Spectral elements for transport dominated equations

Auteurs

• Daniele Funaro

Paramètres

• Mathématiques

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In recent years, there has been increasing interest in developing numerical techniques for approximating differential model problems with multiscale solutions. These problems often feature functions that behave smoothly except in specific regions with sudden, sharp variations, such as internal or boundary layers. When the discretization process lacks sufficient degrees of freedom to finely resolve these layers, stabilization procedures are necessary to prevent oscillatory effects without introducing excessive artificial viscosity. In finite element analysis, methods like streamline diffusion, Galerkin least-squares, and bubble function approaches effectively address transport equations of elliptic type with small diffusive terms, known in fluid dynamics as advection-diffusion equations. This work aims to guide readers in constructing a computational code based on the spectral collocation method using algebraic polynomials. It focuses on approximating elliptic type boundary-value partial differential equations in 2-D, particularly transport-diffusion equations where second-order diffusive terms are significantly overshadowed by first-order advective terms. The applications discussed will highlight cases where nonlinear systems of partial differential equations can be simplified to a sequence of transport-diffusion equations.

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• Année de publication: 1997