Unique continuation for the Helmholtz equation using stabilized finite element methods

Burman, Erik; Nechita, Mihai; Oksanen, Lauri

doi:10.1016/j.matpur.2018.10.003

Cited by 17 publications

(39 citation statements)

References 27 publications

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“…Let us also mention that the method in the present paper draws from our experience on stabilized finite element methods for the elliptic Cauchy problem [7,8], and other types of data assimilation problems, see [10] for elliptic and [9,11] for parabolic cases. In [10] we considered the Helmholtz equation. The convergence estimate there is explicit in the wave number, and exhibits a hyperbolic character in the sense that it relies on a convexity assumption that can viewed as a particular local version of the geometric control condition.…”

Section: Introductionmentioning

confidence: 99%

A finite element data assimilation method for the wave equation

2020

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We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions are used for the approximation in space and backward differentiation is used in time. Stabilizing terms are added on the discrete level. The design of these terms is driven by numerical stability and the stability of the continuous problem, with the objective of minimizing the computational error. Error estimates are then derived that are optimal with respect to the approximation properties of the numerical scheme and the stability properties of the continuous problem. The effects of discretizing the (smooth) domain boundary and other perturbations in data are included in the analysis.

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Section: Introductionmentioning

confidence: 99%

A finite element data assimilation method for the wave equation

2020

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“…The improved local conservation of the full primal dual formulation was observed and the iterative procedure converged to a relative residual of the increment in the L 2 -norm of O(10 −6 ) within up to three iterations. Finally we point out the the method presented herein also can be applied to inverse problems subject to the Helmholtz equation, as those discussed in [19].…”

Section: Discussionmentioning

confidence: 95%

“…Herein we will advocate a different approach based on discretization of the ill-posed physical model in an optimization framework, followed by regularization of the discrete problem. This primal-dual approach was first introduced by Burman in the papers [11,13,12,14], drawing on previous work by Bourgeois and Dardé on quasi reversibility methods [4,5,7,8] and further developed for elliptic data assimilation problems [17], for parabolic data reconstruction problems in [20,18] and finally for unique continuation for Helmholtz equation [19]. For a related method using finite element spaces with C 1 -regularity see [22] and for methods designed for well-posed, but indefinite problems, we refer to [9] and for second order elliptic problems on non-divergence form see [38] and [39].…”

Section: Introductionmentioning

confidence: 99%

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

Burman¹,

Larson²,

Oksanen³

2018

SIAM J. Numer. Anal.

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We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Dardé, Hannukainen and Hyvönen in An H div -based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Numer. Anal., 51(4) 2013. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

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“…This framework combines stabilized finite element methods designed for well-posed problems with variational formulations for data assimilation and sharp stability estimates for the continuous problems based on three balls inequalities or Carleman estimates. Recent developments include finite element data assimilation methods with optimal error estimates for the heat equation [24,23] and design of methods for indefinite or nonsymmetric scalar elliptic problems analyzed using Carleman estimates with explicit dependence on the physical parameters [17,16].…”

Section: Introductionmentioning

confidence: 99%

Data assimilation finite element method for the linearized Navier–Stokes equations in the low Reynolds regime

et al. 2020

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In this paper we are interested in designing and analyzing a finite element data assimilation method for laminar steady flow described by the linearized incompressible Navier-Stokes equation. We propose a weakly consistent stabilized finite element method which reconstructs the whole fluid flow from velocity measurements in a subset of the computational domain. Using the stability of the continuous problem in the form of a three balls inequality, we derive quantitative local error estimates for the velocity. Numerical simulations illustrate these convergences properties and we finally apply our method to the flow reconstruction in a blood vessel.

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Unique continuation for the Helmholtz equation using stabilized finite element methods

Cited by 17 publications

References 27 publications

A finite element data assimilation method for the wave equation

A finite element data assimilation method for the wave equation

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

Data assimilation finite element method for the linearized Navier–Stokes equations in the low Reynolds regime

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