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Efficient least squares discretizations for Unique Continuation and Cauchy problems

In: Multiscale, Nonlinear and Adaptive Approximation II

Author

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  • Rob Stevenson

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

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.

Suggested Citation

  • Rob Stevenson, 2024. "Efficient least squares discretizations for Unique Continuation and Cauchy problems," Springer Books, in: Ronald DeVore & Angela Kunoth (ed.), Multiscale, Nonlinear and Adaptive Approximation II, pages 449-460, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-75802-7_20
    DOI: 10.1007/978-3-031-75802-7_20
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