Unique continuation problems and stabilised finite element methods

Book summary

Numerical analysis for partial differential equations (PDEs) traditionally considers problems that are well-posed in the continuum, for example the boundary value problem for Poisson’s equation. Computational methods such as the finite element method (FEM) then discretise the problem and provide numerical solutions. However, when a part of the boundary is inaccessible for measurements or no information is given on the boundary at all, the continuum problem might be ill-posed and solving it, in this case, requires regularisation.

In this book we consider the unique continuation problem with (possibly noisy) data given in an interior subset of the domain. This is an ill-posed problem also known as data assimilation and is related to the elliptic Cauchy problem. It arises often in inverse problems and control theory. We will focus on two PDEs for which the stability of this problem depends on the physical parameters: the Helmholtz and the convection–diffusion equations. We first prove conditional stability estimates that are explicit in the wave number and in the Péclet number, respectively, by using Carleman inequalities. Under a geometric convexity assumption, we obtain that for the Helmholtz equation the stability constants grow at most linearly in the wave number.

Then we present a discretise-then-regularise approach for the unique continuation problem. We cast the problem into PDE-constrained optimisation with discrete weakly consistent regularisation. The regularisation is driven by stabilised FEMs and we focus on the interior penalty stabilisation. For the Helmholtz and diffusion-dominated problems, we apply the continuum stability estimates to the approximation error and prove convergence rates by controlling the residual through stabilisation. For convection dominated problems, we perform a different error analysis and obtain sharper weighted error estimates along the characteristics of the convective field through the data region, with quasi-optimal convergence rates. The results are illustrated by numerical examples.

Book cover


Chapter 1
1 Ill-posed inverse problems and unique continuation
1.1 Well-posed problems
1.2 Linear inverse problems
1.2.1 Conditionally stable problems
1.3 The Cauchy problem
1.4 Unique continuation

Chapter 2

2 Conditional stability estimates for unique continuation
2.1 A pointwise Carleman estimate for the Laplacian
2.2 Holder stability estimates
2.3 Convection–diffusion operator
2.3.1 Shifting the norms
2.4 Helmholtz operator
2.4.1 Shifting the norms

Chapter 3

3 Discrete regularisation using stabilised finite element methods
3.1 Discretise-then-regularise
3.2 Stabilised finite element methods
3.3 Continuous interior penalty

Chapter 4

4 Helmholtz equation
4.1 Discretisation
4.2 Error estimates
4.2.1 Data perturbations
4.3 Numerical examples

Chapter 5
5 Diffusion-dominated problems
5.1 Discretisation
5.2 Error estimates
5.3 Numerical examples

Ch. 6

6 Convection-dominated problems
6.1 Discretisation
6.1.1 Stability region and weight functions
6.1.2 Discrete commutator property
6.2 Error estimates
6.2.1 Downstream estimates
6.2.2 Upstream estimates
6.3 Numerical examples

A Finite element inequalities
B Pseudodifferential operators


unique continuation, ill-posed problems, stabilised finite element methods, data assimilation


pdf file

Book on publisher website

Cite this book as:

M. Nechita, Unique continuation problems and stabilised finite element methods, Casa Cărții de Știință, Cluj-Napoca, Romania, 2021, ISBN: 978-606-17-1816-0

Book Title

Unique continuation problems and stabilised finite element methods


Casa Cărții de Știință

Print ISBN


Online ISBN

Google scholar

The book on google scholar.

[1] G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella. The stability for the Cauchy problem for elliptic equations. Inverse Problems, 25:123004, 2009.
[2] M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells. The FEniCS project version 1.5. Archive of Numerical Software, 3(100), 2015.
[3] B. Ayuso and L. D. Marini. Discontinuous Galerkin methods for advection-diffusionreaction problems. SIAM J. Numer. Anal., 47(2):1391–1420, 2009.
[4] I. M. Babuska and S. A. Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal., 34(6):2392–2423, 1997.
[5] D. Baskin, E. A. Spence, and J. Wunsch. Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations. SIAM J. Math. Anal., 48(1):229–267, 2016.
[6] R. Becker and B. Vexler. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math., 106(3):349–367, 2007.
[7] F. B. Belgacem. Why is the Cauchy problem severely ill-posed? Inverse Problems, 23(2):823, 2007.
[8] S. Bertoluzza. The discrete commutator property of approximation spaces. C. R. Acad. Sci. Paris S´er. I Math., 329(12):1097–1102, 1999.
[9] M. Boulakia, E. Burman, M. A. Fernandez, and C. Voisembert. Data assimilation finite element method for the linearized Navier-Stokes equations in the low Reynolds regime. Inverse Problems, 36:085003, 2020
[10] L. Bourgeois. Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation. Inverse problems, 22(2):413, 2006.
[11] L. Bourgeois and L. Chesnel. On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates. ESAIM Math. Model. Numer. Anal., 54(2):493–529, 2020.
[12] A. N. Brooks and T. J. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations. Comput. Methods Appl. Mech. Engrg., 32(1-3):199–259, 1982.
[13] E. Burman. A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal., 43(5):2012–2033, 2005.
[14] E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and illposed problems. Part I: elliptic equations. SIAM J. Sci. Comput., 35(6):A2752–A2780, 2013.
[15] E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and illposed problems. Part II: hyperbolic equations. SIAM J. Sci. Comput., 36(4):A1911– A1936, 2014.
[16] E. Burman, A. Feizmohammadi, and L. Oksanen. A finite element data assimilation method for the wave equation. Math. Comp., 89(324):1681–1709, 2020.
[17] E. Burman, J. Guzman, and D. Leykekhman. Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal., 29(2):284–314, 2009.
[18] E. Burman and P. Hansbo. Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems. Comput. Methods Appl. Mech. Engrg., 193(15-16):1437–1453, 2004.
[19] E. Burman and P. Hansbo. Stabilized nonconforming finite element methods for data assimilation in incompressible flows. Math. Comp., 87(311):1029–1050, 2018.
[20] E. Burman, P. Hansbo, and M. G. Larson. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems, 34:035004, 2018
[21] E. Burman, M. Nechita, and L. Oksanen. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl., 129:1–22, 2019.
[22] E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection–diffusion equation. I: diffusion-dominated regime. Numer. Math., 144(451–477), 2020.
[23] E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime. arXiv preprint arXiv:2006.13201, 2020.
[24] E. Burman and L. Oksanen. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505–528, 2018.
[25] E. Burman, H. Wu, and L. Zhu. Linear continuous interior penalty finite element method for Helmholtz equation with high wave number: one-dimensional analysis. Numer. Methods Partial Differential Equations, 32(5):1378–1410, 2016.
[26] T. Carleman. Sur un probleme d’unicite pour les systemes d’equations aux derivees partielles a deux variables independantes. Ark. Mat. Astr. Fys., 26B(17):1–9, 1939.
[27] L. Dede’ and A. Quarteroni. Optimal control and numerical adaptivity for advectiondiffusion equations. ESAIM Math. Model. Numer. Anal., 39(5):1019–1040, 2005.
[28] D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods, volume 69. Springer Science & Business Media, 2011.
[29] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems. Invent. Math., 178(1):119–171, 2009.
[30] J. Douglas and T. Dupont. Interior penalty procedures for elliptic and parabolic Galerkin methods. In Computing methods in applied sciences, pages 207–216. Springer, 1976.
[31] H. W. Engl, M. Hanke, and A. Neubauer. Regularization of inverse problems, volume 375. Springer, 1996.
[32] A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
[33] A. Ern and J.-L. Guermond. Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM Math. Model. Numer. Anal., 40(1):29–48, 2006.
[34] L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, 2010.
[35] J. Hadamard. Lectures on Cauchy’s Problem in Linear Partial Differential Equations.Yale University Press, 1923.
[36] F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251–265, 2012.
[37] M. Hinze, N. Yan, and Z. Zhou. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math., 27(2-3):237–253, 2009.[38] L. Hormander. The analysis of linear partial differential operators, volume III. Springer-Verlag, 1985.
[39] L. Hormander. The analysis of linear partial differential operators, volume IV. SpringerVerlag, 1985.
[40] T. Hrycak and V. Isakov. Increased stability in the continuation of solutions to the Helmholtz equation. Inverse Problems, 20(3):697–712, 2004.
[41] F. Ihlenburg and I. Babuska. Finite element solution of the Helmholtz equation with high wave number. I. The h-version of the FEM. Comput. Math. Appl., 30(9):9–37, 1995.
[42] F. Ihlenburg and I. Babuska. Finite element solution of the Helmholtz equation with high wave number. II. The h-p version of the FEM. SIAM J. Numer. Anal., 34(1):315–358, 1997.
[43] V. Isakov. Inverse problems for partial differential equations, volume 127 of Applied Mathematical Sciences. Springer, 3rd edition, 2017.
[44] F. John. Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math., 13:551–585, 1960.
[45] V. John, P. Knobloch, and J. Novo. Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story? Comput. Vis. Sci., 19(5-6):47–63, 2018
[46] C. Johnson, U. Navert, and J. Pitkaranta. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg., 45:285–312, 1984.
[47] M. V. Klibanov and F. Santosa. A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM J. Appl. Math., 51(6):1653–1675, 1991.
[48] M. V. Klibanov and A. A. Timonov. Carleman estimates for coefficient inverse problems and numerical applications, volume 46. Walter de Gruyter, 2012.
[49] R. Lattes and J. Lions. Methode de quasi-reversibilite et applications, volume 15 of Travaux et Recherches Mathematiques. Dunod, Paris, 1967.
[50] M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskiı. Ill-posed problems of mathematical physics and analysis, volume 64 of Translations of Mathematical Monographs. American Mathematical Society, 1986.
[51] J. Le Rousseau and G. Lebeau. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., 18(3):712–747, 2012.
[52] E. Malinnikova and S. Vessella. Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann., 353(4):1157–1181, 2012.
[53] J. M. Melenk and S. Sauter. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal., 49(3):1210–1243, 2011.
[54] H.-G. Roos and M. Stynes. Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Methods Appl. Math., 15(4):531–550, 2015.
[55] H.-G. Roos, M. Stynes, and L. Tobiska. Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, volume 24. Springer Science & Business Media, 2008.
[56] L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54(190):483–493, 1990.
[57] E. M. Stein. Singular integrals and differentiability properties of functions, volume 30 of Princeton Mathematical Series. Princeton University Press, 1970.
[58] D. Tataru. . In C. B. Croke, M. S. Vogelius, G. Uhlmann, and I. Lasiecka, editors, Geometric Methods in Inverse Problems and PDE Control, pages 239–255. Springer, New York, NY, 2004.
[59] L. N. Trefethen. Quantifying the ill-conditioning of analytic continuation. BIT, 2020.
[60] H. Wu. Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version. IMA J. Numer. Anal., 34(3):1266–1288, 2014.
[61] N. Yan and Z. Zhou. A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math., 223(1):198–217, 2009.
[62] M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, 2012.


Related Posts