HDG-NEFEM with degree adaptivity for Stokes flows

TL;DR

This paper introduces a novel HDG-NEFEM method with degree adaptivity for Stokes flows, eliminating boundary approximation errors and reducing degrees of freedom, leading to more accurate and efficient simulations.

Contribution

It presents the first combination of NEFEM with HDG, providing a boundary-accurate, degree-adaptive solver for Stokes flows with a reliable error estimator.

Findings

Eliminates boundary approximation uncertainties in curved geometries.

Reduces degrees of freedom compared to traditional DG methods.

Demonstrates superior accuracy and efficiency with degree adaptation strategies.

Abstract

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.