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UID:news-1480@dmi.unibas.ch
DTSTAMP:20230418T090618
DTSTART;TZID=Europe/Zurich:20230421T110000
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SUMMARY:Seminar in Numerical Analysis: Omar Lakkis (University of Sussex)
LOCATION:
DESCRIPTION:Least-squares finite element recovery-based methods provide a s
imple and practical way to approximate linear elliptic PDEs in
nondivergence form where standard variational approach either f
ails or requires technically complex modifications.\r\nThis ide
a allows the creation of efficient solvers for fully nonlinear
elliptic equations, the linearization of which leaves us with a
n equation in nondivergence form. An important class of fully n
onlinear elliptic PDEs can be written in Hamilton--Jacobi--Bell
man (Dynamic Programming) form, i.e., as the supremum of a coll
ection of linear operators acting on the unkown.\r\nThe least-s
quares FEM approach, a variant of the nonvariational finite ele
ment method, is based on gradient or Hessian recovery and allow
s the use of FEMs of arbitrary degree. The price to pay for usi
ng higher order FEMs is the loss of discrete-level monotonicity
(maximum principle), which is valid for the PDE and crucial in
proving the convergence of many degree one FEM and finite diff
erence schemes.\r\nSuitable functional spaces and penalties in
the least-squares's cost functional must be carefully crafted i
n order to ensure stability and convergence of the scheme with
a good approximation of the gradient (or Hessian) under the Cor
des condition on the family of linear operators being optimized
.\r\nFurthermore, the nonlinear operator which is not necessari
ly everywhere differentiable, must be linearized in appropriate
functional spaces using semismooth Newton or Howard's policy i
teration method. A crucial contribution of our work, is the pro
of of convergence of the semismooth Newton method at the contin
uum level, i.e., on infinite dimesional functionals spaces. Thi
s allows an easy use of our non-monotone schemes which provides
convergence rates as well as a posteriori error estimates.\r\n
\r\nFor further information about the seminar, please visit t
his webpage [https://dmi.unibas.ch/de/forschung/mathematik/semi
nar-in-numerical-analysis/].
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