Page 450 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 450
MODELING, APPROXIMATION, AND ANALYSIS OF PARTIAL DIFFERENTIAL
EQUATIONS INVOLVING SINGULAR SOURCE TERMS (MS-39)
larized problem. As an application, we show how finite element approximations of a regularized
immersed interface method result in the same rates of convergence of its non-regularized coun-
terpart, provided that the support of the Dirac delta approximation is set to a multiple of the
mesh size, at a fraction of the implementation complexity. Numerical experiments are provided
to support our theories.
Regularizations of the Dirac delta distribution, and applications
Nilima Nigam, nigam@math.sfu.ca
Simon Fraser University, Canada
Coauthors: John Stockie, Bamdad Hosseini
Nilima Nigam Regularizations of the Dirac delta distribution, and applications.’ The need to
approximate singular sources is widespread in numerical analysis. In this talk, we present a
historical overview of, and a framework for constructing approximations of the Dirac delta
distribution. As part of this framework we study their convergence in suitable topologies. This
in turn allows us to examine the consistency error incurred in their use while numerically solving
PDEs. We present numerical experiments in which these ideas are illustrated. This work was
inspired by notable previous works on approximation of singular source terms, and is joint with
Bamdad Hosseini and John Stockie.
Projection in negative norms and the regularization of rough linear
functionals
Sergio Rojas, srojash@gmail.com
Pontificia Universidad Católica de Valparaíso, Chile
Coauthors: Felipe Millar, Ignacio Muga, Kris van der Zee
Rough linear functionals (such as Dirac Delta distributions) often appear on the right-hand side
of variational formulations of PDEs. As they live in negative Sobolev spaces, they dramatically
affect adaptive finite element procedures to approximate the solution of a given PDE.
In this talk, we propose an alternative that, in a first step, computes a projection of the rough
functional over piecewise polynomial spaces, up to a given desired precision in a negative norm
sense. The projection (being Lp-regular) can be used as the right-hand side of a regularized
problem for which adaptive Galerkin methods perform better. A complete error analysis of the
proposed methodology will be shown, together with numerical experiments.
Analysis and approximation of fluids under singular forcing
Abner Salgado, asalgad1@utk.edu
University of Tennessee, United States
Motivated by applications, like modeling of thin structures immersed in a fluid, we develop a
well posedness theory for Newtonian and some non-Newtonian fluids under singular forcing
in Lipschitz domains, and in convex polytopes. The main idea, that allows us to deal with
such forces, is that we study the problem in suitably weighted Sobolev spaces. We develop an
448
EQUATIONS INVOLVING SINGULAR SOURCE TERMS (MS-39)
larized problem. As an application, we show how finite element approximations of a regularized
immersed interface method result in the same rates of convergence of its non-regularized coun-
terpart, provided that the support of the Dirac delta approximation is set to a multiple of the
mesh size, at a fraction of the implementation complexity. Numerical experiments are provided
to support our theories.
Regularizations of the Dirac delta distribution, and applications
Nilima Nigam, nigam@math.sfu.ca
Simon Fraser University, Canada
Coauthors: John Stockie, Bamdad Hosseini
Nilima Nigam Regularizations of the Dirac delta distribution, and applications.’ The need to
approximate singular sources is widespread in numerical analysis. In this talk, we present a
historical overview of, and a framework for constructing approximations of the Dirac delta
distribution. As part of this framework we study their convergence in suitable topologies. This
in turn allows us to examine the consistency error incurred in their use while numerically solving
PDEs. We present numerical experiments in which these ideas are illustrated. This work was
inspired by notable previous works on approximation of singular source terms, and is joint with
Bamdad Hosseini and John Stockie.
Projection in negative norms and the regularization of rough linear
functionals
Sergio Rojas, srojash@gmail.com
Pontificia Universidad Católica de Valparaíso, Chile
Coauthors: Felipe Millar, Ignacio Muga, Kris van der Zee
Rough linear functionals (such as Dirac Delta distributions) often appear on the right-hand side
of variational formulations of PDEs. As they live in negative Sobolev spaces, they dramatically
affect adaptive finite element procedures to approximate the solution of a given PDE.
In this talk, we propose an alternative that, in a first step, computes a projection of the rough
functional over piecewise polynomial spaces, up to a given desired precision in a negative norm
sense. The projection (being Lp-regular) can be used as the right-hand side of a regularized
problem for which adaptive Galerkin methods perform better. A complete error analysis of the
proposed methodology will be shown, together with numerical experiments.
Analysis and approximation of fluids under singular forcing
Abner Salgado, asalgad1@utk.edu
University of Tennessee, United States
Motivated by applications, like modeling of thin structures immersed in a fluid, we develop a
well posedness theory for Newtonian and some non-Newtonian fluids under singular forcing
in Lipschitz domains, and in convex polytopes. The main idea, that allows us to deal with
such forces, is that we study the problem in suitably weighted Sobolev spaces. We develop an
448
