Efficient least squares discretizations for Unique Continuation and Cauchy problems

Rob Stevenson ()
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Rob Stevenson: University of Amsterdam, Korteweg-de Vries (KdV) Institute for Mathematics

A chapter in Multiscale, Nonlinear and Adaptive Approximation II, 2024, pp 449-460 from Springer

Abstract: Abstract 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.

Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-75802-7_20

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DOI: 10.1007/978-3-031-75802-7_20

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