Ultra-weak least squares discretizations for unique continuation and Cauchy problems

TL;DR

This work develops conditional stability estimates for the UC and Cauchy problems of the Poisson equation using ultra-weak variational formulations, and they exploit regularized least squares with discretized dual norms to obtain convergent, stable approximations. By forming uniformly stable trial/test space pairs (e.g., piecewise constants with HTC or Morley test spaces), the method achieves $L_2$-error bounds on subdomains that scale with a fractional exponent $oldsymbol{α}$ of the sum of data-error and best-approximation error, measured in weaker norms. The paper also demonstrates that nonconforming test spaces can replace $C^1$ tests with comparable error behavior, up to a data-oscillation term that can be controlled by companion/smoothing operators. Numerical experiments on UC and Cauchy problems corroborate the theory, showing stable convergence and adaptive refinement benefits, even under data perturbations and challenging perturbations that reflect the ill-posed nature of the problems. Overall, the approach offers a computationally efficient pathway to stable UC and Cauchy solutions with provable error decay in practical discretizations.

Abstract

In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.

Figures (8)

• Figure 1: Illustration with proof of Theorem \ref{['thm:condstabCauchy']}
• Figure 2: HTC macro-element and its 12 DoF.
• Figure 3: Morley element and its 6 DoF.
• Figure 4: DoFs in $X^\delta$ vs. error in $u_\varepsilon^\delta$ in $L_2(G)$-norm for different choices of $\omega \subseteq G\subseteq \Omega$. Left: error for $u(x,y) = e^x\cos(y)$. Right: error for $u(x,y) = \tfrac{1}{4 \pi} \log(x^2 + y^2)$. In both cases $\varepsilon = \hbox{DoFs}^{-\frac{1}{2}}$.
• Figure 5: Left: Plot of $(x,y)\mapsto \min(\max(u(x,y),20),-20)$. Right: Example of a mesh generated by our adaptive routine.

Theorems & Definitions (24)

• Theorem 2.1: Propagation of smallness
• Theorem 2.2: Conditional stability UC ultra-weak
• proof
• remark 2.3
• Theorem 2.4: Conditional stability Cauchy ultra-weak
• proof
• remark 2.5
• Theorem 3.1: 58.7
• Corollary 3.2
• proof