Precision calculation of the bound-electron $g$ factor in molecular hydrogen ions

We calculate the bound-electron $g$ factor for a wide range of rovibrational states of the molecular hydrogen ions H$_2^+$ and HD$^+$. Relativistic and QED corrections of orders up to $\alpha^5$ are taken into account. All contributions are calculated in a nonrelativistic QED framework, except for relativistic corrections of order $(Z\alpha)^4$ and above, which are obtained by calculating the relativistic $g$ factor using a precise minmax finite element solution of the two-center Dirac equation. A relative accuracy of $4-5 \times 10^{-11}$ is achieved for the scalar $g$ factor component, which represents an improvement by more than three orders of magnitude over previous calculations. These results are useful for internal state identification and rovibraional spectroscopy of single molecular hydrogen ions in Penning traps, and open a new avenue towards precision tests of QED.