TY - JOUR
T1 - $hp$-Version Analysis for Arbitrarily Shaped Elements on the Boundary Discontinuous Galerkin Method for Stokes Systems
AU - Karatzas , Efthymios N.
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 528
EP - 559
PY - 2024
DA - 2024/06
SN - 21
DO - http://doi.org/10.4208/ijnam2024-1021
UR - https://global-sci.org/intro/article_detail/ijnam/23201.html
KW - Arbitrarily shaped elements, discontinuous Galerkin finite element method, $hp$-version stability, a-priori error estimates, fluid dynamics.
AB - <p style="text-align: justify;">In the present work, we examine and analyze an $hp$-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on
computational meshes consisting of polytopic elements on the boundary. This approach is based
on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has
been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and $H^1/L^2$-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluid
system enabling the proof of the inf/sup condition and the $hp$- a priori error estimates, while we
investigate the optimal convergence rates numerically. This approach recovers and integrates the
flexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometrical
deformations are taking place by degenerating the edges, facets, of the polytopic elements only
on the boundary, combined with the efficiency of the $hp$-version techniques based on arbitrarily
shaped elements without requiring any mapping from a given reference frame.</p>