A Computational Comparison of Q4 and Q8 Interpolation for Curvature-Preserving Surface Reconstruction

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.