Algebraic stabilizations of convection-diffusion-reaction equations

• Date: 19. 12. 2024
• Time: 14:00 - 15:30
• Address:
  Sokolovská 83, Praha
• Room: K3
• Speaker: Petr Knobloch

Convection-diffusion-reaction equations appear in many mathematical models of physical, technical or biological processes. Often, the diffusion is very small in comparison with the convection or reaction, which causes that the solutions comprise layers. It is well known that standard numerical methods then provide approximate solutions polluted by spurious oscillations unless the underlying mesh resolves the layers. During the last five decades, many various stabilized methods for convection-diffusion-reaction equations have been developed and it turns out that, due to the multiscale character of the problem, accurate approximate solutions can be obtained only if nonlinear approaches are used. However, in general, some spurious oscillations are often still present, which may be not acceptable in many applications. The only possibility to avoid violations of global bounds it to use discretizations satisfying the discrete maximum principle (DMP). An important class of such methods are algebraically stabilized schemes. These methods have been intensively developed in recent years and we will formulate an abstract framework that enables the analysis of algebraically stabilized discretizations for steady-state problems in a unified way. We will present general results on local and global DMPs, existence of solutions, and error estimation. Then, various examples of algebraic stabilizations fitting into the abstract framework will be given and applied to discretizations of steady-state convection-diffusion-reaction equations. The properties of these particular algebraic stabilizations will be discussed both theoretically and by means of numerical examples.