HDG-NEFEM with degree adaptivity for Stokes flows
For computational fluid dynamics researchers, this method improves accuracy and efficiency in simulating Stokes flows with curved boundaries, though it is an incremental combination of existing techniques.
The paper introduces HDG-NEFEM for Stokes flows, eliminating uncertainty from polynomial boundary approximation and reducing degrees of freedom compared to other DG methods. It demonstrates superiority and reliability of degree adaptation for curved boundaries.
The NURBS-enhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDG-NEFEM with degree adaptation is illustrated.