On reconstruction of three-dimensional harmonic functions from discrete data

Abstract

This study presents an approach for reconstruction of harmonic functions in three dimensions from the finite number of field and surface measurements. The approach, based on the Trefftz method, performs reconstruction as the best fit to the data and provides smoothness of the reconstructed function. Two particular algorithms are proposed; the first one uses specific radial basis functions and the second one is of finite element type. Either of them can be applied to analyse different data types but the latter can handle larger problems. The data types considered in this study also cover direct and inverse boundary value problems. Therefore, the proposed approach is universal and capable of dealing with both well-posed and ill-posed formulations. Examples from steady heat conduction and elastostatics are examined in order to investigate the efficiency of the approach.