The purpose of this paper is to consider reconstructions of potential 2D fields from discrete measurements. Two potential processes are addressed, steady flow and heat conduction. In the first case, the flow speed and streamlines are determined from the discrete data on flow directions, in the second case, the temperature and flux are recovered from temperature measurements at discrete points.
The method employs the Trefftz element principle and the collocation. The domain is seen as a combination of elements, where the solution is sought as a linear holomorphic function a priori satisfying the governing equations. Continuity of piecewise holomorphic functions is imposed at collocation points located on the element interfaces. These form the first group of equations. The second group of equations is formed by addressing the measured data, therefore the matrix coefficients may reflect experimental errors. In the case of fluid flow, all equations are homogeneous, therefore one normalising equation is added, which provides existence of a non‐trivial solution. The system is over‐determined; it is solved by the least squares method.
For the heat flow problem, the determination of heat flux is unique, while for the fluid flow, the determined streamlines are unique and the determination of speed contains one free multiplicative positive constant. Several examples are presented to illustrate the methods and investigate their efficiency and sensitivity to noisy data.
The approach can be applied to other 2D potential problems.
The paper studies two novel formulations of the reconstruction problem for 2D potential fields. It is shown that the suggested numerical method is able to deal directly with discrete experimental data.
