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  • 1.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå universitet, Institutionen för matematik och matematisk statistik.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå universitet, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Institutionen för matematik och matematisk statistik.
    Hybridized CutFEM for Elliptic Interface Problems2019In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 41, no 5, p. A3354-A3380Article in journal (Refereed)
    Abstract [en]

    We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.

  • 2.
    Burman, Erik
    et al.
    University College London, London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G
    Umeå University, Umeå, Sweden.
    Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization2018In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 3, article id 035004Article in journal (Refereed)
    Abstract [en]

    Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection– diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson’s equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.

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