Hierarchical high‐order conforming C 1 bases for quadrangular and triangular finite elements

Summary We present three new sets of C 1 hierarchical high‐order tensor‐product bases for conforming finite elements. The first basis is a high‐order extension of the Bogner–Fox–Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C 1 global continuity on the common edges of elements. The second basis uses the tensor product of fifth‐order Hermite polynomials and high‐order functions and achieves global C 1 continuity for meshes of quadrilaterals and C 2 continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one‐dimensional functions defined in barycentric coordinates. It also has global C 1 continuity on edges and C 2 continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h ‐ and p ‐refinements are evaluated by comparing the approximation errors in the L 2 ‐ and energy norms. A plate with singularity is then studied, and h ‐ and p ‐refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non‐distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd.