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arXiv · 2610.32048

Piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation

Rui Ma·Yang Xing·Shuo Zhang
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Abstract

This paper develops a piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on $(d-2)$-faces and of averaged normal derivatives across interior $(d-1)$-faces. A local algebraic characterization of $P_2$ is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a $3^d$-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken $H^2$ error estimate for the biharmonic problem.

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Rui Ma, Yang Xing, Shuo Zhang. 2026-10-23. Piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. https://arxiv.org/abs/2610.32048

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