Data-based adaptive refinement of finite element thin plate spline

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. •Studied four PDE-based error indicators for data-driven techniques.•Demonstrated the performance of resulting adaptively refined grids outperforms uniform grids.•Verified error indicators work for different data distributions including sparse data.•Validated the performance of adaptive refinement on one model problem and two non-trivial bathymetric data sets.•Showed Neumann boundaries conditions are challenging and require further consideration.