Optimal Approximation of Unique Continuation

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-024-09655-w Optimal Approximation of Unique Continuation Erik Burman1 · Mihai Nechita2,3 · Lauri Oksanen4 Received: 13 November 2023 / Revised: 8 April 2024 / Accepted: 9 April 2024 © The Author(s) 2024 Abstract 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. Keywords Unique continuation · Ill-posed problems · Conditional stability · Approximation methods · Finite element methods · Stabilised methods · Regularisation · Error estimates · Optimality · Optimal convergence Communicated by Rob Stevenson. E.B.: supported by the EPSRC Grants EP/T033126/1 and EP/V050400/1. M. N.: this work was supported by the project “The Development of Advanced and Applicative Research Competencies in the Logic of STEAM + Health” /POCU/993/6/13/153310, project co-financed by the European Social Fund through The Romanian Operational Programme Human Capital 2014–2020. L. O.: Co-funded by the European Union (ERC, LoCal, 101086697). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Co-funded by the Research Council of Finland (347715, 3530969). B Erik Burman Mihai Nechita Lauri Oksanen 1 Department of Mathematics, University College London, London WC1E 6BT, UK 2 Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania 3 Department of Mathematics, Babes.-Bolyai University, Cluj-Napoca, Romania 4 Department of Mathematics and Statistics, University of Helsinki, P.O. 68, 00014 Helsinki, Finland 123 Foundations of Computational Mathematics Mathematics Subject Classification 35J05 · 35J15 · 65N12 · 65N20 · 65N21 · 65N30 1 Introduction Arguably one of the most fundamental results in finite element analysis is the best approximation result for the Galerkin method, known as Cea’s lemma [20], which together with approximation estimates for finite element functions results in quasioptimal error estimates for finite element methods [3, 27, 44, 48]. This result, that we will review below, essentially says that if a (2m)-order elliptic problem, m ≥ 1, is approximated with H m -conforming finite elements of local polynomial order p the error in H m -norm is of the order h p+1−m for a sufficiently smooth solution and that this rate is optimal compared to approximation: the best interpolant of the exact solution has similar accuracy. For ill-posed elliptic problems the situation is different. On the continuous level existence can only be guaranteed after regularisation of the problem. The two main approaches are Tikhonov regularisation [45] and quasi-reversibility [31]. These two approaches are strongly related (see for instance [7]). The main effort in the error analysis has been to estimate the perturbation induced by the addition of regularisation, and how to choose the associated regularisation operator or parameter [6, 28, 35, 38, 43]. The error due to approximation in finite dimensional spaces of such regularised problems has also been analysed [24, 36, 41]. There is also a rich literature on projection methods for ill-posed problems where the discretisation serves as regularisation and refinement has to stop as soon as the effect of perturbations in data becomes dominant [22, 23, 26, 30, 42]. These methods are often based on least squares methods and the convergence of the approximate solution to the exact solution for unperturbed data has been proven in several works. There are also different stopping criteria for mesh refinement in order to avoid degeneration due to pollution from perturbations. However no results on rates of convergence where the discretisation errors and the perturbation errors are both included appear in these references. The use of conditional stability (continuous dependence on data under the assumption of a certain a priori bound) to obtain more complete error estimates has been proposed in [9–11, 18] for a class of finite element methods based on weakly consistent regularisation/stabilisation in a primal-dual framework. Here stability is obtained through a combination of consistent stabilisation and Tikhonov regularisation, scaled with the mesh parameter to obtain weak consistency. The upshot is that for this class of methods an error analysis exists, where the computational error is bounded in terms of the mesh parameter and perturbations of data, with constants depending on Sobolev norms of the exact solution. Similarly to the well-posed case, the error estimates for this approach combine the stability of the physical problem with the numerical stability of the computational method and the approximability of the finite element space. Contrary to the well-posed case, numerical stability can not be deduced from the physical stability, but has to be a consequence of the design of the stabilisation terms. This means that the stabilisation in this framework is bespoke, and must be designed to combine optimal (weak) consistency and sufficient numerical stability. There is often 123 Foundations of Computational Mathematics a tension between these two design criteria. As noted above, sometimes Tikhonov regularisation, scaled with the mesh parameter, may be used in the framework. An interesting feature is that the bespoke character also allows for the integration of the dependence of the estimates on physical parameters and different problems regimes [16, 17]. Other physical models that have been considered in this framework include data assimilation for fluids [5, 13, 18], or wave equations [12]. Common for all these references is the fact that the error estimates reflect the stability of the continuous problem and the approximation order of the finite element space, which seems to be an optimality property of the methods. No rigorous proof, however, has been given for this optimality. The objective of the present work is to show, in the model case of unique continuation for Laplace equation, that the proposed error estimates are indeed optimal. For ill-posed PDEs that are conditionally stable, error estimates in terms of the modulus of continu (...truncated)