7,154 research outputs found
Penalty-Free Discontinuous Galerkin Methods for the Stokes and Navier-Stokes Equations
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
Stabilization arising from PGEM : a review and further developments
The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
This paper presents the construction of a correct-energy stabilized finite
element method for the incompressible Navier-Stokes equations. The framework of
the methodology and the correct-energy concept have been developed in the
convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I.
Akkerman, Correct energy evolution of stabilized formulations: The relation
between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric
analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech.
Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding
paper to build a stabilized method within the variational multiscale (VMS)
setting which displays correct-energy behavior. Similar to the
convection--diffusion case, a key ingredient is the proper dynamic and
orthogonal behavior of the small-scales. This is demanded for correct energy
behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin
(SUPG) and the Galerkin/least-squares method (GLS).
The presented method is a Galerkin/least-squares formulation with dynamic
divergence-free small-scales (GLSDD). It is locally mass-conservative for both
the large- and small-scales separately. In addition, it locally conserves
linear and angular momentum. The computations require and employ NURBS-based
isogeometric analysis for the spatial discretization. The resulting formulation
numerically shows improved energy behavior for turbulent flows comparing with
the original VMS method.Comment: Update to postprint versio
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