Discrete maximum principles and a posteriori error estimation for finite element approximations to transport equations
January 14, 2009 3pm, 104 SEC
Abstract
Continuous and discrete maximum principles are formulated for
scalar transport equations. Relevant a priori bounds are
enforced within the framework of algebraic flux correction.
A family of nonlinear high-resolution finite element schemes
is presented and combined with adaptive mesh refinement.
The derivation of computable error indicators is discussed in
some detail. Gradient recovery techniques are revisited and
integrated into a new goal-oriented a posteriori error
estimate. The methodology to be presented builds on the
duality argument and features a node-based approach to
localization of errors in the quantity of interest, as
represented by a linear target functional. A possible
violation of Galerkin orthogonality is taken into account in
a simple and natural way. The use of an averaged gradient
makes it possible to obtain a nonoscillatory distribution of
weighted residuals without introducing jump terms. The
weights are determined using the difference between the
linear and quadratic finite element interpolants of the dual solution.
The benefits of mesh adaptation are illustrated by numerical
results for scalar conservation laws and hyperbolic systems.
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