On an accurate numerical integration for the triangular and tetrahedral spectral finite elements

In the triangular/tetrahedral spectral finite elements, we apply a bilinear/trilinear transformation to map a reference square/cube to a triangle/tetrahedron, which consequently maps the \(\varvec{Q_k}\) polynomial space on the reference element to a finite element space of rational/algebraic functions on the triangle/tetrahedron. We prove that the resulting finite element space, even under this singular referencing mapping, can retain the property of optimal-order approximation. In addition, we prove that the standard Gauss-Legendre numerical integration would provide sufficient accuracy so that the finite element solutions converge at the optimal order. In particular, the finite element method, with singular mappings and numerical integration, preserves \(\varvec{P_k}\) polynomials. That is, the \(\varvec{Q_k}\) finite element solution is exact if the true solution is a \(\varvec{P_k}\) polynomial. Numerical tests are provided, verifying all theoretic findings.