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

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.

💡 Research Summary

This paper presents the construction and analysis of a new family of C1-continuous finite elements on rectangular meshes, designed for solving fourth-order partial differential equations such as the biharmonic (plate bending) equation. The key innovation lies in enriching the standard polynomial space Pk (of total degree ≤ k) with a minimal, carefully selected set of Qk-type “bubble” functions to achieve global C1 continuity while maintaining a relatively low number of degrees of freedom (dofs).

The author builds upon existing elements, particularly the C1-Qk Bell element—a subspace of the Bogner-Fox-Schmit (BFS) element—which imposes a reduced polynomial degree for the normal derivative on edges. From the Bell element’s basis, specific bubble functions (which vanish on the element boundary) are chosen. For k=4, five such bubbles are added to P4 to form a 20-dimensional space V4(T). For k=5, seven bubbles are added to P5 to form a 28-dimensional space. For all k≥6, a consistent set of eight bubbles is added to Pk. The number of enriching functions is minimal, precisely tailored to meet the C1 continuity constraints at each polynomial degree.

The degrees of freedom (dofs) for each element are defined to ensure C1 continuity across element boundaries. They typically consist of: function value, first derivatives (∂x, ∂y), and the mixed derivative (∂xy) at the four vertices; function values at points along the edges; and for higher k (≥6), some normal derivative values at edge points and internal Lagrange nodes. A detailed, step-by-step unisolvency proof is provided for each case (k=4, k=5, k≥6). The proof employs a clever factorization argument: assuming a function in the space has all dofs equal to zero, the conditions force the polynomial part to contain factors corresponding to the edges, eventually leading to the conclusion that the function must be identically zero. This proves that the chosen dofs uniquely determine a function in the local space.

The global finite element space Vh is defined as the set of all functions on the domain which belong to the local space Vk(T) on each rectangle T and are C1 continuous across inter-element boundaries. The discretization of the biharmonic problem using this space leads to a conforming method. The paper proves the well-posedness (unique solvability) of the resulting discrete problem.

The convergence analysis follows standard finite element theory. Because the local space contains Pk and the constructed interpolation operator preserves polynomials of degree k, the method achieves optimal-order error estimates. Under the assumption of full regularity for the biharmonic problem, the error in the H2 norm is of order O(h^{k-1}), and the error in the L2 norm is of order O(h^{k+1}) for k≥6.

The paper concludes by stating that numerical tests have been performed for the new C1-Pk elements of degrees 4 through 8, confirming the theoretical convergence rates and allowing for comparison with the traditional C1-Qk BFS elements. The primary advantage of the new elements is their efficiency: they provide C1 continuity and optimal convergence with significantly fewer degrees of freedom per element compared to their C1-Qk counterparts (e.g., 20 dofs vs. 25 dofs for k=4), leading to smaller linear systems and reduced computational cost.