Integration by table look-up for p-version finite elements on curved tetrahedra

Abstract

We develop a method of evaluating element level matrices and vectors associated with p-version finite element computations on tetrahedral meshes with curved boundaries. The procedure uses optimal interpolation formula [Q. Chen, I. Babuška, Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. Methods Appl. Mech. Engrg. 128 (1995) 405–417; Q. Chen, I. Babuška, Approximate optimal polynomial interpolation points in the tetrahedron, Comput. Methods Appl. Mech. Engrg. 137 (1997) 89–94] for non-polynomial portions of integrands followed by exact integration. Exact integrals of product combinations of polynomial interpolants and shape functions are stored so that different integrals may be efficiently evaluated by a table look-up.We present error-analysis for our scheme applied to curvilinear tetrahedral meshes with exact geometric representation of the boundary of the domain as defined in a geometric-modelling system. Numerical results are presented for problems involving the Helmholtz equation.