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

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 ( h 3 ) versus O ( h 2 ) in L 2 , O ( h 2 ) versus O ( h ) in H 1 , and O ( h ) versus O ( h 0 ) in H 2 , with the H 2 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.