Background: Mean-field methods based on an energy density functional (EDF) are powerful tools used to describe many properties of nuclei in the entirety of the nuclear chart. The accuracy required on energies for nuclear physics and astrophysics applications is of the order of 500 keV and much effort is undertaken to build EDFs that meet this requirement. Purpose: The mean-field calculations have to be accurate enough in order to preserve the accuracy of the EDF. We study this numerical accuracy in detail for a specific numerical choice of representation for the mean-field equations that can accommodate any kind of symmetry breaking. Method: The method that we use is a particular implementation of 3-dimensional mesh calculations. Its numerical accuracy is governed by three main factors: the size of the box in which the nucleus is confined, the way numerical derivatives are calculated and the distance between the points on the mesh. Results: We have examined the dependence of the results on these three factors for spherical doubly-magic nuclei, neutron-rich $^{34}$Ne, the fission barrier of $^{240}$Pu and isotopic chains around Z = 50. Conclusions: Mesh calculations offer the user extensive control over the numerical accuracy of the solution scheme. By making appropriate choices for the numerical scheme the achievable accuracy is well below the model uncertainties of mean-field methods.
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