Wednesday, 06 November, 2013
SPEAKER: Prof. Jean-Luc Guermond, Texas A&M
TITLE: Revisiting first-order viscosity for continuous finite element approximation of nonlinear conservation equations
ABSTRACT: I will revisit the standard standard artificial viscosity based on the operator $-DIV(nu_hGRAD)$, where $nu_h$ is scalar-valued and proportional to some wave-speed and some mesh-size. Some key shortcomings of this formulation will be identified: i.e. what is the local wave-speed? what is the proportionality constant? what is the local mesh-size on anisotropic meshes? I will then construct a first-order viscosity method for the explicit approximation of scalar conservation equations using continuous finite elements on arbitrary grids in any space dimension that does not require any a priori knowledge of quantities
like local wave-speed, proportionality constant, mesh-size. Provided the approximation setting satisfies a local convexity assumption (ie piecewise linears, for instance) and the flux is $calC^1$, the method is proved to satisfy the local maximum principal under a usual CFL condition. The method is independent of
the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions without any particular regularity assumption. Higher-order extensions of the method will be discussed as well.
CAM seminar schedule: http://www.math.utk.edu/~vasili/FTP/AM/CAMseminar.html