Pressure robust discretizations for Navier Stokes equations : divergence-free reconstruction for Taylor-Hood elements and high order hybrid discontinuous Galerkin methods

Abstract

This thesis focuses on a well-known issue of discretization techniques for solving the incompressible Navier Stokes equations. Due to a weak treatment of the incompressibility constraint there are different disadvantages that appear, which can have a major impact on the convergence and physical behaviour of the solutions. First we approximate the equations with a well-known pair of elements and introduce an operator that creates a reconstruction into a proper space to fix the mentioned problems. \newline Afterwards we use an H(div) conforming method that already handles the incompressibility constraint in a proper way. For a stable high order approximation an estimation for the saddlepoint structure of the Stokes equations is needed, known as the Ladyschenskaja-Babuska-Brezzi (LBB) condition. The independency of the estimation from the order of the polynomial degree is shown in this thesis. For that we introduce an H 2-stable extension that preserves polynomials. All operators and schemes are implemented based on the finite element library Netgen/NGSolve and tested with proper examples.