STABLE AND COMPATIBLE POLYNOMIAL EXTENSIONS IN THREE DIMENSIONS AND APPLICATIONS TO THE p AND h-p FINITE ELEMENT METHOD

Polynomial extensions play a vital role in the analysis of the p and h-p finite element method (FEM) and the spectral element method. We construct explicitly polynomial extensions on standard elements: cubes and triangular prisms, which together with the extension on tetrahedrons are used by the p and h-p FEM in three dimensions. These extensions are proved to be stable and compatible with FEM subspaces on tetrahedrons, cubes, and prisms and realize a continuous mapping: H-00(1/2)(T) (or H-00(1/2) (S)) -> H-1(Omega(st)), where Omega(st) denotes one of these standard elements and T and S are their triangular and square faces. Applications of these polynomial extensions to the p and h-p FEM are illustrated.

多项式延拓在p和h - p有限元方法（FEM）以及谱元方法的分析中起着至关重要的作用。我们在标准单元（立方体和三棱柱）上显式地构造多项式延拓，它们与四面体上的延拓一起被三维的p和h - p有限元方法所使用。这些延拓被证明是稳定的，并且与四面体、立方体和棱柱上的有限元子空间兼容，并且实现了一个连续映射：\(H^{00(1/2)}(T)\)（或\(H^{00(1/2)}(S)\)）\(\to H^1(\Omega_{st})\)，其中\(\Omega_{st}\)表示这些标准单元之一，\(T\)和\(S\)是它们的三角形面和正方形面。阐述了这些多项式延拓在p和h - p有限元方法中的应用。