Completions to sparse shape functions for triangular and tetrahedral p-FEM

holds for all v ∈ H Γ1(Ω). Problem (1) will be discretized by means of the hp-version of the finite element method using triangular/tetrahedral elements △s, s = 1, . . . , nel, see e.g. Schwab [1998], Solin et al. [2003]. Let △̂d, d = 2, 3 be the reference triangle (tetrahedron) and Fs : △̂ → △s be the (possibly nonlinear) isoparametric mapping to the element △s. We define the finite element space M := {u ∈ H Γ1(Ω), u |△s= ũ(F s (x, y, z)), ũ ∈ Pp}, where Pp is the space of all polynomials of maximal total degree p. By Ψ = (ψ1, . . . , ψN ), we denote a basis for M. The Galerkin projection of (1) onto M leads to the linear system of algebraic finite element equations