The paper presents local gravity field modelling in spatial domain using the finite element method (FEM). FEM as a numerical method is applied for solving the geodetic boundary value problem with oblique derivative boundary conditions (BC). For such a problem, we derive a new numerical scheme where the oblique derivative BC are considered directly at computational nodes on the discretized Earth’s topography. Then, the developed FEM approach is tested in several artificial testing experiments as well as by a reconstruction of a known harmonic function above the extremely complicated Earth’s topography in the Himalayas. A main numerical experiment is focused on very detailed local gravity field modelling in Slovakia using terrestrial gravity data. The high horizontal resolution 100 \(\times \) 100 m and non-uniform resolution in the radial direction has resulted in a 3D unstructured mesh of finite elements with 5,287,500,000 unknowns. Large-scale parallel computations were performed on a parallel cluster using 1.5 TB of distributed memory. The obtained local quasigeoid model is tested at 403 GNSS-levelling benchmarks. The standard deviation of residuals 2.77 cm, which decreases to 2.54 cm after excluding 7 outliers, indicates its high precision. However, depicted residuals show their low-frequency character with amplitudes about ± 3 cm. As a by-product, the first and second derivatives of the obtained disturbing potential in the radial direction are also evaluated in several altitude levels as well as on the Earth’s surface. Finally, the paper presents a comparison of the obtained FEM solution with the recent local quasigeoid models in Slovakia computed in the spatial as well as spectral domain. It illustrates a practical contribution of the presented FEM approach for precise local gravity field modelling, especially in high mountains.

本文介绍了使用有限元方法 (FEM) 在空间域中进行局部重力场建模。有限元法作为一种数值方法用于求解具有斜导数边界条件 (BC) 的大地测量边界值问题。对于这样的问题，我们推导出一种新的数值方案，其中直接在离散地球地形的计算节点处考虑斜导数 BC。然后，开发的 FEM 方法在几个人工测试实验中进行测试，并通过在喜马拉雅山脉极其复杂的地球地形上方重建已知的谐波函数进行测试。一个主要的数值实验侧重于使用地面重力数据在斯洛伐克进行非常详细的局部重力场建模。高水平分辨率 100 \(\times \)100 m 和径向方向上的非均匀分辨率导致了具有 5,287,500,000 个未知数的有限元 3D 非结构化网格。使用 1.5 TB 分布式内存在并行集群上执行大规模并行计算。获得的局部准地球仪模型在 403 个 GNSS 水平基准测试中进行了测试。残差的标准差为2.77 cm，剔除7个异常值后降低到2.54 cm，说明其精度高。然而，所描绘的残差显示了它们的低频特征，幅度约为 ± 3 cm。作为副产品，所获得的径向干扰电势的一阶和二阶导数也在几个海拔高度以及地球表面进行了评估。最后，该论文将所获得的 FEM 解与斯洛伐克最近在空间和谱域中计算的局部准地球模型进行了比较。它说明了所提出的 FEM 方法对精确局部重力场建模的实际贡献，尤其是在高山中。