Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10489225" target="_blank" >RIV/00216208:11320/24:10489225 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=LkKFRd9jpb" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=LkKFRd9jpb</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1137/22M1488934" target="_blank" >10.1137/22M1488934</a>

Result language

angličtina

Original language name

Finite Element Methods Respecting the Discrete Maximum Principle for Convection-Diffusion Equations

Original language description

Convection -diffusion -reaction equations model the conservation of scalar quantities. From the analytic point of view, solutions 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 the 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 convectiondominated 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 the 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 both 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. Similarly, methods based on algebraic stabilization, both nonlinear and linear, are currently 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 scenario.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10102 - Applied mathematics

Project

<a href="/en/project/GA22-01591S" target="_blank" >GA22-01591S: Mathematical theory and numerical analysis for equations of viscous newtonian compressible fluids</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

SIAM Review

ISSN

0036-1445

e-ISSN

1095-7200

Volume of the periodical

66

Issue of the periodical within the volume

1

Country of publishing house

US - UNITED STATES

Number of pages

86

Pages from-to

3-88

UT code for WoS article

001222180700006

EID of the result in the Scopus database

2-s2.0-85191833722