5 research outputs found
Two families of -rectangle nonconforming finite elements for sixth-order elliptic equations
In this paper, we propose two families of nonconforming finite elements on
-rectangle meshes of any dimension to solve the sixth-order elliptic
equations. The unisolvent property and the approximation ability of the new
finite element spaces are established. A new mechanism, called the exchange of
sub-rectangles, for investigating the weak continuities of the proposed
elements is discovered. With the help of some conforming relatives for the
problems, we establish the quasi-optimal error estimate for the
tri-harmonic equation in the broken norm of any dimension. The
theoretical results are validated further by the numerical tests in both 2D and
3D situations
Nonstandard finite element de Rham complexes on cubical meshes
We propose two general operations on finite element differential complexes on
cubical meshes that can be used to construct and analyze sequences of
"nonstandard" convergent methods. The first operation, called DoF-transfer,
moves edge degrees of freedom to vertices in a way that reduces global degrees
of freedom while increasing continuity order at vertices. The second operation,
called serendipity, eliminates interior bubble functions and degrees of freedom
locally on each element without affecting edge degrees of freedom. These
operations can be used independently or in tandem to create nonstandard
complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show
that the resulting elements provide convergent, non-conforming methods for
problems requiring stronger regularity and satisfy a discrete Korn inequality.
We discuss potential benefits of applying these elements to Stokes, biharmonic
and elasticity problems.Comment: 31 page