We present a new disparity functional to measure and improve the geometric accuracy of a curved high-order mesh that approximates a target geometry model. We have devised the disparity to account for compound models, be independent of the entity parameterization, and allow trimmed entities. The disparity depends on the physical mesh and the auxiliary parametric meshes. Since it is two times differentiable on all these variables, we can minimize it with a second-order method. Its minimization with the parametric meshes as design variables measures the geometric accuracy of a given mesh. Furthermore, the minimization with both the physical and parametric meshes as design variables improves the geometric accuracy of an initial mesh. We have numerical evidence that the obtained meshes converge to the target geometry (unitary normal) algebraically, in terms of the element size, with order 2p ($2p-1$, respectively), where p is the polynomial degree of the mesh. Although we obtain meshes with non-interpolative boundary nodes, we propose a post-process to enforce, if required by the application, meshes with interpolative boundary nodes and featuring the same order of geometric accuracy. In conclusion, we can obtain super-convergent orders, at least for sufficiently smooth parametric curve (surface) entities, for meshes of polynomial degrees up to 4 (3, respectively). In perspective, this super-convergence might enable using a lower polynomial degree to approximate the geometry than to approximate the solution without hampering the required geometric accuracy for high-order analysis.

我们提出了一种新的视差函数来测量和提高近似目标几何模型的弯曲高阶网格的几何精度。我们设计了差异来解释复合模型，独立于实体参数化，并允许修剪实体。视差取决于物理网格和辅助参数网格。由于它在所有这些变量上都是两倍可微的，我们可以用二阶方法将其最小化。它以参数网格作为设计变量的最小化测量给定网格的几何精度。此外，将物理和参数网格作为设计变量的最小化提高了初始网格的几何精度。我们有数值证据表明所获得的网格在代数上收敛到目标几何（酉法线），p ($2磷-1$，分别），其中p是网格的多项式次数。尽管我们获得了具有非插值边界节点的网格，但如果应用程序需要，我们提出了一个后处理来强制使用具有插值边界节点并具有相同几何精度顺序的网格。总之，我们可以获得超收敛阶数，至少对于足够平滑的参数曲线（曲面）实体，对于多项式次数高达 4（分别为 3）的网格。从角度来看，这种超收敛可能允许使用比逼近解更低的多项式次数来逼近几何，而不会妨碍高阶分析所需的几何精度。