ANID

Permanent URI for this collection

https://repositorio.uc.cl/handle/11534/74801

Browse

Browsing ANID by browse.metadata.categoria "Matemática física y química"

Now showing 1 - 5 of 5
  • Thumbnail Image
    Item
    A robust DPG method for large domains
    (2021) Führer, Thomas; Heuer, Norbert
    We observe a dramatic lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincare-type inequality. We show how robustness can be re-established by using appropriately scaled test norms. As model cases we study the Poisson problem and the Kirchhoff-Love plate bending model, and also include fully discrete variants where optimal test functions are approximated. Numerical experiments for both model problems, including an-isotropic domains and mixed boundary conditions, confirm our findings.
  • Thumbnail Image
    Item
    Analysis of Backward Euler Primal DPG Methods
    (2021) Führer, Thomas; Heuer, Norbert; Karkulik, Michael
    We analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in a natural norm and in the L-2 norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.
  • Thumbnail Image
    Item
    Lipschitz stability for backward heat equation with application to fluorescence microscopy
    (SIAM PUBLICATIONS, 2021) Arratia, Pablo; Courdurier Bettancourt, Matías Alejandro; Cueva, Evelyn; Osses, Axel; Palacios Farias, Benjamin Pablo
    In this work we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in R-n, from measurements along a positive time interval and over an open set containing its support. We employ a nonconstructive method which ensures the existence of the stability constant, but it is not explicit in terms of the parameters of the problem. The main ingredients in our method are the compactness of support of the initial condition and the explicit dependency of solutions to the heat equation with respect to it. By means of Carleman estimates we obtain an analogous result for the case when the observation is made along an exterior region omega x (t, T), such that the unobserved part R-n\omega is bounded. In the latter setting, the method of Carleman estimates gives a general conditional logarithmic stability result when initial temperatures belong to a certain admissible set, without the assumption of compactness of support and allowing an explicit stability constant. Furthermore, we apply these results to deduce similar stability inequalities for the heat equation in R and with measurements available on a curve contained in Rx[0,infinity), leading to the derivation of stability estimates for an inverse problem arising in 2D fluorescence microscopy. In order to further understand this Lipschitz stability, in particular, the magnitude of its stability constant with respect to the parameters of the problem, a numerical reconstruction is presented based on the construction of a linear system for the inverse problem in fluorescence microscopy. We investigate the stability constant by analyzing the condition number of the corresponding matrix.
  • Thumbnail Image
    Item
    MINRES for Second-Order PDEs with Singular Data
    (2022) Führer Thomas; Heuer, Norbert; Karkulik, Michael
    Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov-Galerkin (DPG) method with optimal test functions usually exclude singular data, e.g., non-square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of H-1 loads and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.
  • Thumbnail Image
    Item
    Ultraweak formulation of linear PDEs in nondivergence form and DPG approximation
    (2021) Führer, Thomas
    We develop and analyze an ultraweak formulation of linear PDEs in nondivergence form where the coefficients satisfy the Cordes condition. Based on the ultraweak formulation we propose discontinuous Petrov-Galerkin (DPG) methods. We investigate Fortin operators for the fully discrete schemes and provide a posteriori estimators for the methods under consideration. Numerical experiments are presented in the case of uniform and adaptive mesh-refinement. (C) 2020 Elsevier Ltd. All rights reserved.