Page 449 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 449
MODELING, APPROXIMATION, AND ANALYSIS OF PARTIAL DIFFERENTIAL
EQUATIONS INVOLVING SINGULAR SOURCE TERMS (MS-39)
istic scenario, demonstrating that the method can robustly and efficiently handle the one-way
coupling between complex fluid microstructures and the elastic matrix.
Some remarks on a two phase problem in the Heisenberg group
Fausto Ferrari, fausto.ferrari@unibo.it
Dipartimento di Matematica dell’Università di Bologna,
Piazza di Porta S. Donato, 5, 40126 Bologna, Italia
We will discuss about the existence of an Alt-Caffarelli-Friedman monotonicity formula for a
two phase problem associated with a Bernoulli type functional in the Heisenberg group.
Finite element approximation of Stokes equations with non-smooth data
Lucia Gastaldi, lucia.gastaldi@unibs.it
University of Brescia, Italy
The lid driven cavity flow is a well known model used frequently to test the finite element
approximation of the Stokes problem. Actually, this model does not meet the regularity re-
quirements for the boundary datum which is only in L2, so that it cannot be the trace of the
velocity which belongs to H1. In this talk, we analyze the finite element approximation of the
Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution we
first approximate the boundary datum by a smooth one and then apply a standard finite element
method to the regularized problem. We prove almost optimal order error estimates for two reg-
ularization procedures in the case of general data in fractional order Sobolev spaces, and for the
Lagrange interpolation (with appropriate modifications at the discontinuities) for piece- wise
smooth data. Our results apply in particular to the classic lid-driven cavity problem improving
the existing error estimates.
Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and
efficiency, and show some numerical examples which suggest that optimal order of convergence
is obtained by an adaptive procedure based on our estimator.
The results reported in this talk have been obtained, in collaboration with Ricardo Duràn
and Ariel Lombardi.
A priori error estimates of regularized elliptic problems
Wenyu Lei, wenyu.lei@sissa.it
SISSA-International School for Advanced Studies, Italy
Coauthor: Luca Heltai
Approximations of the Dirac delta distribution are commonly used to create sequences of
smooth functions approximating nonsmooth (generalized) functions, via convolution. We show
a-priori rates of convergence of this approximation process in standard Sobolev norms, with
minimal regularity assumptions on the approximation of the Dirac delta distribution. The appli-
cation of these estimates to the numerical solution of elliptic problems with singularly supported
forcing terms allows us to provide sharp H1 and L2 error estimates for the corresponding regu-
447
EQUATIONS INVOLVING SINGULAR SOURCE TERMS (MS-39)
istic scenario, demonstrating that the method can robustly and efficiently handle the one-way
coupling between complex fluid microstructures and the elastic matrix.
Some remarks on a two phase problem in the Heisenberg group
Fausto Ferrari, fausto.ferrari@unibo.it
Dipartimento di Matematica dell’Università di Bologna,
Piazza di Porta S. Donato, 5, 40126 Bologna, Italia
We will discuss about the existence of an Alt-Caffarelli-Friedman monotonicity formula for a
two phase problem associated with a Bernoulli type functional in the Heisenberg group.
Finite element approximation of Stokes equations with non-smooth data
Lucia Gastaldi, lucia.gastaldi@unibs.it
University of Brescia, Italy
The lid driven cavity flow is a well known model used frequently to test the finite element
approximation of the Stokes problem. Actually, this model does not meet the regularity re-
quirements for the boundary datum which is only in L2, so that it cannot be the trace of the
velocity which belongs to H1. In this talk, we analyze the finite element approximation of the
Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution we
first approximate the boundary datum by a smooth one and then apply a standard finite element
method to the regularized problem. We prove almost optimal order error estimates for two reg-
ularization procedures in the case of general data in fractional order Sobolev spaces, and for the
Lagrange interpolation (with appropriate modifications at the discontinuities) for piece- wise
smooth data. Our results apply in particular to the classic lid-driven cavity problem improving
the existing error estimates.
Finally, we introduce and analyze an a posteriori error estimator. We prove its reliability and
efficiency, and show some numerical examples which suggest that optimal order of convergence
is obtained by an adaptive procedure based on our estimator.
The results reported in this talk have been obtained, in collaboration with Ricardo Duràn
and Ariel Lombardi.
A priori error estimates of regularized elliptic problems
Wenyu Lei, wenyu.lei@sissa.it
SISSA-International School for Advanced Studies, Italy
Coauthor: Luca Heltai
Approximations of the Dirac delta distribution are commonly used to create sequences of
smooth functions approximating nonsmooth (generalized) functions, via convolution. We show
a-priori rates of convergence of this approximation process in standard Sobolev norms, with
minimal regularity assumptions on the approximation of the Dirac delta distribution. The appli-
cation of these estimates to the numerical solution of elliptic problems with singularly supported
forcing terms allows us to provide sharp H1 and L2 error estimates for the corresponding regu-
447
