88,356 research outputs found
Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm
We consider the approximation in the reaction-diffusion norm with continuous
finite elements and prove that the best error is equivalent to a sum of the
local best errors on pairs of elements. The equivalence constants do not depend
on the ratio of diffusion to reaction. As application, we derive local error
functionals that ensure robust performance of adaptive tree approximation in
the reaction-diffusion norm.Comment: 21 pages, 1 figur
(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods
We consider ultraweak variational formulations for (parametrized) linear
first order transport equations in time and/or space. Computationally feasible
pairs of optimally stable trial and test spaces are presented, starting with a
suitable test space and defining an optimal trial space by the application of
the adjoint operator. As a result, the inf-sup constant is one in the
continuous as well as in the discrete case and the computational realization is
therefore easy. In particular, regarding the latter, we avoid a stabilization
loop within the greedy algorithm when constructing reduced models within the
framework of reduced basis methods. Several numerical experiments demonstrate
the good performance of the new method
An optimal adaptive Fictitious Domain Method
We consider a Fictitious Domain formulation of an elliptic partial
differential equation and approximate the resulting saddle-point system using
an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the
approximation of an elliptic problems performed using adaptive finite element
methods. We prove that the overall method converges with the best possible rate
and illustrate numerically our theoretical findings
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