Trygve Karper

Kenneth Karlsen, Trygve Karper

We propose a mixed finite element method for the motion of a strongly viscous, ideal, and isentropic gas. At the boundary we impose a Navier-slip condition such that the velocity equation can be posed in mixed form with the vorticity as an auxiliary variable. In this formulation we design a finite element method, where the velocity and vorticity is approximated with the div- and curl- conforming Nedelec elements, respectively, of the first order and first kind. The mixed scheme is coupled to a standard piecewise constant upwind discontinuous Galerkin discretization of the continuity equation. For the time discretization, implicit Euler time stepping is used. Our main result is that the numerical solution converges to a weak solution as the discretization parameters go to zero. The convergence analysis is inspired by the continuous analysis of Feireisl and Lions for the compressible Navier-Stokes equations. Tools used in the analysis include an equation for the effective viscous flux and various renormalizations of the density scheme.

Helge Kristian Jenssen, Trygve Karper

We establish existence of global-in-time weak solutions to the one dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas (pressure $p=Kθ/τ$, internal energy $e=c_v θ$), when the viscosity $μ$ is constant and the heat conductivity $κ$ depends on the temperature $θ$ according to $κ(θ) = \bar κθ^β$, with $0\leqβ<{3/2}$. This choice of degenerate transport coefficients is motivated by the kinetic theory of gasses. Approximate solutions are generated by a semi-discrete finite element scheme. We first formulate sufficient conditions that guarantee convergence to a weak solution. The convergence proof relies on weak compactness and convexity, and it applies to the more general constitutive relations $μ(θ) = \bar μθ^α$, $κ(θ) = \bar κθ^β$, with $α\geq 0$, $0 \leq β< 2$ ($\bar μ, \bar κ$ constants). We then verify the sufficient conditions in the case $α=0$ and $0\leqβ<{3/2}$. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.