A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime

[1] R. Becker and B. Vexler. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math., 106(3):349–367, 2007.
[2] S. C. Brenner. Poincare-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal., 41(1):306–324, 2003.

[3] 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.

[4] E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput., 35(6):A2752 A2780, 2013.

[5] E. Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris, 352(7-8):655–659, 2014.

[6] 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.

[7] 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. in preparation, 2018.

[8] E. Burman, M. Nechita, and L. Oksanen. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl., 2018.

[9] E. Burman and L. Oksanen. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505–528, 2018.

[10] L. Dede’ and A. Quarteroni. Optimal control and numerical adaptivity for advection-diffusion equations. M2AN Math. Model. Numer. Anal., 39(5):1019–1040, 2005.

[11] 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.

[12] A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.

[13] A. Ern and J.-L. Guermond. Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal., 40(1):29 48, 2006.

[14] F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251–265, 2012.

[15] 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.

[16] 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.

[17] E. Malinnikova and S. Vessella. Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann., 353(4):1157–1181, 2012.

[18] P. Monk and E. Suli. The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal., 36(1):251–274, 1999.

[19] 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.

[20] M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.[32] M.D. Zeiler, ADADELTA : An adaptive learning rate method, 2012,http://arxiv.org/abs/1212.5701.