TITLE:
A Computational Comparison of Q4 and Q8 Interpolation for Curvature-Preserving Surface Reconstruction
AUTHORS:
Kuldeep S. Rawat
KEYWORDS:
Q4 Interpolation, Q8 Serendipity Finite Element, Bramble-Hilbert Lemma, Curvature Preservation, Sobolev Norms, Surface Reconstruction
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.16 No.3,
July
29,
2026
ABSTRACT: The four-node quadrilateral element (Q4) and the eight-node serendipity quadrilateral element (Q8) are standard interpolation models in numerical approximation and finite element analysis. This paper compares Q4 and Q8 interpolation for curvature-preserving surface reconstruction using a Hessian Frobenius-norm proxy and a Monge-patch formulation of surface normals. Interpolation error bounds derived from the Bramble-Hilbert lemma establish that Q8 achieves one higher order of Sobolev convergence than Q4 at every norm level: O(h3) versus O(h2) in L2, O(h2) versus O(h) in H1, and O(h) versus O(h0) in H2, with the H2 gap directly governing curvature fidelity. These theoretical rates are not directly measured by the single-element tests on analytic surfaces, which confirm polynomial reproduction and approximation behavior consistent with the predicted Q8 advantage; a mesh-refinement study on assembled elements is needed to verify the convergence orders computationally. Single-element experiments confirm exact polynomial reproduction for dome and saddle surfaces and show that, for a transcendental ripple surface, Q8 reduces height RMSE from 0.3310 to 0.0843 and normal-angle error from 37.47˚ to 15.02˚. The results support a practical selection rule: Q4 is appropriate for height-only interpolation on locally planar or bilinear patches, while Q8 is preferred whenever surface normals or curvature are important.