P-FEM Quadrature Error Analysis on Tetrahedra

Abstract

In the p-FEM and the closely related spectral method, the solution of an elliptic boundary value problems is approximated by piecewise (mapped) polynomials of degree p on a fixed mesh T. In practice, the entries of the p-FEM stiffness matrix cannot be evaluated exactly due to variable coefficients and/or non-affine element maps and one has to resort to numerical quadrature to obtain a fully discrete method. Computationally, choosing shape functions that are related to the quadrature formula employed can significantly improve the computational complexity. For example, for tensor product elements (i.e., quadrilaterals, hexahedra) choosing tensor product Gauss-Lobatto quadrature with q + 1 = p + 1 points in each spatial direction and taking as shape functions the Lagrange interpolation polynomials (of degree p) in the Gauss-Lobatto points effectively leads to a spectral method. The quadrature error analysis for the p-FEM/spectral method is available even for this case of minimal quadrature (see, e.g., [5, 6] and reference there). Key to the error analysis is a one-dimensional discrete stability result for the Gauss-Lobatto quadrature due to [2] (corresponding to α = 0 in Lemma 2 below) that can readily be extended to quadrilaterals/hexahedra by tensor product arguments.