Murtazo Nazarov: A maximum-principle preserving continuous finite element method for scalar conservation equations

Abstract: We will give a brief overview of the stabilized finite element methods and in particular the standard artificial viscosity based on the operator $-\DIV(\nu_h\GRAD)$, where $\nu_h$ is a scalar-valued and proportional to some wave-speed and some mesh-size. We will discuss about some key shortcomings of this formulation. For example what is the local wave-speed? What is proportionality constants? And what is the local mesh-size on anisotropic meshes? We 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 linear, for instance) and the flux is $\calC1$, 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.