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

Abstract

In this paper, the second order boundary value problem −∇ · (A(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 A is a constant function on each triangle and each triangle has straight edges, we prove that the element stiffness matrix has not more than 25 2 p 2 nonzero matrix entries. An application for preconditioning is given. Numerical examples show the advantages of the proposed basis.

Sparsity optimized high order finite element functions for H(div) on simplices

“…These relations have been proved in [10], [8] and [9]. In the present paper the main relations required for proving the orthogonality of our basis function in H(div) are…”

“…Carrying out such computations manually, as done e.g. for the scalar H 1 -conforming triangle in [10], is very paper and time consuming. Hence, we present explicit computations only for some illustrative examples, while the general proof was carried out symbolically.…”

“…Note that these are the polynomials used in the definition of the H 1 -cell based basis functions φ C ij = u i v ij in [10] and [9] for a = 1, respectively. The set of all interior basis functions is denoted by…”

“…For the proof of the sparsity of the mass matrix we will need additional relations which are proven in [10,8,9] and summarized in the appendix A. Jacobi-and integrated Jacobi-type polynomials can be evaluated efficiently by three term recurrences as summarized in the appendix.…”

“…The set of divergence-free basis functions [Ψ (1) ] are chosen as the curl of the hierarchic H(curl)-conforming completion functions as presented in [35]. But in this work, we rely on products of mixed-weighted Jacobi-polynomials (instead of Legendre-type polynomials), where the weights are chosen in analogy to the H 1 -conforming basis suggested in [10,8].…”