Alleviating missing boundary conditions in elliptic partial differential equations using interior point measurements

TL;DR

This work addresses the challenge of recovering the Poisson solution when boundary data is unknown by leveraging interior point measurements. The authors formulate optimal recovery via a harmonic decomposition $u=u_0+u_{\mathcal{H}}$, where the harmonic part is reconstructed from Riesz representers $\{\phi_i\}$ and coefficients $U_i$ so that $u_{\mathcal{H}}^*=\sum_i U_i\phi_i$, with the measurements enforcing $\phi_i$ through a FE-constructed matrix $G$. They develop a near-optimal FE algorithm that approximates the Riesz representers and coefficients, derive refined error estimates for pointwise, $H^1$, and $L_\infty$ norms, and demonstrate improved convergence rates when measurement points lie in the interior $\Omega_d$ as opposed to the boundary. The results are complemented by saddle-point FE implementations and numerical experiments showing how domain regularity, measurement placement, and the number of measurements affect the recovery accuracy. Overall, the paper advances interior-boundary recovery strategies for elliptic PDEs, providing rigorous error bounds and practical FE procedures for accurate reconstruction under missing boundary information.

Abstract

We consider an optimal recovery problem for the Poisson problem when the boundary data is unknown. Compensating information is provided in the form of a finite number of measurements of the solution. A finite element algorithm for this problem was given in Binev et al. (2024), where measurements were assumed to be either bounded linear functionals of the solution or point measurements at locations lying anywhere in the closure of the computational domain. In contrast, we focus on the case of point measurements at locations lying in the interior of the domain. This lowers the regularity requirements placed on the solution. Also, a key ingredient in the recovery process is the finite element approximation of Riesz representers associated with the measurements. Our main result is a pointwise error estimate for the Riesz representers. We apply this to obtain improved estimates which measure the performance of the recovery algorithm in various norms.

Figures (6)

• Figure 7.1: Overrefinement errors $\| \phi_9 - \phi_k\|_{X}$ using polynomial degree $\zeta=1$ (left) and $\zeta=2$ (right) for the Riesz Representer $\phi \approx \phi_9$ associated with the measurement location $(0.5, 0.5)$ on the disc $\Omega = B(0, 1)$.
• Figure 7.2: Overrefinement errors $\| \phi_{10} - \phi_k\|_{X}$ using polynomial degree $\zeta=1$ (left) and $\zeta=2$ (right) for the Riesz Representer $\phi \approx \phi_{10}$ associated with the measurement location $(0.5, 0.5)$ on the disc $\Omega = B(0, 1)$.
• Figure 7.3: Overrefinement errors $\| \phi_{9}-\phi_k\|_X$ using polynomials of degree $\zeta=1$ for the Riesz representer $\phi \approx \phi_9$ associated with the measurement location $(-0.47, 0.47)$ on the 2D L-shaped domain $\Omega = (0, 1)^2\setminus ((0, 1) \times (-1, 0))$.
• Figure 7.4: Overrefinement errors $\|\phi_9 - \phi_8\|_{X}$ using polynomials of degree $\zeta=1$ (left) and $\zeta=2$ (right) with $X=H^1(\Omega)$ and $X=L_{\infty}(\Omega)$ and $\Omega = B(0, 1)$ as the measurement location approaches the boundary ($d \rightarrow 0^+$).
• Figure 7.5: Overrefinement errors $\|\phi_{10}-\phi_k\|$ using polynomials of degree $\zeta=1$ (left) and $\zeta=2$ (right) for the Riesz representer $\phi \approx \phi_{10}$ associated with the measurement location $(0, \sqrt{2}/2)\in \Gamma$.

Theorems & Definitions (35)

• Lemma 3.3
• Theorem 3.4: Near-optimal recovery in $X$
• proof
• Proposition 4.1: Localization
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Proposition 4.4: Inverse estimates
• proof