The thin plate spline, as introduced by Duchon, interpolates a smooth surface through scattered data. It is computationally expensive when there are many data points. The finite element thin plate spline (TPSFEM) possesses similar smoothing properties and is efficient for large data sets. Its efficiency is further improved by adaptive refinement that adapts the precision of the finite element grid. Adaptive refinement processes and error indicators developed for partial differential equations may not apply to the TPSFEM as it incorporates information about the scattered data. This additional information results in features not evident in partial differential equations. An iterative adaptive refinement process and five error indicators were adapted for the TPSFEM. We give comprehensive depictions of the process in this article and evaluate the error indicators through a numerical experiment with a model problem and two bathymetric surveys in square and L-shaped domains.

翻译：薄板样条由Duchon提出，可通过散乱数据插值生成光滑曲面。当数据点数量较多时，其计算成本较高。有限元薄板样条（TPSFEM）具有相似的平滑特性，且适用于大规模数据集。通过自适应细化调整有限元网格精度，可进一步提升其效率。针对偏微分方程开发的自适应细化流程与误差指示器并不完全适用于TPSFEM，因其需整合散乱数据信息。这些额外信息会产生偏微分方程中未显现的特征。本文针对TPSFEM改进了迭代式自适应细化流程及五种误差指示器，全面描述了该流程，并通过模型问题的数值实验及方形域与L形域中的两次水深测量数据对误差指示器进行了评估。