1,165 research outputs found
POD model order reduction with space-adapted snapshots for incompressible flows
We consider model order reduction based on proper orthogonal decomposition
(POD) for unsteady incompressible Navier-Stokes problems, assuming that the
snapshots are given by spatially adapted finite element solutions. We propose
two approaches of deriving stable POD-Galerkin reduced-order models for this
context. In the first approach, the pressure term and the continuity equation
are eliminated by imposing a weak incompressibility constraint with respect to
a pressure reference space. In the second approach, we derive an inf-sup stable
velocity-pressure reduced-order model by enriching the velocity reduced space
with supremizers computed on a velocity reference space. For problems with
inhomogeneous Dirichlet conditions, we show how suitable lifting functions can
be obtained from standard adaptive finite element computations. We provide a
numerical comparison of the considered methods for a regularized lid-driven
cavity problem
Trefftz Difference Schemes on Irregular Stencils
The recently developed Flexible Local Approximation MEthod (FLAME) produces
accurate difference schemes by replacing the usual Taylor expansion with
Trefftz functions -- local solutions of the underlying differential equation.
This paper advances and casts in a general form a significant modification of
FLAME proposed recently by Pinheiro & Webb: a least-squares fit instead of the
exact match of the approximate solution at the stencil nodes. As a consequence
of that, FLAME schemes can now be generated on irregular stencils with the
number of nodes substantially greater than the number of approximating
functions. The accuracy of the method is preserved but its robustness is
improved. For demonstration, the paper presents a number of numerical examples
in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering
of electromagnetic (acoustic) waves, and wave propagation in a photonic
crystal. The examples explore the role of the grid and stencil size, of the
number of approximating functions, and of the irregularity of the stencils.Comment: 28 pages, 12 figures; to be published in J Comp Phy
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