Automatically adaptive, stabilized finite element method via residual minimization for heterogeneous, anisotropic advection–diffusion–reaction problems

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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605 3D Heat Advection-Diffusion Analysis Using Stabilized Finite Element Method

The Proceedings of Conference of Kyushu Branch ◽

10.1299/jsmekyushu.2001.54.159 ◽

2001 ◽

Vol 2001.54 (0) ◽

pp. 159-160

Author(s):

Hiroshi KANAYAMA ◽

Daisuke TAGAMI ◽

Leong Yew SIAH ◽

Takahiro ARAKI

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Heat Advection ◽

Stabilized Finite Element ◽

Stabilized Finite Element Method ◽

Advection Diffusion ◽

Diffusion Analysis ◽

Element Method

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

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-02-2018-0077 ◽

2018 ◽

Vol 28 (11) ◽

pp. 2688-2715 ◽

Cited By ~ 2

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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