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：Duchon提出的薄板样条通过散乱数据插值生成光滑曲面。当数据点数量庞大时，其计算成本高昂。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。偏微分方程中开发的自适应细化流程和误差指标可能不适用于TPSFEM，因其整合了散乱数据信息。这些附加信息产生了偏微分方程中不存在的特征。本文为TPSFEM适配了迭代式自适应细化流程及五种误差指标。我们对该流程进行了全面描述，并通过模型问题的数值实验以及方形和L形域内的两个水深测量数据对误差指标进行了评估。