1.138],

$$\begin{aligned} \Vert \nabla v_h\Vert _{L^2(K)} \le C h^{-1} \Vert v_h\Vert _{L^2(K)},\quad \forall v_h \in \mathbb {P}_1(K). \end{aligned}$$

(16)

We also recall the following continuous trace inequality, see for instance [18],

$$\begin{aligned} \Vert v\Vert _{L^2(\partial K)} \le C( h^{-\frac{1}{2}} \Vert v\Vert _{L^2(K)} + h^{\frac{1}{2}} \Vert \nabla v\Vert _{L^2(K)}),\quad \forall v \in H^1(K), \end{aligned}$$

(17)

and the discrete one

$$\begin{aligned} \Vert \nabla v_h \cdot n\Vert _{L^2(\partial K)} \le C h^{-\frac{1}{2}} \Vert \nabla v_h\Vert _{L^2(K)},\quad \forall v_h \in \mathbb {P}_1(K). \end{aligned}$$

(18)

Using the Cauchy–Schwarz inequality together with (18) and (16) we get

$$\begin{aligned} s_\Omega (u_h,v_h)&= \gamma \mu (1+Pe(h)) \sum _{F \in \mathcal {F}_i} \int _{F}h \llbracket \nabla u_h \cdot n \rrbracket _F \llbracket \nabla v_h \cdot n \rrbracket _F ~\text{ d }s\\&\le C \mu (1+Pe(h)) h^{-2} \Vert u_h\Vert _{L^2(\Omega )} \Vert v_h\Vert _{L^2(\Omega )}, \end{aligned}$$

hence

$$\begin{aligned} s(u_h,v_h) \le C \mu (1+Pe(h)) h^{-2} \Vert u_h\Vert _{L^2(\Omega )} \Vert v_h\Vert _{L^2(\Omega )}. \end{aligned}$$

Combining this with the Cauchy–Schwarz inequality and the inequalities (16) and (17), we obtain

$$\begin{aligned} -s_*(z_h,w_h) \le C \mu (1+Pe(h)) h^{-2} \Vert z_h\Vert _{L^2(\Omega )} \Vert w_h\Vert _{L^2(\Omega )}. \end{aligned}$$

Again due to the Cauchy–Schwarz inequality, and trace and inverse inequalities, we have

$$\begin{aligned} a_h(u_h,w_h)&= (\beta \cdot \nabla u_h, w_h )_\Omega + \mu \sum _{F \in \mathcal {F}_i} \int _{F}h \llbracket \nabla u_h \cdot n \rrbracket _F w_h \,\mathrm {d}s\\&\le C \mu (1+Pe(h)) h^{-2} \Vert u_h\Vert _{L^2(\Omega )} \Vert w_h\Vert _{L^2(\Omega )}, \end{aligned}$$

Collecting the above estimates we have

$$\begin{aligned} \Upsilon _h \le C \mu (1+Pe(h)) h^{-2}, \end{aligned}$$

and we conclude by (15). $\square$

3.1 Error estimates for the weakly consistent regularization

The error analysis proceeds in two main steps:

• (i) First we prove that the stabilizing terms and the data fitting term must vanish at an optimal rate for smooth solutions, with constant independent of the physical stability (Proposition 3).

• (ii) Then we show that the residual of the PDE is bounded by the stabilizing terms and the data fitting term. Using this result together with the first step and the continuous stability estimates in Sect. 2, we prove $L^2$- and $H^1$-convergence results (Theorems 1 and 2).

To quantify stabilization and data fitting for $(v_h,w_h) \in [V_h]^2$ we introduce the norm

$$\begin{aligned} \Vert (v_h,w_h)\Vert ^2_s:= s(v_h,v_h) + s_*(w_h,w_h). \end{aligned}$$

We also define the “continuity norm” on $H^{\frac{3}{2}+\epsilon }(\Omega )$, for any $\epsilon >0$,

$$\begin{aligned} \Vert v \Vert _\sharp := \Vert |\beta |^\frac{1}{2} h^{-\frac{1}{2}} v \Vert _{\Omega } + \Vert \mu ^{\frac{1}{2}} \nabla v\Vert _\Omega + \Vert \mu ^{\frac{1}{2}} h^{\frac{1}{2}} \nabla v \cdot n\Vert _{\partial \Omega }. \end{aligned}$$

Using standard approximation properties and the trace inequality (17), we have

$$\begin{aligned} \Vert u - \pi _h u \Vert _\sharp \le C ( \mu ^{\frac{1}{2}} h + |\beta |^\frac{1}{2} h^{\frac{3}{2}} ) |u|_{H^2(\Omega )}. \end{aligned}$$

Using (13) and interpolation

$$\begin{aligned} \Vert (u - \pi _h u,0)\Vert ^2_s&= s(u -\pi _h u,u -\pi _h u) = s_\Omega (\pi _h u,\pi _h u) + s_\omega (u -\pi _h u,u -\pi _h u) \\&\le C (\mu h^2 + |\beta | h^3) |u|^2_{H^2(\Omega )}, \end{aligned}$$

where we used that $s_\Omega (u,v_h) = 0$, since $u \in H^2(\Omega )$. Hence it follows that for $u \in H^2(\Omega )$

$$\begin{aligned} \Vert (u - \pi _h u,0)\Vert _s + \Vert u - \pi _h u \Vert _\sharp \le C (\mu ^{\frac{1}{2}} h + |\beta |^{\frac{1}{2}} h^{\frac{3}{2}}) |u|_{H^2(\Omega )}. \end{aligned}$$

(19)

Observe that, when $Pe(h)<1$, the first term dominates and the estimate is O(h), whereas when $Pe(h)>1$ the bound is $O(h^{\frac{3}{2}})$. We note in passing that the same estimates hold for the nodal interpolant.

Lemma 3

(Consistency) Let $u \in H^2(\Omega )$ be a solution to (1) and $(u_h,z_h) \in [V_h]^2$ the solution to (14), then

$$\begin{aligned} a_h(\pi _h u - u_h,w_h) + s_*(z_h,w_h) = a_h(\pi _h u - u,w_h), \end{aligned}$$

and

$$\begin{aligned} -a_h(v_h,z_h) + s(\pi _h u - u_h,v_h) = s_\Omega (\pi _h u - u,v_h) + s_\omega (\pi _h u - \tilde{U}_\omega ,v_h), \end{aligned}$$

for all $(v_h,w_h)\in [V_h]^2$.

Proof

The first claim follows from the definition of $a_h$, since

$$\begin{aligned} a_h(u_h,w_h) - s_*(z_h,w_h) = (f,w_h)_\Omega = (\beta \cdot \nabla u - \mu \Delta u, w_h)_\Omega = a_h(u,w_h), \end{aligned}$$

where in the last equality we integrated by parts. The second claim follows similarly from

$$\begin{aligned} a_h(v_h,z_h) + s(u_h,v_h) = s_\omega (\tilde{U}_\omega ,v_h), \end{aligned}$$

leading to

$$\begin{aligned} -a_h(v_h,z_h) + s(\pi _h u - u_h,v_h)&= s(\pi _h u,v_h) - s_\omega (\tilde{U}_\omega ,v_h)\\&= s_\Omega (\pi _h u - u,v_h) + s_\omega (\pi _h u - \tilde{U}_\omega ,v_h). \end{aligned}$$

$\square$

Lemma 4

(Continuity) Assume the low Péclet regime (11) and that $|\beta |_{1,\infty } \le C |\beta |$. Let $v \in H^2(\Omega )$ and $w_h \in V_h$, then

$$\begin{aligned} a_h(v,w_h) \le C \Vert v \Vert _\sharp \Vert (0,w_h)\Vert _s. \end{aligned}$$

Proof

Writing out the terms of $a_h$ and integrating by parts in the advective term leads to

$$\begin{aligned} a_h(v,w_h)&= - (v,\beta \cdot \nabla w_h)_\Omega - (v \nabla \cdot \beta , w_h)_\Omega + \left\langle v \beta \cdot n,w_h \right\rangle _{\partial \Omega } \\&\quad +\,( \mu \nabla v , \nabla w_h )_{\Omega }- \left\langle \mu \nabla v \cdot n , w_h \right\rangle _{\partial \Omega }. \end{aligned}$$

Using the Cauchy–Schwarz inequality and the trace inequality (17) for v, we see that

$$\begin{aligned} \left\langle v \beta \cdot n,w_h \right\rangle _{\partial \Omega }+ ( \mu \nabla v, \nabla w_h )_{\Omega }- \left\langle \mu \nabla v \cdot n, w_h \right\rangle _{\partial \Omega } \le C \Vert v \Vert _\sharp \Vert (0,w_h)\Vert _s. \end{aligned}$$

By the Cauchy–Schwarz inequality and a discrete Poincaré inequality for $w_h$, see e.g. [6], we bound

$$\begin{aligned} -(v \nabla \cdot \beta , w_h)_\Omega \le C |\beta |_{1,\infty } \Vert v\Vert _\Omega \Vert w_h\Vert _\Omega \le C \frac{|\beta |_{1,\infty }}{|\beta |} Pe(h)^\frac{1}{2} \Vert v\Vert _\sharp \Vert (0,w_h)\Vert _s. \end{aligned}$$

Under the assumption $|\beta |_{1,\infty } \le C |\beta |$, we get

$$\begin{aligned} -(v \nabla \cdot \beta , w_h)_\Omega \le C Pe(h)^\frac{1}{2} \Vert v\Vert _\sharp \Vert (0,w_h)\Vert _s. \end{aligned}$$

We bound the remaining term by

$$\begin{aligned} -(v,\beta \cdot \nabla w_h)_\Omega \le |\beta |^\frac{1}{2} h^{\frac{1}{2}} \Vert v\Vert _\sharp \Vert \nabla w_h\Vert _\Omega \le C Pe(h)^\frac{1}{2} \Vert v\Vert _\sharp \Vert (0,w_h)\Vert _s. \end{aligned}$$

Finally, exploiting the low Péclet regime $Pe(h)<1$, we obtain the conclusion. $\square$

Proposition 3

(Convergence of regularization) Assume the low Péclet regime (11) and that $|\beta |_{1,\infty } \le C |\beta |$. Let $u \in H^2(\Omega )$ be a solution to (1) and $(u_h,z_h) \in [V_h]^2$ the solution to (14), then

$$\begin{aligned} \Vert (\pi _h u - u_h,z_h)\Vert _s \le C (\mu ^{\frac{1}{2}} h + |\beta |^{\frac{1}{2}} h^{\frac{3}{2}}) (|u|_{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ). \end{aligned}$$

Proof

Denoting $e_h = \pi _h u - u_h$, it holds by definition that

$$\begin{aligned} \Vert (e_h,z_h)\Vert _s^2 = a_h(e_h,z_h) + s_*(z_h,z_h) - a_h(e_h,z_h) + s(e_h,e_h). \end{aligned}$$

Using both claims in Lemma 3 we may write

$$\begin{aligned} \Vert (e_h,z_h)\Vert _s^2 = a_h(\pi _h u - u,z_h) +s_\Omega (\pi _h u - u,e_h)+s_\omega (\pi _h u - \tilde{U}_\omega ,e_h). \end{aligned}$$

Lemma 4 gives the bound

$$\begin{aligned} a_h(\pi _h u - u,z_h) \le C \Vert \pi _h u - u\Vert _\sharp \Vert (0,z_h)\Vert _s. \end{aligned}$$

The other terms are simply bounded using the Cauchy–Schwarz inequality as follows

$$\begin{aligned} s_\Omega (\pi _h u - u,e_h)+s_\omega (\pi _h u - \tilde{U}_\omega ,e_h) \le (\Vert (\pi _h u - u,0)\Vert _s + (\mu ^{\frac{1}{2}} + |\beta |^{\frac{1}{2}} h^{\frac{1}{2}})\Vert \delta \Vert _\omega ) \Vert (e_h,0)\Vert _s. \end{aligned}$$

Collecting the above bounds we have

$$\begin{aligned} \Vert (e_h,z_h)\Vert _s^2 \le C(\Vert \pi _h u - u\Vert _\sharp +\Vert (\pi _h u - u,0)\Vert _s + (\mu ^{\frac{1}{2}} + |\beta |^{\frac{1}{2}} h^{\frac{1}{2}}) \Vert \delta \Vert _\omega ) \Vert (e_h,z_h)\Vert _s, \end{aligned}$$

and the claim follows by applying the approximation (19). $\square$

Lemma 5

(Covergence of the convective term) Assume the low Péclet regime (11) and that $|\beta |_{1,\infty } \le C |\beta |$. Let $u \in H^2(\Omega )$ be a solution to (1), $(u_h,z_h) \in [V_h]^2$ the solution to (14) and $w\in H^1_0(\Omega )$, then

$$\begin{aligned} (\beta \cdot \nabla u_h, w-\pi _h w)_\Omega \le C (\mu + |\beta |) ( h \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega )\Vert w\Vert _{H^1(\Omega )}, \end{aligned}$$

Proof

Denote by $\beta _h\in [V_h]^n$ a piecewise linear approximation of $\beta$ that is $L^\infty$-stable and for which

$$\begin{aligned} \Vert \beta - \beta _h\Vert _{0,\infty } \le C h |\beta |_{1,\infty }, \end{aligned}$$

and recall the approximation estimate in [7, Theorem 2.2]

$$\begin{aligned} \inf _{x_h \in V_h} \Vert h^{\frac{1}{2}} (\beta _h \cdot \nabla u_h - x_h)\Vert _\Omega \le C \left( \sum _{F \in \mathcal {F}_i} \Vert h \llbracket \beta _h \cdot \nabla u_h \rrbracket \Vert _{F}^2\right) ^{\frac{1}{2}} \le C |\beta |^\frac{1}{2} s_\Omega (u_h,u_h)^{\frac{1}{2}}. \end{aligned}$$

(20)

We also use Proposition 3 and the jump inequality (13) to estimate

$$\begin{aligned} s_{\Omega }(u_h,u_h)^{\frac{1}{2}}&\le s_{\Omega }(u_h-\pi _h u,u_h-\pi _h u)^{\frac{1}{2}} + s_{\Omega }(\pi _h u,\pi _h u)^{\frac{1}{2}} \\&\le C (\mu ^{\frac{1}{2}} h + |\beta |^{\frac{1}{2}} h^{\frac{3}{2}}) (|u|_{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ) + C (\mu ^\frac{1}{2} + |\beta |^\frac{1}{2} h^\frac{1}{2}) h|u|_{H^2(\Omega )}, \end{aligned}$$

obtaining

$$\begin{aligned} s_{\Omega }(u_h,u_h)^{\frac{1}{2}} \le C (\mu ^{\frac{1}{2}} h + |\beta |^{\frac{1}{2}} h^{\frac{3}{2}}) (|u|_{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ). \end{aligned}$$

(21)

We now write

$$\begin{aligned} (\beta \cdot \nabla u_h, w-\pi _h w)_\Omega = (\beta _h \cdot \nabla u_h, w-\pi _h w)_\Omega + ((\beta -\beta _h) \cdot \nabla u_h, w-\pi _h w)_\Omega , \end{aligned}$$

and using orthogonality, (20), (21), interpolation and (11), we bound the first term by

$$\begin{aligned} (\beta _h \cdot \nabla u_h, w-\pi _h w)_\Omega&\le C |\beta |^\frac{1}{2} h^{-\frac{1}{2}} s_\Omega (u_h,u_h)^{\frac{1}{2}} h \Vert w\Vert _{H^1(\Omega )} \\ {}&\le C |\beta |^\frac{1}{2} h^\frac{1}{2} (\mu ^{\frac{1}{2}} + |\beta |^{\frac{1}{2}} h^{\frac{1}{2}}) (h|u|_{H^2(\Omega )} + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )} \\ {}&\le C (\mu + |\beta | h) (h|u|_{H^2(\Omega )} + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )}. \end{aligned}$$

We now use the Poincaré-type inequality (12) and interpolation to bound the second term

$$\begin{aligned} ((\beta -\beta _h) \cdot \nabla u_h, w-\pi _h w)_\Omega&\le C h^2 |\beta |_{1,\infty } \Vert \nabla u_h\Vert _\Omega \Vert w\Vert _{H^1(\Omega )} \\ {}&\le C h |\beta |_{1,\infty } (\mu ^\frac{1}{2} + |\beta |^\frac{1}{2} h^\frac{1}{2})^{-1} s(u_h,u_h)^{\frac{1}{2}} \Vert w\Vert _{H^1(\Omega )} \\ {}&\le C h |\beta |_{1,\infty } ( h |u|_{H^2(\Omega )} + \Vert u\Vert _\Omega + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )} \\ {}&\le C h |\beta |_{1,\infty } ( \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )} \end{aligned}$$

since due to Proposition 3 and inequality (13)

$$\begin{aligned} s(u_h,u_h)^\frac{1}{2}&\le s(u_h-\pi _h u,u_h-\pi _h u)^\frac{1}{2} + s_\Omega (\pi _h u,\pi _h u)^{\frac{1}{2}} + s_\omega (\pi _h u,\pi _h u)^{\frac{1}{2}} \\ {}&\le C (\mu ^{\frac{1}{2}} + |\beta |^{\frac{1}{2}} h^{\frac{1}{2}}) ( h |u|_{H^2(\Omega )} + \Vert \delta \Vert _\omega + \Vert u\Vert _\Omega ). \end{aligned}$$

Under the assumption $|\beta |_{1,\infty } \le C |\beta |$, we collect the above bounds to get

$$\begin{aligned} (\beta \cdot \nabla u_h, w-\pi _h w)_\Omega \le C (\mu + |\beta |) ( h \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega )\Vert w\Vert _{H^1(\Omega )}. \end{aligned}$$

$\square$

We now combine these results with the conditional stability estimates from Sect. 2 to obtain error bounds for the discrete solution. For this purpose, we consider an open bounded set $B\subset \Omega$ that contains the data region $\omega$ such that $B{\setminus } \omega$ does not touch the boundary of $\Omega$. Then the estimates in Lemma 2 and Corollary 2 hold true by a covering argument, see e.g. [19], and we obtain local error estimates in B. For global unique continuation from $\omega$ to the entire $\Omega$, however, the stability deteriorates and it is of a different nature: the modulus of continuity for the given data is not of Hölder type $|\cdot |^\kappa$ any more, but of a logarithmic kind $|\log (\cdot )|^{-\kappa }$.

Theorem 1

($L^2$-error estimate) Assume the low Péclet regime (11) and that $|\beta |_{1,\infty } \le C |\beta |$. Consider $\omega \subset B \subset \Omega$ such that $\overline{B{\setminus }\omega } \subset \Omega$. Let $u \in H^2(\Omega )$ be a solution to (1) and $(u_h,z_h) \in [V_h]^2$ the solution to (14), then there is $\kappa \in (0,1)$ such that

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(B)} \le C h^{\kappa } e^{C \tilde{Pe}^2} (\Vert u\Vert _{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ), \end{aligned}$$

where $\tilde{Pe} = 1+|\beta |/\mu$.

Proof

Let us consider the residual defined by $\left\langle r,w \right\rangle = a_h(u_h,w)-\left\langle f,w \right\rangle$, for $w\in H^1_0(\Omega )$. Using (14) we obtain

$$\begin{aligned} \left\langle r,w \right\rangle&= a_h(u_h,w-\pi _h w) - \left\langle f,w-\pi _h w \right\rangle + a_h(u_h,\pi _h w) - \left\langle f,\pi _h w \right\rangle \\&= a_h(u_h,w-\pi _h w) - \left\langle f,w-\pi _h w \right\rangle + s_*(z_h,\pi _h w). \end{aligned}$$

We split the first term in the right-hand side into convective and non-convective terms, and for the latter we integrate by parts on each element K and use Cauchy–Schwarz followed by the trace inequality (17) to get

$$\begin{aligned}&( \mu \nabla u_h , \nabla (w-\pi _h w) )_{\Omega }- \left\langle \mu \nabla u_h \cdot n , w-\pi _h w \right\rangle _{\partial \Omega } \\&\quad = \sum _{F \in \mathcal {F}_i} \int _{F}\mu \llbracket \nabla u_h \cdot n \rrbracket _F (w-\pi _h w) \,\mathrm {d}s\\&\quad \le C \mu (\mu + |\beta |h)^{-\frac{1}{2}} s_\Omega (u_h,u_h)^{\frac{1}{2}} (h^{-1} \Vert w-\pi _h w\Vert _{L^2(\Omega )} + \Vert w-\pi _h w\Vert _{H^1(\Omega )}). \end{aligned}$$

Using (21) and interpolation we obtain

$$\begin{aligned}&(\mu \nabla u_h , \nabla (w-\pi _h w) )_{\Omega }-\left\langle \mu \nabla u_h \cdot n , w-\pi _h w \right\rangle _{\partial \Omega }\\&\quad \le C \mu ( h |u|_{H^2(\Omega )} + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )}. \end{aligned}$$

We bound the convective term in $a_h(u_h,w-\pi _h w)$ by Lemma 5, hence obtaining

$$\begin{aligned} a_h(u_h,w-\pi _h w) \le C (\mu + |\beta |) ( h \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )}. \end{aligned}$$

The next term in the residual is bounded by

$$\begin{aligned} \left\langle f,w-\pi _h w \right\rangle \le \Vert f \Vert _{L^2(\Omega )} \Vert w-\pi _h w \Vert _{L^2(\Omega )} \le C h \Vert f \Vert _{L^2(\Omega )} \Vert w \Vert _{H^1(\Omega )}. \end{aligned}$$

The last term left to bound from the residual is

$$\begin{aligned} s_*(z_h,\pi _h w) \le \Vert (0,z_h) \Vert _s \Vert (0, \pi _h w)\Vert _s, \end{aligned}$$

and using (18) for the jump term, together with the $H^1$-stability of $\pi _h$, we see that

$$\begin{aligned} \Vert (0, \pi _h w)\Vert _s&\le C( \mu ^\frac{1}{2} \Vert \nabla (\pi _h w)\Vert _\Omega + (\mu ^{\frac{1}{2}} + |\beta |^{\frac{1}{2}} h^{\frac{1}{2}}) \Vert \nabla (\pi _h w)\Vert _\Omega + (\mu h^{-1} + |\beta |)^\frac{1}{2} \Vert \pi _h w\Vert _{\partial \Omega } )\\&\le C (\mu ^{\frac{1}{2}} + |\beta |^{\frac{1}{2}} h^{\frac{1}{2}} ) \Vert w\Vert _{H^1(\Omega )}, \end{aligned}$$

where for the boundary term we used that $w|_{\partial \Omega } = 0$ together with interpolation and (17). Bounding $\Vert (0,z_h) \Vert _s$ by Proposition 3, we get

$$\begin{aligned} s_*(z_h,\pi _h w) \le C (\mu + |\beta |h) ( h |u|_{H^2(\Omega )} + \Vert \delta \Vert _\omega ) \Vert w\Vert _{H^1(\Omega )}. \end{aligned}$$

Collecting the above estimates we bound the residual norm by

$$\begin{aligned} \Vert r\Vert _{H^{-1}(\Omega )}&\le C (\mu + |\beta |)(h \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega ) + C h \Vert f \Vert _{L^2(\Omega )}\\&\le C (\mu + |\beta |) (h \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega ). \end{aligned}$$

We now use the stability estimate in Lemma 2 to write

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(B)} \le C e^{C \tilde{Pe}^2} \left( \Vert u-u_h\Vert _{L^2(\omega )} + \frac{1}{\mu } \Vert r\Vert _{H^{-1}(\Omega )} \right) ^{\kappa } \Vert u-u_h\Vert _{L^2(\Omega )}^{1-\kappa }. \end{aligned}$$

By Proposition 3 we have

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(\omega )}&\le \Vert u-\pi _h u\Vert _{L^2(\omega )} + \Vert u_h - \pi _h u\Vert _{L^2(\omega )}\\&\le C h^2 |u|_{H^2(\Omega )} + C h |u|_{H^2(\Omega )} + C \Vert \delta \Vert _\omega .\\&\le C (h |u|_{H^2(\Omega )} + \Vert \delta \Vert _\omega ). \end{aligned}$$

Using (12) and Proposition 3 again, we bound

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(\Omega )}&\le \Vert u-\pi _h u\Vert _{L^2(\Omega )} + \Vert u_h-\pi _h u\Vert _{L^2(\Omega )}\\&\le C h^2 |u|_{H^2(\Omega )} + C (\mu ^{\frac{1}{2}} h + |\beta |^{\frac{1}{2}} h^{\frac{3}{2}})^{-1} s(u_h-\pi _h u,u_h-\pi _h u)^{\frac{1}{2}}\\&\le C (|u|_{H^2(\Omega )} + h^{-1}\Vert \delta \Vert _\omega ). \end{aligned}$$

Hence we conclude by

$$\begin{aligned} \Vert u-u_h\Vert _{L^2(B)}&\le C e^{C \tilde{Pe}^2} \left( h \Vert u\Vert _{H^2(\Omega )} + \Vert \delta \Vert _\omega \right) ^{\kappa } \left( |u|_{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega \right) ^{1-\kappa }\\&\le C e^{C \tilde{Pe}^2} h^{\kappa } (\Vert u\Vert _{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ), \end{aligned}$$

where we have absorbed the $\tilde{Pe} = 1 + |\beta |/\mu$ factor into the exponential factor $e^{C\tilde{Pe}^2}$. $\square$

Remark 2

Let us briefly discuss the effect of decreasing the size of the data region $\omega$ by considering the case of balls, that is $\omega = B(x_0, r_1)$ and $B = B(x_0, r_2)$, with $x_0\in \Omega$ and $r_1<r_2$. Notice from Remark 1 that the exponent $\kappa$ is an increasing function of the radius $r_1$ and that decreasing the size of the data region $\omega$ implies that the convergence rate $h^\kappa$ decreases as well. Bounding the radius $r_2$ away from zero and letting $r_1\rightarrow 0$ implies that the exponent $\kappa \rightarrow 0$. The continuum three-ball inequality then becomes the trivial inequality $\Vert u\Vert _{L^2(B)} \le \Vert u\Vert _{L^2(\Omega )}$ and the method does not converge any more.

Theorem 2

($H^1$-error estimate) Assume the low Péclet regime (11) and that $|\beta |_{1,\infty } \le C |\beta |$ and ${{\,\mathrm{ess\,sup}\,}}_{\Omega } \nabla \cdot \beta \le 0$. Consider $\omega \subset B \subset \Omega$ such that $\overline{B{\setminus } \omega } \subset \Omega$. Let $u \in H^2(\Omega )$ be a solution to (1), and $(u_h,z_h) \in [V_h]^2$ the solution to (14), then there is $\kappa \in (0,1)$ such that

$$\begin{aligned} \Vert u-u_h\Vert _{H^1(B)} \le C h^{\kappa } e^{C \tilde{Pe}^2} (\Vert u\Vert _{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ), \end{aligned}$$

where $\tilde{Pe} = 1+|\beta |/\mu$.

Proof

Letting $e_h = u-u_h$, we combine the proof of Theorem 1 with the stability estimate in Corollary 2 to obtain

$$\begin{aligned} \left\| e_h \right\| _{H^1(B)}&\le C e^{C \tilde{Pe}^2} \left( \left\| e_h \right\| _{L^2(\omega )} + \frac{1}{\mu } \left\| r \right\| _{H^{-1}(\Omega )}\right) ^\kappa \left( \left\| e_h \right\| _{L^2(\Omega )} + \frac{1}{\mu } \left\| r \right\| _{H^{-1}(\Omega )} \right) ^{1-\kappa }\\&\le C e^{C \tilde{Pe}^2} h^{\kappa } (\Vert u\Vert _{H^2(\Omega )} + h^{-1} \Vert \delta \Vert _\omega ). \end{aligned}$$

$\square$

4 Numerical experiments

We illustrate the theoretical results with some numerical examples. The implementation of the stabilized FEM (14) has been carried out in FreeFem++ [15] on uniform triangulations with alternating left and right diagonals. The mesh size is taken as the inverse square root of the number of nodes. The parameters in $s_\Omega$ and $s_*$ are set to $\gamma = 10^{-5}$ and $\gamma _* = 1$. We also rescale the boundary term in $s_*$ by the factor 50, drawing on results from different numerical experiments. In this section we denote $e_h = \pi _h u - u_h$.

We consider $\Omega$ to be the unit square and the exact solution with global unit $L^2$-norm

$$\begin{aligned} u(x,y) = 30x(1-x)y(1-y). \end{aligned}$$

We take the diffusion coefficient $\mu = 1$ and investigate two cases for the convection field: the coercive case of the constant field

$$\begin{aligned} \beta _c=(1,0), \end{aligned}$$

and the case

$$\begin{aligned} \beta _{nc} = 100 (x+y, y-x), \end{aligned}$$

plotted in 2, for which $\nabla \cdot \beta = 200$ and $\Vert \beta \Vert _{0,\,\infty } = 200$. This makes the (well-posed) problem strongly non-coercive with a medium high Péclet number. The latter example was also considered in [8] for numerical experiments on a non-coercive convection–diffusion equation with Cauchy data.

We consider the following domains for data assimilation, shown in Fig. 1,

$$\begin{aligned}&\omega = (0.2,0.45)\times (0.2,0.45),\quad B = (0.2,0.45)\times (0.55,0.8), \end{aligned}$$

(22)

$$\begin{aligned}&\omega = (0,0.125)\times (0.4,0.6) \cup (0.875,1)\times (0.4,0.6),\quad B = (0.25,0.75)\times (0.4,0.6), \nonumber \\ \end{aligned}$$

(23)

$$\begin{aligned}&\omega = \Omega {\setminus } [0,0.875]\times [0.125,0.875],\quad B = \Omega {\setminus } [0,0.125]\times [0.125,0.875]. \end{aligned}$$

(24)

Fig. 1

Computational domains

Fig. 2

Left: convection field $\beta _{nc}$. Right: condition number $\mathcal {K}_2$ for domains (22), $\beta =\beta _c$; the dotted lines are proportional to $h^{-3}$ and $h^{-4}$

The condition number upper bound in Proposition 2 is illustrated for a particular case in Fig. 2, where we plot the condition number $\mathcal {K}_2$ versus the mesh size h, together with reference dotted lines proportional to $h^{-3}$ and $h^{-4}$. For five meshes with $2^N$ elements on each side, $N=3,\ldots ,7$, the approximate rates for $\mathcal {K}_2$ are $-3.03$, $-3.16$, $-3.2$, $-3.34$.

Fig. 3

Convergence for domains (22). Left: $\beta = \beta _c$. Right: $\beta = \beta _{nc}$

The results in Fig. 3 for the domains (22) strongly agree with the convergence rates expected from Theorems 1 and 2 for the relative errors in B computed in the $L^2$- and $H^1$-norms, and with the rates for $\Vert (e_h,z_h)\Vert _s$ given in Proposition 3.

The numerical approximation improves when considering the setting in (23), in which data is given both downstream and upstream, as reported in Fig. 4. The convergence is almost linear and the size of the errors is considerably reduced in the non-coercive case.

The resolution increases all the more when data is given near a big part of the boundary $\partial \Omega$, as for the computational domains (24) considered in Fig. 5. In this configuration of the set $\omega$, for both convective fields $\beta _{c}$ and $\beta _{nc}$, the $L^2$-errors decrease below $10^{-4}$ with superlinear rates on the same meshes considered in Figs.  3 and 4.

Fig. 4

Convergence for domains (23). Left: $\beta = \beta _c$. Right: $\beta = \beta _{nc}$

Fig. 5

Convergence for domains (24). Left: $\beta = \beta _c$. Right: $\beta = \beta _{nc}$

Comparing the geometries in (22) and (23) we also expect to see different effects of the two convective fields $\beta _{c}$ and $\beta _{nc}$. Notice that for both geometries the horizontal magnitude of $\beta _{nc}$ is greater than that of $\beta _{c}$. In (22) the solution is continued in the crosswind direction for both $\beta _{c}$ and $\beta _{nc}$, and a stronger convective field is not expected to improve the reconstruction. On the other side, in (23) information is propagated both downstream and upstream, and a stronger convective field can improve the resolution, despite the increase in the Péclet number. Indeed, we can see in Fig. 3 that for the geometry in (22) the numerical approximation is better for $\beta _c$ than for $\beta _{nc}$, while Fig. 4 shows better results for $\beta _{nc}$ than for $\beta _{c}$ in the case of (23), especially for the $L^2$-error.

Fig. 6

Convergence for perturbed $\tilde{U}_\omega$ in domains (22), $\beta = \beta _{c}$

To exemplify the noisy data $\tilde{U}_\omega = u\vert _{\omega } + \delta$, we perturb the restriction of u to $\omega$ on every node of the mesh with uniformly distributed values in $[-h^\frac{1}{2}, h^\frac{1}{2}]$, respectively $[-h,h]$. Recall that by the error estimates in Sect. 3 the contribution of the perturbation $\delta$ is bounded by $h^{-1} \Vert \delta \Vert _\omega$. It can be seen in Fig.  6 that the perturbations are strongly visible for an $O(h^\frac{1}{2})$ amplitude, but not for an O(h) one.

Appendices

Appendix A

Denote by $(\cdot , \cdot )$, $|\cdot |$, div, $\nabla$ and $D^2$ the inner product, norm, divergence, gradient and Hessian in the Euclidean setting of $\Omega \subset \mathbb {R}^n$. We recall the following identity [4, Lemma 1].

Lemma 6

Let $\ell , w \in C^2(\Omega )$ and $\sigma \in C^1(\Omega )$. We define $v = e^\ell w$ and

$$\begin{aligned} a = \sigma - \Delta \ell , \quad q = a + |\nabla \ell |^2, \quad b = -\sigma v - 2(\nabla v, \nabla \ell ), \quad B = (|\nabla v|^2 - q v^2) \nabla \ell . \end{aligned}$$

Then

$$\begin{aligned} e^{2 \ell } (\Delta w)^2/2&=(\Delta v + q v)^2/2 + b^2/2\\&+ a |\nabla v|^2 + 2 D^2 \ell (\nabla v, \nabla v) + \left( -a |\nabla \ell |^2 + 2 D^2 \ell (\nabla \ell , \nabla \ell )\right) v^2 \\ {}&+ \mathrm{div}(b \nabla v + B) + R, \end{aligned}$$

where $R = (\nabla \sigma , \nabla v)v + \left( \mathrm{div} (a \nabla \ell ) - a\sigma \right) v^2.$

Proof of Proposition 1

Let $\ell = \tau \phi$ and let $\lambda >0$ such that $|D^2 \rho (X,X)| \le \lambda |X|^2$. Recalling that $\phi = e^{\alpha \rho }$ and using the product rule we have that

$$\begin{aligned} D^2\phi (X,X) = \alpha \phi (\alpha (\nabla \rho , X)^2 + D^2 \rho (X,X)), \end{aligned}$$

hence

$$\begin{aligned} D^2\phi (X,X) \ge \alpha \phi D^2 \rho (X,X) \ge - \alpha \lambda \phi |X|^2. \end{aligned}$$

Combining this with the previous equality, we obtain

$$\begin{aligned} D^2 \phi (\nabla \phi , \nabla \phi ) \ge \alpha ^3 \phi ^3(\alpha |\nabla \rho |^4 - \lambda |\nabla \rho |^2). \end{aligned}$$

Choosing $\epsilon >0$ such that $\epsilon \le |\nabla \rho |^2 \le \epsilon ^{-1}$ it holds

$$\begin{aligned} D^2 \phi (\nabla \phi , \nabla \phi ) \ge \alpha ^3 \phi ^3(\alpha \epsilon ^2 - \lambda \epsilon ^{-1}). \end{aligned}$$

Since

$$\begin{aligned} 2 D^2 \ell (\nabla v, \nabla v) \ge -2 \alpha \lambda \phi \tau |\nabla v|^2, \end{aligned}$$

by choosing $\sigma = \Delta \ell + 3 \alpha \lambda \phi \tau$, i.e. $a = 3 \alpha \lambda \phi \tau$ in Lemma 6 we obtain the bounds

$$\begin{aligned}&a |\nabla v|^2 + 2 D^2 \ell (\nabla v, \nabla v) \ge \alpha \lambda \phi \tau |\nabla v|^2,\\&(-a |\nabla \ell |^2 + 2 D^2 \ell (\nabla \ell , \nabla \ell ))v^2 \ge (2 \alpha \epsilon ^2 - (3 + 2\lambda ) \epsilon ^{-1}) (\alpha \phi \tau )^3 v^2. \end{aligned}$$

We now bound

$$\begin{aligned} (\nabla \sigma , \nabla v)v = (\nabla (\Delta \ell ), \nabla v)v + 3\alpha \lambda (\nabla \ell , \nabla v)v \ge - \left( |\nabla (\Delta \phi )| + 3\alpha \lambda |\nabla \phi | \right) \tau |\nabla v| |v| \end{aligned}$$

and

$$\begin{aligned} (\mathrm{div}(a\nabla \ell ) - a\sigma )v^2 = ((\nabla a, \nabla \ell ) - a^2)v^2 \ge (3\alpha \lambda |\nabla \phi |^2- 9 \alpha ^2 \lambda ^2 \phi ^2) \tau ^2 v^2. \end{aligned}$$

Combining these lower bounds with

$$\begin{aligned} \tau |\nabla v| |v| \le \frac{1}{2} (|\nabla v|^2 + \tau ^2 |v|^2), \end{aligned}$$

and taking $\alpha$ large enough, we obtain from Lemma 6 that

$$\begin{aligned} C e^{2 \tau \phi } (\Delta w)^2 \ge (a_1 \tau ^3 - a_2 \tau ^2)v^2 + (b_1 \tau - b_0)|\nabla v|^2 + \mathrm{div}(b \nabla v + B), \end{aligned}$$

(25)

with $a_j,b_j > 0$ depending only on $\alpha$, $\phi$ and $\lambda$. Taking $\tau$ large enough and using the elementary inequality

$$\begin{aligned} |\nabla v|^2 = e^{2\tau \phi } |\tau w \nabla \phi + \nabla w|^2 \ge e^{2\tau \phi } \frac{1}{2} |\nabla w|^2 - e^{2\tau \phi }|\nabla \phi |^2 \tau ^2 w^2, \end{aligned}$$

we conclude by integrating over K and using the divergence theorem. $\square$

Appendix B

We briefly recall herein the definition of semiclassical pseudodifferential operators and semiclassical Sobolev spaces. We then discuss the composition rule of two such operators, which is also called symbol calculus, and some estimates that are used in the proof of Lemma 2. This presentation is based on [21, Chapter 4] and [17, Section 2], to which we refer the reader for more details.

We shall use the following standard notation. For $\xi \in \mathbb {R}^n$ we set $\langle \xi \rangle = (1+|\xi |^2)^{\frac{1}{2}}$, and for a multi-index $\alpha =(\alpha _1,\ldots ,\alpha _n)\in \mathbb {N}^n$ let $|\alpha |=\alpha _1+\ldots \alpha _n$, $\alpha ! = \alpha _1! \cdots \alpha _n!$, $\xi ^\alpha = \xi _1^{\alpha _1}\cdots \xi _n^{\alpha _n}$. Let also $\partial ^\alpha = \partial _{x_1}^{\alpha _1}\cdots \partial _{x_n}^{\alpha _n}$, $D=\frac{1}{i} \partial$ and $D^\alpha = \frac{1}{i^{|\alpha |}} \partial ^\alpha$. The Schwartz space $\mathcal {S}(\mathbb {R}^n)$ is the set of rapidly decreasing $C^\infty$ functions and its dual $\mathcal {S}'(\mathbb {R}^n)$ is the set of tempered distributions. The semiclassical parameter $\hbar$ is assumed to be small: $\hbar \in (0,\hbar _0)$ with $\hbar _0 \ll 1$.

The semiclassical Fourier transform is a rescaled version of the standard Fourier transform. It is given by

$$\begin{aligned} \mathcal {F}_\hbar \varphi (\xi ) := \int _{\mathbb {R}^n} e^{-\frac{i}{\hbar }x\cdot \xi } \varphi (x) \mathrm {d}x \end{aligned}$$

and its inverse is

$$\begin{aligned} \mathcal {F}^{-1}_\hbar \psi (x) := \frac{1}{(2\pi \hbar )^n} \int _{\mathbb {R}^n} e^{\frac{i}{\hbar }x\cdot \xi } \psi (\xi ) \mathrm {d}\xi . \end{aligned}$$

The following properties hold: $\mathcal {F}_\hbar ((\hbar D_x)^\alpha \varphi ) = \xi ^\alpha \mathcal {F}_\hbar \varphi$ and $(\hbar D_\xi )^\alpha \mathcal {F}_\hbar \varphi (\xi ) = \mathcal {F}_\hbar ((-x)^\alpha \varphi )$.

1.1 B.1. Symbol classes

For $m\in \mathbb {R}$ the symbol class $S^m$ consists of functions $a(x,\xi ,\hbar ) \in C^\infty (\mathbb {R}^n \times \mathbb {R}^n)$ such that for all multi-indices $\alpha ,\tilde{\alpha }\in \mathbb {N}^n$ there exists a constant $C_{\alpha ,\tilde{\alpha }}>0$ uniform in $\hbar \in (0,\hbar _0)$ such that

$$\begin{aligned} |\partial _x^\alpha \partial _\xi ^{\tilde{\alpha }} a(x,\xi ,\hbar )| \le C_{\alpha ,\tilde{\alpha }} \langle \xi \rangle ^{m-|\tilde{\alpha }|}, \quad x\in \mathbb {R}^n,\, \xi \in \mathbb {R}^n. \end{aligned}$$

Symbols in $S^m$ thus behave roughly as polynomials of degree m. We write that $a\in \hbar ^N S^{m}$ if

$$\begin{aligned} |\partial _x^\alpha \partial _\xi ^{\tilde{\alpha }} a(x,\xi ,\hbar )| \le C_{\alpha ,\tilde{\alpha }} \hbar ^N \langle \xi \rangle ^{m-|\tilde{\alpha }|}, \quad x\in \mathbb {R}^n,\, \xi \in \mathbb {R}^n. \end{aligned}$$

Lemma

(Asymptotic series) Let $m\in \mathbb {R}$ and the symbols $a_j \in S^{m-j}$ for $j=0,1,\ldots$. Then there exists a symbol $a\in S^m$ such that $a\sim \sum \limits _{j=0}^{\infty } \hbar ^j a_j$, that is for every $N\in \mathbb {N}$,

$$\begin{aligned} a-\sum _{j=0}^{N} \hbar ^j a_j \in \hbar ^{N+1} S^{m-N-1}. \end{aligned}$$

The symbol a is unique up to $\hbar ^\infty S^{-\infty }$, in the sense that the difference of two such symbols is in $\hbar ^NS^{-M}$ for all $N,M\in \mathbb {R}$. The principal symbol of a is given by $a_0$.

1.2 B.2. Pseudodifferential operators

Using these symbol classes we can define semiclassical pseudodifferential operators ($\psi$DOs). For a symbol $a\in S^m$ we define the corresponding semiclassical $\psi$DO of order m, $Op(a):\mathcal {S}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$,

$$\begin{aligned} Op(a)u(x) := \frac{1}{(2\pi \hbar )^n} \int _{\mathbb {R}^n} \int _{\mathbb {R}^n} e^{\frac{i}{\hbar }(x-y)\cdot \xi } a(x,\xi , \hbar )u(y) \mathrm {d}y \mathrm {d}\xi . \end{aligned}$$

This is also called quantization of the symbol. Op(a) can be extended to $\mathcal {S}'(\mathbb {R}^n)$ and $Op(a):\mathcal {S}'(\mathbb {R}^n)\rightarrow \mathcal {S}'(\mathbb {R}^n)$ continuously. Note that $Op(a)u(x) = \mathcal {F}^{-1}_\hbar (a(x,\cdot )\mathcal {F}_\hbar u(\cdot ))$ and that the operator corresponding to the symbol $a(x,\xi ) = \sum _{|\alpha |\le N} a_\alpha (x) \xi ^\alpha$ is $Op(a)u = \sum _{|\alpha |\le N} a_\alpha (x) (\hbar D)^\alpha u$. Notice that each derivative of this operator scales with $\hbar$.

For the present paper the most important example is the second order differential operator $A = -\hbar ^2 \Delta + \hbar ^2 \sum _{j=1}^{n} \beta _j(x) \partial _j$. Its symbol is given by $a(x,\xi ,\hbar ) = |\xi |^2 + i\hbar \sum _{j=1}^{n} \beta _j(x) \xi _j$, and its principal symbol is $a_0(x,\xi ,\hbar ) = |\xi |^2$.

1.3 B.3. Semiclassical Sobolev spaces

For $s\in \mathbb {R}$ the semiclassical Sobolev spaces $H_\text {scl}^s(\mathbb {R}^n)$ are algebraically equal to the standard Sobolev spaces $H^s(\mathbb {R}^n)$ but are endowed with different norms

$$\begin{aligned} \left\| u \right\| _{H_\text {scl}^s(\mathbb {R}^n)} = \left\| J^s u \right\| _{L^2(\mathbb {R}^n)}, \end{aligned}$$

where the semiclassical Bessel potential is defined by $J^s = Op(\langle \xi \rangle ^s)$. Informally,

$$\begin{aligned} J^s = (1-\hbar ^2 \Delta )^{s/2}, \quad s \in \mathbb {R}. \end{aligned}$$

For example, $\left\| u \right\| _{H_\text {scl}^1(\mathbb {R}^n)}^2 = \left\| u \right\| _{L^2(\mathbb {R}^n)}^2 + \left\| \hbar \nabla u \right\| _{L^2(\mathbb {R}^n)}^2$. A semiclassical $\psi$DO of order m is continuous from $H_\text {scl}^{s}(\mathbb {R}^n)$ to $H_\text {scl}^{s-m}(\mathbb {R}^n)$.

1.4 B.4. Composition

Composition of semiclassical $\psi$DOs can be analysed using the following calculus.

Theorem

(Symbol calculus) Let $a\in S^m$ and $b\in S^{m'}$. Then $Op(a)\circ Op(b) = Op(a\#b)$ for a certain $a\#b\in S^{m+m'}$ that admits the following asymptotic series

$$\begin{aligned} a\#b(x,\xi ,\hbar ) \sim \sum _{\alpha } \frac{\hbar ^{|\alpha |} i^{|\alpha |}}{\alpha !} D^\alpha _\xi a(x,\xi ,\hbar ) D^\alpha _x b(x,\xi ,\hbar ). \end{aligned}$$

The commutator and disjoint support estimates (4) and (5) follow, respectively, from the following.

Corollary

(Commutator and disjoint support) Let $a\in S^m$ and $b\in S^{m'}$. Then

• (i) $a\#b - b\#a \in \hbar S^{m+m'-1}.$

• (ii) If ${{\,\mathrm{supp}\,}}(a)\cap {{\,\mathrm{supp}\,}}(b) = \emptyset$, then $a\#b\in \hbar ^{\infty } S^{-\infty }$, i.e. $a\#b \in \hbar ^NS^{-M}$ for all $N,M\in \mathbb {R}$.

Proof

• (i) The principal symbol of $a\#b$, that is the first term in its asymptotic series, is ab. The second term is $\frac{\hbar }{i} \sum _{j=1}^n \partial _{\xi _j}a(x,\xi ,\hbar ) \partial _{x_j}b(x,\xi ,\hbar )$. We thus have that the principal symbol of the commutator $[Op(a),Op(b)] = Op(a\#b - b\#a)$ is given by

  $$\begin{aligned} \frac{\hbar }{i} \sum _{j=1}^n (\partial _{\xi _j}a \partial _{x_j}b - \partial _{x_j}a \partial _{\xi _j}b) \in \hbar S^{m+m'-1}. \end{aligned}$$

• (ii) If ${{\,\mathrm{supp}\,}}(a)\cap {{\,\mathrm{supp}\,}}(b) = \emptyset$, then each term in the asymptotic series of $a\#b$ vanishes. $\square$