A 𝐶1-𝑃7 Bell Finite Element on Triangle

Abstract

We construct a C 1 C^{1} - P 7 P_{7} Bell finite element by restricting its normal derivative from a P 6 P_{6} polynomial to a P 5 P_{5} polynomial, and its second normal derivative from a P 5 P_{5} polynomial to a P 4 P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P 6 P_{6} polynomial space. We show the method converges at order 7 in L 2 L^{2} -norm. By eliminating all degrees of freedom on edges of C 1 C^{1} - P 7 P_{7} Argyris finite element, the global degrees of freedom of the new element are reduced substantially from 27 ⁢ V 27V to 12 ⁢ V 12V asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the C 1 C^{1} - P 6 P_{6} Argyris finite element is 19 ⁢ V 19V , the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.