BAIL 2006 Book of Abstracts - Institut für Numerische und ...

P. HOUSTON: Discontinuous Galerkin Finite Element Methods for CFD: A Posteriori
Error Estimation and Adaptivity
✬
✫
Discontinuous Galerkin Finite Element Methods for CFD: A Posteriori Error Estimation and
Adaptivity
Paul Houston
School of Mathematical Sciences, University of Nottingham, UK.
In recent years there has been considerable interest in the mathematical design and
practical application of nonconforming finite element methods that are based on discontinuous
piecewise polynomial approximation spaces; such approaches are referred to as
discontinuous Galerkin (DG) methods. The main advantages of these methods lie in their
conservation properties, their ability to treat a wide range of problems within the same
unified framework, and their great flexibility in the mesh-design. Indeed, DG methods can
easily handle non-matching grids and non-uniform, even anisotropic, polynomial approximation
degrees, which makes them ideally suited for application within adaptive finite
element software.
In this talk we present an overview of some recent developments concerning the a posteriori
error analysis and adaptive mesh design of h– and hp–version DG finite element
methods for the numerical approximation of second–order elliptic boundary value problems.
In particular, we consider the derivation of computable upper and lower bounds
on the error measured in terms of an appropriate (mesh–dependent) energy norm. The
proofs of the upper bounds are based on rewriting the method in a non-consistent manner
using polynomial lifting operators and employing an appropriate decomposition result
for the underlying discontinuous spaces. Applications to the numerical approximation
of second–order linear elliptic problems, including Poisson’s equation, Stokes equations,
nearly–incompressible elasticity, and the time harmonic eddy current problem, as well
as second–order quasilinear boundary value problems, which typically arise in the modelling
of non-Newtonian flows, will be considered. Numerical experiments confirming the
reliability and efficiency of the proposed a posteriori error bounds within an automatic
mesh refinement algorithm employing both local mesh subdivision and local polynomial
enrichment will be presented.
This research has been carried out in collaboration with Dominik Schötzau (University
of British Columbia), Thomas Wihler (University of Minnesota), and Ilaria Perugia
(University of Pavia).
References
[1] P. Houston, I. Perugia, and D. Schötzau. An posteriori error indicator for discontinuous
Galerkin discretizations of H(curl)–elliptic partial differential equations.
Submitted to IMA J. Numer. Anal.
[2] P. Houston, D. Schötzau, and T. Wihler. Energy norm a posteriori error estimation
for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci.
Comp., 22(1):357–380, 2005.
[3] P. Houston, D. Schötzau, and T. Wihler. An hp-adaptive mixed discontinuous
Galerkin FEM for nearly incompressible linear elasticity. Comput. Methods Appl.
Mech. Engrg., (to appear).
[4] P. Houston, D. Schötzau, and T. P. Wihler. Energy norm a posteriori error estimation
of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math.
Models Methods Appl. Sci., (to appear).
1
Speaker: HOUSTON, P. 4 BAIL 2006
✩
✪