Space-Time Spectral Element Methods for One-Dimensional Nonlinear Advection-Diffusion Problems

1995 ◽  
Vol 119 (1) ◽  
pp. 62-74 ◽  
Author(s):  
Pinhas Bar-Yoseph ◽  
Eduard Moses ◽  
Uzi Zrahia ◽  
Alexander L. Yarin
Keyword(s):  
Spectral Element ◽  
Space Time ◽  
Spectral Element Methods ◽  
One Dimensional ◽  
Advection Diffusion ◽  
Nonlinear Advection ◽  
Diffusion Problems

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

2013 ◽  
Vol 11 (8) ◽  
Author(s):  
Pavel Bochev ◽  
Kara Peterson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
One Dimensional ◽  
Stabilized Finite Element ◽  
Edge Elements ◽  
Stabilized Finite Element Method ◽  
Stabilized Methods ◽  
Advection Diffusion ◽  
Diffusion Problems ◽  
Element Method

AbstractWe 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.

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