Kim S. Bey

hp-Version discontinuous Galerkin methods for hyperbolic conservation laws

Abstract The development of hp -version discontinuous Galerkin methods for hyperbolic conservation laws is presented in this work. A priori error estimates are derived for a model class of linear hyperbolic conservation laws. These estimates are obtained using a new mesh-dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation. The results generalize and extend previous results on mesh-dependent norms to hp -version discontinuous Galerkin methods. A posteriori error estimates which provide bounds on the actual error are also developed in this work. Numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution.

A parallel hp -adaptive discontinuous Galerkin method for hyperbolic conservation laws

This paper describes a parallel adaptive strategy based on discontinuous hp-finite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and deliver super-linear rates of convergence, thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems.

Application of integrated fluid-thermal structural analysis methods

Hypersonic vehicles operate in a hostile aerothermal environment which has a significant impact on their aerothermostructural performance. Significant coupling occurs between the aerodynamic flow field, structural heat transfer, and structural response creating a multidisciplinary interaction. Interfacing state-of-the-art disciplinary analysis methods is not efficient, hence interdisciplinary analysis methods integrated into a single aerothermostructural analyzer are needed. The NASA Langley Research Center is developing such methods in an analyzer called LIFTS (Langley Integrated Fluid-Thermal-Structural) analyzer. The evolution and status of LIFTS is reviewed and illustrated through applications.

Numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates

In this paper, we consider a nonlinear hierarchical finite element method for heat conduction problems over two- or three-dimensional plates. Problems considered are nonlinear because the heat conductivity parameter depends upon the temperature itself. This paper explores a new technique recently proposed by the first author which transforms a nonlinear parabolic problem to a linear problem at the discrete level. We present several numerical examples which demonstrate the efficiency of the current technique.

A discontinuous Galerkin Method for parabolic problems with modified hp-finite element approximation technique

Abstract A recent paper [Hideaki Kaneko, Kim S. Bey, Gene J.W. Hou, Discontinuous Galerkin finite element method for parabolic problems, preprint November 2000, NASA] is generalized to a case where the spatial region is taken in R 3 . The region is assumed to be a thin body, such as a panel on the wing or fuselage of an aerospace vehicle. The traditional h - as well as hp -finite element methods are applied to the surface defined in the x – y variables, while, through the thickness, the technique of the p -element is employed. Time and spatial discretization scheme developed in Kaneko et al. (2000), based upon an assumption of certain weak singularity of ∥ u t ∥ 2 , is used to derive an optimal a priori error estimate for the current method.