Summary Curved geometries and the corresponding near‐surface fields typically require a large number of linear computational elements. High‐order numerical solvers have been primarily used with low‐order meshes. There is a need for curved, high‐order computational elements. Typical near‐surface meshes consist of hexahedral and/or prismatic elements. The present work studies the employment of quadratic meshes that are relatively coarse for field simulations. Directionally quadratic high‐order elements are proposed for the near‐surface field regions. The quadratic meshes are compared with the conventional low‐order ones in terms of accuracy and efficiency. The cases considered include closed surface volume calculations, as well as computation of gradients of several analytic fields. A special method of adaptive local quadratic meshes is proposed and evaluated. Truncation error analysis for quadratic grids yields comparison with the conventional linear hexahedral/prismatic meshes, which are subject to typical distortions such as stretching, skewness, and torsion.