On the solvability of the nonlinear problems in an algebraically stabilized finite element method for evolutionary transport-dominated equations

John, Volker; Knobloch, Petr; Korsmeier, Paul

doi:10.1090/mcom/3576

Cited by 8 publications

(3 citation statements)

References 21 publications

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“…While all the aforementioned approaches attempt to stabilize the finite element method by adding additional terms to the Galerkin finite element discretization, flux-corrected transport (FCT) schemes have been developed [8,[37][38][39] as a different technique which is a nonlinear scheme working on the algebraic level by modifying the algebraic equation obtained from the Galerkin finite element method. Stabilization methods were further investigated and developed for timedependent DCR models, see for instance [12,21,33,34], to just mention a few. However, most of these finite element stabilization techniques deal with the cases where the convection terms are linear and their implementation to nonlinear convection terms still calls for further investigation.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Fuest

2021

Nonlinearity

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Systems of the type u t = ∇ ⋅ ( D 1 ( u ) ∇ u − S 1 ( u ) ∇ v ) + f 1 ( u , v ) , v t = ∇ ⋅ ( D 2 ( v ) ∇ v + S 2 ( v ) ∇ u ) + f 2 ( u , v ) ( ⋆ ) can be used to model pursuit-evasion relationships between predators and prey. Apart from local kinetics given by f 1 and f 2, the key components in this system are the taxis terms −∇ ⋅ (S 1(u)∇v) and +∇ ⋅ (S 2(v)∇u); that is, the species are not only assumed to move around randomly in space but are also able to partially direct their movement depending on the nearby presence of the other species. In the present article, we construct global weak solutions of (⋆) for certain prototypical nonlinear functions D i , S i and f i , i ∈ {1, 2}. To that end, we first make use of a fourth-order regularisation to obtain global solutions to approximate systems and then rely on an entropy-like identity associated with (⋆) for obtaining various a priori estimates.

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Section: Numerical Simulationsmentioning

confidence: 99%

Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Fuest

2021

Nonlinearity

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“…A comparison of the linear and the nonlinear schemes are present in [JS08] where it is shown that the linear FEM-FCT has a better ratio of accuracy and efficiency. The solvability of the nonlinear FEM-FCT schemes has been presented recently in [JKK21] and the existence and uniqueness of the solution are shown in [JK21] where the analysis from [BJK16] is extended. This paper presents the finite element error analysis of the FEM-FCT schemes with backward Euler as the time-stepping.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Flux Corrected Transport Schemes for Evolutionary Convection-Diffusion-Reaction Equations

Jha,

Ahmed

2021

Preprint

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“…In [59], the existence of a solution of (8.26)-(8.28) is proven for arbitrary time steps. The existence and uniqueness of a solution for sufficiently small time steps is shown in [61].…”

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confidence: 99%

Finite element methods respecting the discrete maximum principle for convection-diffusion equations

Barrenechea¹,

John²,

Knobloch³

2022

Preprint

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Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation.

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On the solvability of the nonlinear problems in an algebraically stabilized finite element method for evolutionary transport-dominated equations

Cited by 8 publications

References 21 publications

Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Analysis of Flux Corrected Transport Schemes for Evolutionary Convection-Diffusion-Reaction Equations

Finite element methods respecting the discrete maximum principle for convection-diffusion equations

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