Error estimates for the Laplace interpolation on convex polygons

In the natural element method (NEM), the Laplace interpolation error estimate on convex planar polygons is proved in this study. The proof is based on bounding gradients of the Laplace interpolation for convex polygons which satisfy certain geometric requirements, and has been divided into several parts that each part is bounded by a constant. Under the given geometric assumptions, the optimal convergence estimate is obtained. This work provides the mathematical analysis theory of the NEM. Some numerical examples are selected to verify our theoretical result.