Posts by Mihai Nechita


We consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including both convergence with respect to discretisation and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the approximation space. A proof is given that no approximation can converge at a better rate than that given by the definition without increasing the sensitivity to perturbations, thus justifying the concept. A recently introduced class of primal-dual finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be optimal in the sense of this definition.


Erik Burman
Department of Mathematics, University College London, London, WC1E 6BT, UK

Mihai Nechita
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
Department of Mathematics, Babeṣ-Bolyai University, Cluj-Napoca, Romania

Lauri Oksanen
Department of Mathematics and Statistics, University of Helsinki, P.O. 68, 00014, Helsinki, Finland


Unique continuation; Ill-posed problems; Conditional stability; Approximation methods; Finite element methods; Stabilised methods; Regularisation; Error estimates; Optimality; Optimal convergence

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E. Burman, M. Nechita, L. Oksanen, Optimal approximation of unique continuation. Found. Comput. Math. (2024).


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Foundations of Computational Mathematics

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Optimal approximation of unique continuation

Abstract We consider numerical approximations of ill-posed elliptic problems with conditional stability. The notion of optimal error estimates is defined including…