28 research outputs found
Residual estimates for post-processors in elliptic problems
In this work we examine a posteriori error control for post-processed
approximations to elliptic boundary value problems. We introduce a class of
post-processing operator that `tweaks' a wide variety of existing
post-processing techniques to enable efficient and reliable a posteriori bounds
to be proven. This ultimately results in optimal error control for all manner
of reconstruction operators, including those that superconverge. We showcase
our results by applying them to two classes of very popular reconstruction
operators, the Smoothness-Increasing Accuracy-Enhancing filter and
Superconvergent Patch Recovery. Extensive numerical tests are conducted that
confirm our analytic findings.Comment: 25 pages, 17 figure
Recovery methods for evolution and nonlinear problems
Functions in finite dimensional spaces are, in general, not smooth enough to be differentiable in the classical sense and ārecoveredā versions of their first and second derivatives must be sought for certain applications. In this work we make use of recovered derivatives for applications in finite element schemes for two different purposes. We thus split this Thesis into two distinct parts.
In the first part we derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error for fully discrete schemes of the linear heat equation. To our knowledge this is the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique introduced as an aposteriori analog to the elliptic (Ritz) projection.
Our theoretical results are backed up with extensive numerical experimentation aimed at (1) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (2) deriving an adaptive method based on our estimators.
An extra novelty is an implementation of a coarsening error āpreindicatorā, with a complete implementation guide in ALBERTA (versions 1.0ā2.0).
In the second part of this Thesis we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galƫrkin type using conforming finite elements and applied directly to the nonvariational(or nondivergence) form of a second order linear elliptic problem. The key tools are an
appropriate concept of the āfinite element Hessianā based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method
is illustrated with computational experiments on linear PDEs in nonvariational form.
We then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newtonās method resulting in a sequence of nonvariational elliptic problems which are then approximated with the nonvariational finite element method. This method is applicable to general fully nonlinear PDEs who admit a unique solution without constraint.
We also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the MongeāAmpĆØre equation
based on using āfinite element convexityā as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation
Recommended from our members
Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
Gradient recovery in adaptive finite element methods for parabolic problems
We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ)
estimators to control the spatial error, for fully discrete schemes for the
linear heat equation. This appears to be the first completely rigorous
derivation of ZZ estimators for fully discrete schemes for evolution problems,
without any restrictive assumption on the timestep size. An essential tool for
the analysis is the elliptic reconstruction technique.
Our theoretical results are backed with extensive numerical experimentation
aimed at (a) testing the practical sharpness and asymptotic behaviour of the
error estimator against the error, and (b) deriving an adaptive method based on
our estimators. An extra novelty provided is an implementation of a coarsening
error "preindicator", with a complete implementation guide in ALBERTA.Comment: 6 figures, 1 sketch, appendix with pseudocod
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Advanced Computational Engineering
The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications
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Schnelle Lƶser fĆ¼r partielle Differentialgleichungen
The workshop Schnelle LoĢser fuĢr partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds