New HDG Methods for Fluid Dynamics and Continuum Mechanics

Description

The long-term goal of the P.I. is to develop and study, theoretically as well ascomputationally,efficient, robust and superconvergent numerical methods for solving partial differentialequations arising in fluid dynamics and continuum mechanics. In this proposal, the P.I.will concentrate on the study and development of a new, emerging numerical scheme -hybridizable discontinuous Galerkin (HDG) methods.HDG methods are characterized by being easier to implement, having enhanced stability,convergence properties and robustness, and by displaying an improved flexibility forhandling arbitrarily-shaped domains. The only globally coupled degrees of freedom ofHDG mehtods are those of a numerical approximation defined on the boundaries of theelements. Compared with convectional DG methods, HDG methods significantly reducethe degrees of freedom and can be more efficiently implemented. On the other hand,HDG methods use stabilization functions along interfaces of meshes to obtain extrastability, such that they have better stability than mixed finite element methods (HDGmethods can be considered as stabilized mixed finite element methods). In addition,post-processed solution of HDG methods achieves one order higher convergence rate forprimary variable (like displacement in linear elasticity problem) than mixed finiteelement methods. These features make HDG methods suitable for solving partialdifferential equations arising in fluid dynamics and continuum mechanics.The P.I. is interested in studying and developing a new class of HDG methods, which isrobust, efficient and superconvergent without post-processing, for a variety of equationsarising in fluid dynamics and continuum mechanics.