New shape functions for triangular p-FEM using integrated Jacobi polynomials

Abstract

In this paper, the second order boundary value problem −∇·(**(x,y)∇u)=f is discretized by the Finite Element Method using piecewise polynomial functions of degree p on a triangular mesh. On the reference element, we define integrated Jacobi polynomials as interior ansatz functions. If ** is a constant function on each triangle and each triangle has straight edges, we prove that the element stiffness matrix has not more than ** nonzero matrix entries. An application for preconditioning is given. Numerical examples show the advantages of the proposed basis.