Rectangular $C^1$-$P_k$ finite elements with $Q_k$-bubble enrichment

Shangyou Zhang

paper

Rectangular $C^{1}$ - $P_{k}$ finite elements with $Q_{k}$ -bubble enrichment

arXiv:2512.12152

Abstract

We enrich the $P_{k}$ polynomial space by $5$ ( $k = 4$ ), or $7$ ( $k = 5$ ), or 8 (all $k \geq 6$ ) $Q_{k}$ bubble functions to obtain a family of $C^{1}$ - $P_{k}$ ( $k \geq 4$ ) finite elements on rectangular meshes. We show the uni-solvency, the $C^{1}$ -continuity and the quasi-optimal convergence. Numerical tests on the new $C^{1}$ - $P_{k}$ , $k = 4, 5, 6, 7$ and $8$ , elements are performed.

Rectangular C1-Pk​ finite elements with Qk​-bubble enrichment

Abstract

Rectangular $C^{1}$ - $P_{k}$ finite elements with $Q_{k}$ -bubble enrichment