[Dottorcomp] Seminari di Matematica Applicata. Mercoledì 17 maggio. Sônia Gomes e Marcus Sarkis

[Lorenzo Tamellini]

Buongiorno a tutt*,

si avvisa che mercoledì 17 maggio presso la sala conferenze del CNR-IMATI
di Pavia si terranno due seminari:
*ore 15:* Sônia Gomes*:* An efficient construction of divergence-free
spaces in the context of exact finite element de Rham sequences
*ore 16:* Marcus Sarkis: Fast solvers for high-order mixed FEMs

Cordiali saluti,
Lorenzo Tamellini

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*Speaker:* Sônia M. Gomes (State University of Campinas, Brazil)

*Title:* An efficient construction of divergence-free spaces in the context
of exact finite element de Rham sequences
*Abstract.* Exact finite element de Rham complexes relate
conforming subspaces in H1, H(curl), H(div), and L2 and in a simple way by
means of differential operators (gradient, curl, and divergence). The
characteristics of such strong couplings are crucial for the design of
stable and conservative discretization of mixed formulations for a variety
of multiphysics systems. This work explores these aspects for the
construction of divergence-free vector shape functions in a  robust fashion
allowing stable and faster simulations of mixed formulations of
incompressible porous media flows. The resolution of the associated
saddle-point problem can be reduced to two consecutive computation steps:
one for the  flux and the next one for the pressure (for cases where it is
required). The formulation for the flux is immediately equivalent to a
standard Galerkin variational problem with positive definite linear system
and reduced number of degrees of freedom. Pressure is obtained by a
post-processing algorithm. This reduced divergence-free model can also be
extended to applications in the context of element-wise divergence-constant
fluxes. The resulting schemes are verified by means of numerical tests with
known smooth solutions and applied to a benchmark problem to confirm the
expected theoretical and computational performance results. This is a joint
work with Philippe R. B. Devloo, Jeferson W. D. Fernandes, Francisco
Orlandini, and Nathan Shauer.

==========================================

*Speaker:* Marcus Sarkis (Department of Mathematical Sciences, Worcester
Polytechnic Institute, USA)
*Title:* Fast solvers for high-order mixed FEMs

*Abstract:* In this work, we propose iterative schemes to solve saddle
point problems arising
in the context of mixed finite elements. We focus the talk on the RTk
Raviart
Thomas of order k discretization for the transmission problem. The strategy
starts by applying the technique developed by Devloo, Fernandes, Gomes,
Orlandini
and Shauer 2022-CMAME where RTk divergence free basis functions
for the whole region can be constructed locally in each element based on
edges
and interior degrees of fredom is each element. This freediv subpace is
augmented with RT0 in order to contains all the divfree subspace of the
RTk.
We then develop three preconditioners which map to divfree subpaces and so
the
conjugated gradient method can be applied. We present theoretical results
and
numerical experiments.
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