A Petrov–Galerkin finite element method for the fractional advection–diffusion equation

Lian, Yuan; Ying, Yuping; Tang, Shaoqiang; Lin, Stephen; Wagner, Gregory J.; Liu, Wing Kam

doi:10.1016/j.cma.2016.06.013

Cited by 34 publications

(10 citation statements)

References 12 publications

(10 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One way to relieve such a problem is by adding an additional artificial diffusion (balancing diffusion) term into the system. This yields the so-called "stabilization methods" (Codina 1998;Franca et al 1992;Lian et al 2016) which have been developed for Galerkin methods over the past decades. In this paper, one of the widely used stabilization methods, namely the finite increment calculus (FIC) method, which introduces a nonlinear balancing artificial diffusion term in (39) is adopted.…”

Section: B1 Finite Element Approximation Of Nonlinear Poisson Equatimentioning

confidence: 99%

See 1 more Smart Citation

Modeling of Symbiotic Bacterial Biofilm Growth with an Example of the Streptococcus–Veillonella sp. System

et al. 2021

View full text Add to dashboard Cite

We present a multi-dimensional continuum mathematical model for modeling the growth of a symbiotic biofilm system. We take a dual-species namely, the Streptococcus–Veillonella sp. biofilm system as an example for numerical investigations. The presented model describes both the cooperation and competition between these species of bacteria. The coupled partial differential equations are solved by using an integrative finite element numerical strategy. Numerical examples are carried out for studying the evolution and distribution of the bio-components. The results demonstrate that the presented model is capable of describing the symbiotic behavior of the biofilm system. However, homogenized numerical solutions are observed locally. To study the homogenization behavior of the model, numerical investigations regarding on how random initial biomass distribution influences the homogenization process are carried out. We found that a smaller correlation length of the initial biomass distribution leads to faster homogenization of the solution globally, however, shows more fluctuated biomass profiles along the biofilm thickness direction. More realistic scenarios with bacteria in patches are also investigated numerically in this study.

show abstract

Section: B1 Finite Element Approximation Of Nonlinear Poisson Equatimentioning

confidence: 99%

“… 1992 ; Lian et al. 2016 ) which have been developed for Galerkin methods over the past decades. In this paper, one of the widely used stabilization methods, namely the finite increment calculus (FIC) method, which introduces a nonlinear balancing artificial diffusion term in ( 39 ) is adopted.…”

Section: Dimensioless Form Of Governing Equationsmentioning

confidence: 99%

Modeling of Symbiotic Bacterial Biofilm Growth with an Example of the Streptococcus–Veillonella sp. System

et al. 2021

View full text Add to dashboard Cite

show abstract

“…A fast finite difference approach is proposed in 35 to solve a class of initial‐boundary problems of time‐space fractional convection‐diffusion equations. Authors of 36 presented an in‐depth numerical analysis of spatial fractional advection‐diffusion equations utilizing the finite element method. An H 1 ‐Galerkin mixed finite element method for solving time fractional reaction‐diffusion equation is presented in 37 .…”

Section: Introductionmentioning

confidence: 99%

The Crank‐Nicolson/interpolating stabilized element‐free Galerkin method to investigate the fractional Galilei invariant advection‐diffusion equation

Abbaszadeh

Dehghan

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

Recently, finding a stable and convergent numerical procedure to simulate the fractional partial differential equations (PDEs) is one of the interesting topics. Meanwhile, the fractional advection‐diffusion equation is a challenge model numerically and analytically. This paper develops a new meshless numerical procedure to simulate the fractional Galilei invariant advection‐diffusion equation. The fractional derivative is the Riemann‐Liouville fractional derivative sense. At the first stage, a difference scheme with the second‐order accuracy has been employed to get a semi‐discrete plan. After this procedure, the unconditional stability has been investigated, analytically. At the second stage, a meshless weak form based upon the interpolating stabilized element‐free Galerkin (ISEFG) method has been used to achieve a full‐discrete scheme. As for the full‐discrete scheme, the order of convergence is scriptOfalse(τ2+rm+1false). Two examples are studied, and simulation results are reported to verify the theoretical results.

show abstract

“…One way to relieve such a problem is by adding an additional artificial diffusion (balancing diffusion) term into the system. This yields the so-called “stabilization methods” (Codina, 1998; Franca et al, 1992; Lian et al, 2016) which have been developed for Galerkin methods over the past decades. In this paper, one of the widely used stabilization methods, namely the finite increment calculus (FIC) method, which introduces a nonlinear balancing artificial diffusion term in (38) is adopted.…”

mentioning

confidence: 99%

Modeling of symbiotic bacterial biofilm growth with an example of theStreptococcus-Veillonellasp. system

Feng

Neuweiler

Nogueira

et al. 2020

Preprint

View full text Add to dashboard Cite

We present a multi-dimensional continuum mathematical model for modeling the growth of a symbiotic biofilm system. We take a dual-species namely, the Streptococcus - Veillonella sp. biofilm system as an example for numerical investigations. The presented model describes both the cooperation and competition between these species of bacteria. The coupled partial differential equations are solved by using an integrative finite element numerical strategy. Numerical examples are carried out for studying the evolution and distribution of the bio-components. The results demonstrate that the presented model is capable of describing the symbiotic behavior of the biofilm system. However, homogenized numerical solutions are observed locally. To study the homogenization behavior of the model, numerical investigations regarding on how random initial biomass distribution influences the homogenization process are carried out. We found that a smaller correlation length of the initial biomass distribution leads to faster homogenization of the solution globally, however, shows more fluctuated biomass profiles along the biofilm thickness direction. More realistic scenarios with bacteria in patches are also investigated numerically in this study.

show abstract

A Petrov–Galerkin finite element method for the fractional advection–diffusion equation

Cited by 34 publications

References 12 publications

Modeling of Symbiotic Bacterial Biofilm Growth with an Example of the Streptococcus–Veillonella sp. System

Modeling of Symbiotic Bacterial Biofilm Growth with an Example of the Streptococcus–Veillonella sp. System

The Crank‐Nicolson/interpolating stabilized element‐free Galerkin method to investigate the fractional Galilei invariant advection‐diffusion equation

Modeling of symbiotic bacterial biofilm growth with an example of theStreptococcus-Veillonellasp. system

Contact Info

Product

Resources

About