[Dottorcomp] Seminari di Matematica Applicata, Lunedi 20 Maggio 2024, Pablo Brubeck Martinez, Oxford

[Luca Franco Pavarino]

Seminari di Matematica Applicata, Dipartimento di Matematica "F. Casorati"
e Istituto del CNR IMATI "E. Magenes" di Pavia.

Lunedì 20 Maggio 2024, alle ore 16.00 precise, presso l'aula Beltrami del
dipartimento di Matematica,

Pablo Brubeck Martinez
del Mathematical Institute, University of Oxford,
https://www.maths.ox.ac.uk/people/pablo.brubeckmartinez

terrà un seminario dal titolo:

Fast solvers for high-order FEM on simplices via sparsity-promoting bases

Abstract.
We present new high-order finite elements discretizing the L2 de Rham
complex on triangular and tetrahedral meshes. The finite elements
discretize the same spaces as usual, but with different basis functions.
They allow for fast linear solvers based on static condensation and space
decomposition methods. The new elements build upon the definition of
degrees of freedom for interpolation given by Demkowicz et al. [1], and
consist of integral moments on a symmetric reference simplex with respect
to a numerically computed polynomial basis that is orthogonal in both the
L2- and H(d)-inner products (d = grad, curl, or div). On the reference
symmetric simplex, the resulting stiffness matrix has diagonal interior
block, and does not couple together the interior and interface degrees of
freedom. Thus, on the reference simplex, the Schur complement resulting
from elimination of interior degrees of freedom is simply the interface
block itself.

This sparsity is not preserved on arbitrary cells mapped from the reference
cell. Nevertheless, the interior-interface coupling is weak because it is
only induced by the geometric transformation. We devise a preconditioning
strategy by neglecting the interior-interface coupling.  We precondition
the interface Schur complement with the interface block, and simply apply
point-Jacobi to precondition the interior block. We further precondition
the interface block by applying a space decomposition method with small
subdomains constructed around vertices, edges, and faces. This allows us to
solve the canonical Riesz maps in H(grad), H(curl), and H(div), at very
high order.  We empirically demonstrate iteration counts that are robust
with respect to the polynomial degree.

References
[1] L. Demkowicz, P. Monk, L. Vardapetyan, and W. Rachowicz. De Rham
diagram for hp finite element spaces. Comput. Math. Appl., 39(7-8):29--38,
2000.

Il seminario verrà anche trasmesso su zoom al link:
https://us02web.zoom.us/j/84168000397?pwd=OEo3TXZjS1pDQlFOa29OSkpYcjFLdz09
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Tutti gli interessati sono invitati a partecipare.
Cordiali saluti,
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