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

Shangyou Zhang

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

Numerical Analysis 2025-12-16 v1 Numerical Analysis

Authors: Shangyou Zhang

Abstract

We enrich the $P_k$ polynomial space by $5$ ( $k=4$ ), or $7$ ( $k=5$ ), or 8 (all $k\ge 6$ ) $Q_k$ bubble functions to obtain a family of $C^1$ - $P_k$ ( $k\ge 4$ ) 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.

Cite

@article{arxiv.2512.12152,
  title  = {Rectangular $C^1$-$P_k$ finite elements with $Q_k$-bubble enrichment},
  author = {Shangyou Zhang},
  journal= {arXiv preprint arXiv:2512.12152},
  year   = {2025}
}

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