2012/08/29 by Andrew Gillette, Gillette, Andrew
Engineering · #41A10 #41A25 #65D05 #65N30 #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1208.5973
openalex publication_date 2012/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce new Hermite-style and Bernstein-style geometric decompositions\nof the cubic order serendipity finite element spaces S3(I2) and S3(I3),\nas defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11\n(2011), 337--344]. The serendipity spaces are substantially smaller in\ndimension than the more commonly used bicubic and tricubic Hermite tensor\nproduct spaces - 12 instead of 16 for the square and 32 instead of 64 for the\ncube - yet are still guaranteed to obtain cubic order \a priori error\nestimates in H1 norm when used in finite element methods. The basis\nfunctions we define have a canonical relationship both to the finite element\ndegrees of freedom as well as to the geometry of their graphs; this means the\nbases may be suitable for applications employing isogeometric analysis where\ndomain geometry and functions supported on the domain are described by the same\nbasis functions. Moreover, the basis functions are linear combinations of the\ncommonly used bicubic and tricubic polynomial Bernstein or Hermite basis\nfunctions, allowing their rapid incorporation into existing finite element\ncodes.\n