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CATEGORIES:Applied Mathematics Seminar Series
SUMMARY:Galerkin methods for (fully) nonlinear elliptic eq
uations - Omar Lakkis (Sussex)
DTSTART:20140324T140000Z
DTEND:20140324T150000Z
UID:TALK1316AT
URL:/talk/index/1316
DESCRIPTION:joint work with Tristan Pryer (Reading)\n\nExcept
some notable cases\, most nonlinear elliptic equat
ions are not variational. Traditional finite elem
ent methods (FEMs)\, as a subclass of Galerkin met
hods\, are variational and thus not suitable for t
he approximation of solutions to fully nonlinear e
lliptic equations. Finite difference methods are m
ore natural\, especially where maximum principle p
lays a role. However\, finite differences have som
e serious drawbacks\, like the need to choose larg
e stencils and the difficulties with handling geom
etries.\n\nThis has lead us to develop a novel num
erical technology based on standard finite element
spaces\, but eschewing the variational approach.
We call this method the NVFEM (Nonvariational FEM
). The key is to make a proper use of the distrib
utional Hessian of a FE function by introducing a
“recovered” approximation to it. The Newton-Raphs
on step for a fully nonlinear equation can thus tu
rned into an unbalanced mixed problem\, where the
dual variable approximates the Hessian of the solu
tion. The resulting method is surprisingly simple
yet powerful. Although the analysis of the full m
ethod in general has still to be fully understood\
, the availability of a posteriori error estimates
allows adaptive methods to be used in the approxi
mation of singular (continuous but not differentia
ble) viscosity solutions.\n\nIn this talk\, I will
first review a bit of fully nonlinear elliptic PD
Es. I will then present what I’ve learned about t
he history and the state of the art of numerical m
ethods for fully nonlinear equations and finally I
will place our work within the context of other m
ethods. I will close with some numerical examples
of solvers for the Monge-Ampère\, Pucci and infin
ite-harmonic equation\, including adaptive methods
for singular solutions.
LOCATION:Watson LRA
CONTACT:Alexandra Tzella
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