I am using the Falstad simulator.
I do not understand why am I getting a DC current through a capacitor in simulation.
This must be an error from the simulator. Please, let me know if I am sane, I don't get why there is current flowing.
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Looks like you found a convergence corner case.
Here's my sim (clicky here):
Point being, avoid direct capacitor (and inductor) loops to fixed sources. These always cause trouble with SPICE and its derivatives.