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

TL;DR

This paper develops ultra-weak least squares discretizations for solving unique continuation and Cauchy problems related to the Poisson equation, providing stability estimates and error bounds in weaker norms with practical finite element implementations.

Contribution

It introduces a novel ultra-weak variational approach with discretized dual norms, enabling stable numerical approximations for ill-posed problems without requiring $C^1$ finite element spaces.

Findings

Error bounds in weaker norms are established for the numerical solutions.

Nonconforming finite element test spaces can be used effectively.

Numerical experiments confirm the theoretical stability and accuracy.

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.