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A parameter-free stabilized finite element method for scalar advection-diffusion problems

Pavel Bochev; Kara Peterson

Open Mathematics (2013)

  • Volume: 11, Issue: 8, page 1458-1477
  • ISSN: 2391-5455

Abstract

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We formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming edge elements to expand the resulting edge fluxes into an exponentially fitted flux field inside each element. Substitution of the nodal flux by this new flux completes the formulation of the method. Utilization of edge elements to define the numerical flux and the lack of stabilization parameters differentiate our approach from other stabilized methods. Numerical studies with representative advection-diffusion test problems confirm the excellent stability and robustness of the new method. In particular, the results show minimal overshoots and undershoots for both internal and boundary layers on uniform and non-uniform grids.

How to cite

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Pavel Bochev, and Kara Peterson. "A parameter-free stabilized finite element method for scalar advection-diffusion problems." Open Mathematics 11.8 (2013): 1458-1477. <http://eudml.org/doc/269431>.

@article{PavelBochev2013,
abstract = {We formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming edge elements to expand the resulting edge fluxes into an exponentially fitted flux field inside each element. Substitution of the nodal flux by this new flux completes the formulation of the method. Utilization of edge elements to define the numerical flux and the lack of stabilization parameters differentiate our approach from other stabilized methods. Numerical studies with representative advection-diffusion test problems confirm the excellent stability and robustness of the new method. In particular, the results show minimal overshoots and undershoots for both internal and boundary layers on uniform and non-uniform grids.},
author = {Pavel Bochev, Kara Peterson},
journal = {Open Mathematics},
keywords = {Advection-diffusion; Upwind stabilization; Exponentially fitted flux; Finite element method; Edge elements; advection-diffusion; upwind stabilization; exponentially fitted flux; finite element method; edge elements; numerical examples; boundary layers},
language = {eng},
number = {8},
pages = {1458-1477},
title = {A parameter-free stabilized finite element method for scalar advection-diffusion problems},
url = {http://eudml.org/doc/269431},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Pavel Bochev
AU - Kara Peterson
TI - A parameter-free stabilized finite element method for scalar advection-diffusion problems
JO - Open Mathematics
PY - 2013
VL - 11
IS - 8
SP - 1458
EP - 1477
AB - We formulate and study numerically a new, parameter-free stabilized finite element method for advection-diffusion problems. Using properties of compatible finite element spaces we establish connection between nodal diffusive fluxes and one-dimensional diffusion equations on the edges of the mesh. To define the stabilized method we extend this relationship to the advection-diffusion case by solving simplified one-dimensional versions of the governing equations on the edges. Then we use H(curl)-conforming edge elements to expand the resulting edge fluxes into an exponentially fitted flux field inside each element. Substitution of the nodal flux by this new flux completes the formulation of the method. Utilization of edge elements to define the numerical flux and the lack of stabilization parameters differentiate our approach from other stabilized methods. Numerical studies with representative advection-diffusion test problems confirm the excellent stability and robustness of the new method. In particular, the results show minimal overshoots and undershoots for both internal and boundary layers on uniform and non-uniform grids.
LA - eng
KW - Advection-diffusion; Upwind stabilization; Exponentially fitted flux; Finite element method; Edge elements; advection-diffusion; upwind stabilization; exponentially fitted flux; finite element method; edge elements; numerical examples; boundary layers
UR - http://eudml.org/doc/269431
ER -

References

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