HDG Methods for Incompressible Navier-Stokes Flow

Outline

HDG Methods for Incompressible Navier-Stokes Flow

In this project, we consider the derivation and analysis of finite element methods for the approximate solution of the Navier-Stokes equations, which describe the motion of liquid and gaseous substances.

We investigate the stability and approximation properties of primal hybrid discontinuous Galerkin methods, which, due to their local conservation properties and the uncomplicated stabilization of convective transport terms, render themselves as promising candidates for the solution of flow problems. Moreover, the hybrid discontinuous approach facilitates the elimination of local degrees of freedom, the implementation of locally adaptive solvers, and allows for a high level of parallelism.

In general, smooth parts of the solution can be efficiently approximated by higher order polynomials, whereas in regions of lower smoothness, e.g., in the vicinity of reentrant corners, one can expect better results by using locally refined meshes. Besides the approximation properties of the underlying ansatz spaces, the upper bound for the discretization error considerably depends on certain constants appearing in the stability bounds. Hence, we explicitly keep track of the dependence of these constants with respect to the (local) mesh-size and the (local) polynomial degree of approximation. In a rigoroushp-analysis, we investigate a priori and a posteriori error estimates for meshes with hanging nodes, consisting of different types of elements (hybrid meshes).

Furthermore, we demonstrate how hybridization techniques can be employed to couple interface problems between conforming finite element discretizations on subdomains. The resulting hybrid mortar methods can be embedded into the framework of domain decomposition algorithms.

Project-related publications

  • H. Egger and C. Waluga. hp-Analysis of a Hybrid DG Method for Navier-Stokes Flow
    (in preparation)
  • C. Waluga. Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems
    Dissertation, RWTH Aachen (download)
  • H. Egger and C. Waluga. A Hybrid Discontinuous Galerkin Method for Darcy-Stokes Problems
    to appear in Proceedings of the 20th Domain Decomposition Conference, 2012 (preprint).
  • H. Egger and C. Waluga. hp-Analysis of a Hybrid DG Method for Stokes Flow
    IMA Journal of Numerical Analysis, 2012 (link, preprint).
  • H. Egger and C. Waluga. A Hybrid Mortar Method for Incompressible Flow
    International Journal of Numerical Analysis and Modeling, 9(4):793–812, 2012 (download, preprint).
  • C. Waluga and H. Egger. An Implementation of Hybrid Discontinuous Galerkin Methods in DUNE
    In Advances in DUNE, Springer, 2012 (link, preprint).

Presentation of project results

  • H. Egger, C. Waluga. hp estimates for hybrid DG methods for incompressible flow. 25th Chemnitz FEM Symposium 2012. Chemnitz, Germany. September 24-26, 2012
  • H. Egger, C. Waluga. Improved hp estimates for DG approximations with applications in incompressible flow. International Conference on Spectral and High Order Methods (ICOSAHOM 2012). Gammarth, Tunisia. June 25-29, 2012
  • C. Waluga. H. Egger. Hybrid DG Methods for the Incompressible Navier-Stokes Equations. Oberseminar über Angewandte Mathematik, Universität Freiburg. Freiburg, Germany. Juli 26, 2011
  • C. Waluga. H. Egger. Hybrid discontinuous Galerkin methods for incompressible flow problems. SIAM Conference on Computational Science and Engineering (CSE11). Reno NV, USA. February 28-March 4, 2011 (announcement, abstract)
  • C. Waluga. H. Egger. Hybrid discontinuous Galerkin methods for incompressible flow. 20th International Workshop on Domain Decomposition Methods (DD20). UC San Diego in La Jolla, CA, USA. February 7-11, 2011
  • C. Waluga. H. Egger. Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems. Forschungsseminar Numerik, TU Chemnitz. Chemnitz, Germany. January 25, 2011
  • C. Waluga. H. Egger. A posteriori error estimation for a hybridized discontinuous Galerkin method for incompressible flow. AICES Annual Progress Report. Aachen, Germany. December 3, 2010
  • C. Waluga. Hybridized DG methods: theory and implementation in DUNE. DUNE User Meeting 2010. Stuttgart, Germany. October 6-8, 2010
  • C. Waluga, H. Egger. A posteriori error estimation for a hybridized discontinuous Galerkin method for incompressible flow. 23rd Chemnitz FEM Symposium 2010. Lichtenwalde, Germany. September 27-29, 2010 (abstract)
  • C. Waluga, H. Egger. Finite Element Methods for Interface Problems in Computational Fluid Dynamics. ASIM-Workshop: Trends in Computational Science and Engineering, Research Center Jülich. March 3-5, 2010 (poster)
  • C. Waluga, H. Egger. Hybrid Mortar Methods for Stokes Interface Problems. Doctoral School for Numerical Simulations in Technical Sciences, TU Graz. February 4, 2010 (abstract)
  • C. Waluga, H. Egger. Hybrid Finite Element Methods for Stokes Problems on Non-Matching Meshes. IGPM Oberseminar, RWTH Aachen. November 19, 2009 (abstract)
  • C. Waluga, H. Egger. A Hybrid Finite Element Method for Stokes Problems on Non-Matching Meshes. 22nd Chemnitz FEM Symposium 2009. Oberwiesenthal, Germany. September 28-30, 2009 (abstract)