Institute of Mathematics

Talk

Modul:   MAT870  Zurich Colloquium in Applied and Computational Mathematics

Ill-posed problems and stabilized Finite Element Methods

Prof. Dr. Erik Burman talk

Date: 18.09.19  Time: 16.15 - 17.45  Room: ETH HG E 1.2

In this talk we will consider some ill-posed elliptic equations and their discretiza- tion using nite element methods. The standard approach to ill-posed problems is to regularize the continuous problem so that existence and uniqueness is guaranteed. The regularized prob- lem can then be solved using standard nite element methods. When using this strategy, in order to optimize accuracy, the regularization parameter must be chosen as a function both of the stability properties of the ill-posed problem, the mesh parameter and perturbations in data. Here we will propose a di erent approach [1], where the ill-posed pde is discretized in an optimization framework, prior to regularization. To ensure discrete well-posedness we add stabilizing terms to the formulation, drawing on experience from stabilized FEM and discon- tinuous Galerkin methods. The error in the resulting nite element reconstructions is then analyzed using Carleman estimates on the continuous problem. This results in approximations that are optimal with respect to the approximation order of the nite element space and the stability of the computed quantity. The mesh parameter here plays the role of the regular- ization parameter. Mesh resolution can be chosen independently of the stability properties of the physical problem, but must match perturbations in data, in a way made explicit in the estimates. Some examples of problems analyzed in this framework will be presented, selected from recent work on the Helmholtz equation [4], the convection diffusion equation [5], Stokes' equations [2] and Darcy's equation [3]. Keywords: ill-posed problems, data assimilation, stabilized finite element methods Mathematics Subject Classifications (2010): 65N30, 35R25, 65N20 [1] E. Burman. Stabilised finite element methods for ill-posed problems with conditional stability. Building bridges: connections and challenges in modern approaches to numerical partial differential equations, 93127, Lect. Notes Comput. Sci. Eng., 114, Springer, [Cham], 2016., Dec. 2015. [2] E. Burman, P. Hansbo, Stabilized nonconforming finite element methods for data assimilation in incompressible flows. Math. Comp. 87, no. 311, 2018. [3] E. Burman, M. G. Larson, L. Oksanen. Primal dual mixed finite element methods for the elliptic Cauchy problem, arXiv:1712.10172, Siam J. Num. Anal., to appear, 2018. [4] E. Burman, M. Nechita, L. Oksanen. Unique continuation for the Helmholtz equation us- ing stabilized finite element methods. Journal de Mathematiques Pures et Appliquees, https://doi.org/10.1016/j.matpur.2018.10.003. [5] E. Burman, M. Nechita, L. Oksanen. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime. arXiv:1811.00431, 2018.