Discontinuous Galerkin finite element methods for (non)conservative partial differential equations

Abstract

The first research topic in this thesis is the development of discontinuous Galerkin (DG) finite element methods for partial differential equations containing nonconservative products, which are present in many two-phase flow models. For this, we combine the theory of Dal Maso, LeFloch and Murat, in which a definition is given for nonconservative products even where the solution field is discontinuous. This theory also provides the mathematical foundation for a new DG finite element method. For this new DG method, we show standard (p+1)-order convergence results using p-th order basis-functions for test-cases of which we know the exact solution. We also show its ability to deal with more complex test cases. Finally, we apply the method to a depth-averaged two-phase flow model of which the numerical results are qualitatively validated against results obtained from a laboratory experiment. The second topic of this thesis is multigrid. The use of multigrid is of great importance to obtain efficient solvers for fully 3D two-phase flow models. As an initial step to improve the efficiency of solving space-time DG discretizations, we have developed, analyzed and tested optimized multigrid methods using explicit Runge-Kutta type smoothers for the 2D advection-diffusion equation. Many physical models describing fluid motion contain second (and higher) order derivatives. Obtaining a DG discretization for these higher order derivatives is non-trivial and many different DG methods exist to deal with these terms. As final topic of this thesis we introduce an alternative derivation of DG methods based on Borel measures. This alternative derivation gives a consistent treatment of derivative terms by assigning a measure to derivatives when the flow field is discontinuous. We investigate the various DG weak formulations arising from this technique by considering the 2D compressible Navier-Stokes equations for the viscous flows over a cylinder and a NACA0012 airfoil.