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.
Highlights
Work on mesh and degree adaptivity schemes for the finite element method [30,44,65] already showed the advantages of adaptive schemes to achieve a required accuracy in an economic manner
Despite traditional discontinuous Galerkin (DG) methods have not been able to consistently prove its superiority against low-order techniques traditionally employed in industry, the recently proposed hybridisable DG (HDG) [11] has shown its superiority compared to traditional DG methods [9,27,33]
The ability to substantially reduce the number of degrees of freedom combined with the possibility to obtain a post-processed solution that converges at a faster rate to the exact solution are the two main properties of HDG methods behind its superiority compared to other DG methods [10,12,38,56]
Summary
Work on mesh and degree adaptivity schemes for the finite element method [30,44,65] already showed the advantages of adaptive schemes to achieve a required accuracy in an economic manner. The third approach, despite not considered useful from a practical point of view, consists of changing the geometry representation of the computational domain to represent with the same degree of polynomials both the geometry and the solution at each iteration of the degree adaptive process. This approach is not considered of interest from a practical point of view because it requires communication with the CAD model at each iteration and re-generation of nodal distributions for curved elements.
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