Penalty-Free Discontinuous Galerkin Methods for the Stokes and Navier-Stokes Equations

摘要

This thesis formulates and analyzes low-order penalty-free discontinuous Galerkin methods for solving the incompressible Stokes and Navier-Stokes equations. Some symmetric and non-symmetric discontinuous Galerkin methods for incompressible Stokes and Navier-Stokes equations require penalizing jump terms for stability and convergence of the methods. These discontinuous Galerkin methods are called interior penalty methods as the penalizing jump terms involve a penalty parameter. It is known that the penalty parameter has to be large enough to prove coercivity of the bilinear form and therefore to obtain existence of the solution for the symmetric case. The momentum equation is satisfied locally on each mesh element, and it depends on the penalty parameter. Setting the penalty parameter equal to zero yields a singular linear system, if piecewise linears are used. To overcome this instability, this thesis discusses an enrichment of the velocity space with locally supported quadratic functions called bubbles. First, the penalty-free non-symmetric discontinuous Galerkin method is analyzed for the Stokes equations. Second, the main contribution of this thesis is the analysis of both symmetric and non-symmetric penalty-free discontinuous Galerkin methods for the incompressible Varier-Stokes equations. Since a direct application of the generalized Lax-Milgram theorem is not possible, the numerical solution is shown to be the solution as a fixed-point of a problem-related map. A priori error estimate is derived.