A new family of high regularity elements

Abstract

In this article, we propose a new family of high regularity finite element spaces. The global approximation spaces are obtained in two steps. We first build an open cover of the computational domain and local approximation spaces on each patch of the cover. Then we construct partition of unity functions subordinate to the open cover depending on the regularity requirement. The basis functions of the global space is given by the products of the local basis functions and the corresponding partition of unity functions. The method can be used to construct finite element spaces of any desired regularity. Approximation properties and implementation details are discussed. Numerical examples for the biharmonic equation are presented to show the effectiveness of the proposed method.

A high accuracy spectral method based on min/max principle for biharmonic eigenvalue problems on a spherical domain

“…However, most of the existing studies were concerned with the second-order elliptic eigenvalue problems and there are relatively few works treating the biharmonic eigenvalue problems. In recent years, the numerical methods of eigenvalue problems were mainly based on finite element methods which include conforming finite elements [3,11,20,25], nonconforming finite elements [1,18,21,23], and mixed finite elements [5,10,17,19]. For the conforming finite element method, it requires globally continuously differentiable finite element spaces, which are difficult to construct and implement (in particular for three dimensional problems).…”