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HDG Methods for Incompressible Navier-Stokes Flow

Investigator: Christian Waluga -- Advisors: Herbert Egger, Wolfgang Dahmen

Outline

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 rigorous hp-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

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