Computational unique continuation with finite dimensional Neumann trace

TL;DR

This work addresses computational unique continuation for the Poisson equation with Neumann data constrained to a finite-dimensional boundary space. It develops a self-contained Lipschitz stability analysis in the global $H^1(\Omega)$-norm and leverages this to derive optimal a posteriori and a priori error estimates for a primal-dual stabilized finite element method, including perturbation and flux-approximation analyses. The framework extends finite-dimensional trace techniques from Dirichlet to Neumann data, enabling robust error control without relying on global logarithmic stability, and is complemented by numerical experiments that validate convergence rates and resilience to perturbations. Collectively, the results provide a rigorous, scalable approach for accurate UC approximations under finite-dimensional Neumann data and offer practical guidance for flux recovery on the boundary.

Abstract

We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.

Figures (7)

• Figure 1: A uniform triangular decomposition on $\Omega=[0,1]\times[0,1]$ with $n=6$.
• Figure 1: Patch $F$ on $\partial\Omega$ together with associated bulk patch $T$. $\varphi_F|_K\in \mathbb{P}_1$ for all $K\in \mathcal{T}_h$, and $\varphi_F(x_T)=\Theta (h)$ and vanishes on other nodes.
• Figure 1: The error $\lVert u-u_h\rVert_{H^1}$ as a function of mesh size $h$ with the reference rate $h$ presented by the dash line. (a) $u_h$ is computed with parameter $\gamma = 1,\ 10^{-1},\ 10^{-2},\ 0$ with triangles pointing down, up, left, and right, respectively. (b) $u_h$ is obtained with different $\mathcal{V}_N$ where $N = 1,\ 8,\ 16,\ 64$ with triangles pointing down, up, left, and square, respectively.
• Figure 2: The error $\lVert u-u_h\rVert_{H^1}$ as a function of mesh size $h$. Here $N=1,2$ with triangles pointing down and up. Reference rate $h$ is presented by the dash line.
• Figure 3: The error $\lVert u-u_h\rVert_{H^1}$ as a function of mesh size $h$. Here $\varepsilon= 0.12, 0.06,0$ with triangles pointing down, up and left. Reference rate $h$ is presented by the dash line.

Theorems & Definitions (38)

• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Theorem 3.3
• Proof 3
• Lemma 4.1
• Proof 4
• Remark 4.2
• Lemma 4.3