Two families of C1-Pk Fraeijs de Veubeke-Sander finite elements on quadrilateral meshes

TL;DR

This paper introduces two new families of $C^1$-$P_k$ finite elements on quadrilateral meshes, extending existing methods to higher polynomial degrees with improved accuracy and efficiency, suitable for interface problems.

Contribution

The paper develops two novel $C^1$-$P_k$ finite element families on quadrilaterals, including a full space and a condensed version with fewer unknowns, both with proven convergence.

Findings

Both element families are unisolvent and optimally convergent.

The condensed element achieves higher accuracy with fewer degrees of freedom.

Numerical tests demonstrate the effectiveness and advantages over existing $C^1$-$P_k$ elements.

Abstract

We extend the $C^1$-$P_3$ Fraeijs de Veubeke-Sander finite element to two families of $C^1$-$P_k$ ($k>3$) macro finite elements on general quadrilateral meshes. On each quadrilateral, four $P_k$ polynomials are defined on the four triangles subdivided from the quadrilateral by its two diagonal lines. The first family of $C^1$-$P_k$ finite elements is the full $C^1$-$P_k$ space on the macro-mesh. Thus the element can be applied to interface problems. The second family of $C^1$-$P_k$ finite elements condenses all internal degrees of freedom by moving them to the four edges. Thus the second element method has much less unknowns but is more accurate than the first one. We prove the uni-solvency and the optimal order convergence. Numerical tests and comparisons with the $C^1$-$P_k$ Argyris are provided.