Is it permissible for a wife to pay her Zakat on her own money secretly and without her husband's knowledge, and what should the wife do if her husband finds out and causes her a problem?

You are not permitted to neglect or delay paying Zakat once it becomes due. There is no obedience to your husband in this matter, for Zakat is one of the pillars of Islam, and withholding it is among the grave sins. Know that Zakat does not diminish wealth but rather increases it, as Allah Almighty says: "But whatever thing you spend [in His cause] - He will compensate it; and He is the best of providers." Hasten to pay Zakat, whether with your husband's knowledge or without it, for he has no authority over your money. If you pay it secretly, there is no harm. If he finds out, explain to him that this is an obligation enjoined by Allah, and it is a cause for blessing and an increase in wealth.