A new stabilized finite element method for reaction-diffusion problems: The source-stabilized Petrov-Galerkin method

2008 ◽  
Vol 75 (13) ◽  
pp. 1607-1630 ◽  
Author(s):  
F. Ilinca ◽  
J.-F. Hétu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Galerkin Method ◽  
Reaction Diffusion ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Diffusion Problems ◽  
Element Method

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.

Finite element modeling of nonlinear reaction–diffusion–advection systems of equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2688-2715 ◽  
Author(s):  
Sanjay Komala Sheshachala ◽  
Ramon Codina
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Diffusion Reaction ◽  
Content Type ◽  
Numerical Examples ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Nonlinear Reaction ◽  
Element Method

Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.

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