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SUMMARY:Ill-Posed Problems and Stabilized Finite Element Methods
DTSTART:20191105T161500
DTEND:20191105T171500
DTSTAMP:20260407T074530Z
UID:dc02e0466aadcea9fb3a34d79ed68f8764456d21571c7f3e423bac50
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Erik BURMAN\, University College London\nComputational M
 athematics Seminar\n\nAbstract:\nIn this talk we will consider some ill-po
 sed elliptic equations and their discretiza-\ntion using nite element meth
 ods. The standard approach to ill-posed problems is to regularize\nthe con
 tinuous problem so that existence and uniqueness is guaranteed. The regula
 rized prob-\nlem can then be solved using standard nite element methods. W
 hen using this strategy\, in\norder to optimize accuracy\, the regularizat
 ion parameter must be chosen as a function both\nof the stability properti
 es of the ill-posed problem\, the mesh parameter and perturbations in\ndat
 a. Here we will propose a di erent approach [1]\, where the ill-posed pde 
 is discretized in\nan optimization framework\, prior to regularization. To
  ensure discrete well-posedness we add\nstabilizing terms to the formulati
 on\, drawing on experience from stabilized FEM and discon-\ntinuous Galerk
 in methods. The error in the resulting nite element reconstructions is the
 n\nanalyzed using Carleman estimates on the continuous problem. This resul
 ts in approximations\nthat are optimal with respect to the approximation o
 rder of the nite element space and the\nstability of the computed quantity
 . The mesh parameter here plays the role of the regular-\nization paramete
 r. Mesh resolution can be chosen independently of the stability properties
  of\nthe physical problem\, but must match perturbations in data\, in a wa
 y made explicit in the\nestimates. Some examples of problems analyzed in t
 his framework will be presented\, selected\nfrom recent work on the Helmho
 ltz equation [4]\, the convection{di usion equation [5]\, Stokes'\nequatio
 ns [2] and Darcy's equation [3].\n\nReferences\n[1] E. Burman. Stabilised 
 nite element methods for ill-posed problems with conditional stability. Bu
 ilding\nbridges: connections and challenges in modern approaches to numeri
 cal partial di erential equations\, 93127\,\nLect. Notes Comput. Sci. Eng.
 \, 114\, Springer\, [Cham]\, 2016.\, Dec. 2015.\n[2] E. Burman\, P. Hansbo
 \, Stabilized nonconforming nite element methods for data assimilation in 
 incom-\npressible\nows. Math. Comp. 87\, no. 311\, 2018.\n[3] E. Burman\, 
 M. G. Larson\, L. Oksanen. Primal dual mixed nite element methods for the 
 elliptic Cauchy\nproblem\, arXiv:1712.10172\, Siam J. Num. Anal.\, to appe
 ar\, 2018.\n[4] E. Burman\, M. Nechita\, L. Oksanen. Unique continuation f
 or the Helmholtz equation us-\ning stabilized nite element methods. Journa
 l de Mathematiques Pures et Appliquees\,\nhttps://doi.org/10.1016/j.matpur
 .2018.10.003.\n[5] E. Burman\, M. Nechita\, L. Oksanen. A stabilized nite 
 element method for inverse problems subject to the\nconvection-di usion eq
 uation. I: di usion-dominated regime. arXiv:1811.00431\, 2018.
LOCATION:MA A3 30
STATUS:CONFIRMED
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