Efficient least squares discretizations for Unique Continuation and Cauchy problems

We consider least squares discretizations of Unique Continuation and Cauchy problems for the Poisson equation based on ultra-weak variational formulations. The dual norm that is present in the (regularized) least squares functional cannot be evaluated exactly, and so has to be discretized which leads to a saddle-point formulation. For uniformly stable pairs of ‘trial’ and ‘test’ finite element spaces, approximations are obtained that are quasi-best in view of the available conditional stability estimates. Compared to standard variational formulations, conditional stability estimates that corresponds to ultra-weak formulations result in better convergence rates with the same error-norm. Globally C1 finite element test spaces to accommodate the ultraweak formulation will be avoided by the application of nonconforming test spaces. Thanks to the ultra-weak formulation, both Neumann and Dirichlet boundary conditions are natural ones, which in particular enables a convenient discretization of the Cauchy problem.