Convergence of a mixed method for a semi-stationary compressible Stokes system

TL;DR

This paper introduces and proves the convergence of a finite element method combining mixed Nedelec spaces and discontinuous Galerkin schemes for semi-stationary compressible Stokes systems with Navier-slip boundary conditions.

Contribution

It presents a novel combined finite element and discontinuous Galerkin approach for compressible Stokes systems and provides a rigorous convergence proof to weak solutions.

Findings

The method converges to a weak solution of the system.

Strong spatial compactness of the velocity field is established.

Discontinuous Galerkin approximations are shown to converge strongly.

Abstract

We propose and analyze a finite element method for a semi-stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nedelec spaces of the first kind. The continuity equation is approximated by a standard piecewise constant upwind discontinuous Galerkin scheme. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only div or curl conforming. (ii) To prove that the discontinuous Galerkin approximations converge strongly, which is required in view of the nonlinear pressure function. Tools involved in the analysis include a higher integrability estimate for the discontinuous Galerkin approximations, a discrete equation for the effective viscous flux, and various renormalized formulations of the discontinuous Galerkin scheme.