Harmonic maps between smooth surfaces are often approximated by “discrete harmonic” maps of triangulated surfaces. This leads to natural questions such as, “How well does a discrete harmonic map approximate a smooth harmonic map?” and, “How could we make the approximation better?” In general, using finer surface triangulations produces better approximations. This leads us to consider methods of refining surface triangulations given an existing discrete harmonic map. Here we present one such method. While it is easily stated, we will see that the problems it leads to—and the solutions to these problems—can be surprisingly complex.

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