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UID:4687@test.i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20170411T110000
DTEND;TZID=Europe/Paris:20170411T120000
DTSTAMP:20170327T090000Z
URL:https://test.i2m.univ-amu.fr/events/discontinuous-galerkin-finite-elem
ent-approximation-of-hamilton-jacobi-bellman-equations-with-cordes-coeffic
ients/
SUMMARY: Discontinuous Galerkin finite element approximation of Hamilton–
Jacobi–Bellman equations with Cordes coefficients -
DESCRIPTION:Elliptic and parabolic Hamilton—Jacobi—Bellman equations ar
e an important class of second-order fully nonlinear PDEs\, with applicati
ons to stochastic optimal control problems in engineering and finance. It
is known that existing finite difference and finite element methods based
on discrete maximum principles can often be guaranteed to converge to the
viscosity solution in the small mesh limit. However\, the requirement for
a discrete maximum principle imposes severe restrictions on the choice of
mesh\, the order of convergence and the size of the stencil for strongly a
nisotropic problems\, which can limit the computational efficiency on prac
tical mesh sizes. This motivates the search for more flexible high-order m
ethods that achieve the key properties of consistency\, stability and conv
ergence without discrete maximum principles. In this talk\, we will presen
t how these challenges are overcome in the context of fully nonlinear seco
nd-order elliptic and parabolic Hamilton–Jacobi–Bellman equations that
satisfy a structural property named the Cordes condition. We construct an
hp-version discontinuous Galerkin finite element method which is motivate
d by the PDE theory of the problem. Both the continuous and discrete analy
ses are based on a variational strong monotonicity argument which establis
hes well-posedness of the fully nonlinear HJB PDE in the class of strong s
olutions\, and of the discrete numerical scheme. We show that the numerica
l method is consistent and stable\, with error bounds that are optimal in
the mesh size\, and suboptimal in the polynomial degrees\, as standard for
hp-version DGFEM. For parabolic problems\, the discretisation is extended
by a high-order DG time-stepping method\, permitting high-order approxima
tion in both time and space. Numerical experiments demonstrate the accurac
y and efficiency of the numerical scheme on problems featuring strongly an
isotropic diffusion coefficients and singular solutions\, including expone
ntial convergence rates under hp-refinement. This is a joint work with Pro
f. Endre Süli\, University of Oxford.https://who.rocq.inria.fr/Iain.Smear
s/
CATEGORIES:Séminaire Analyse Appliquée (AA)
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