Is it permissible to get to know a girl and meet her with the intention of marriage after her initial refusal, and what is the legitimate way to convince her?

The legitimate path to marriage is to propose to the girl through her family. A young man meeting a girl with the intention of marriage without this (proposal) is a temptation from Satan leading to the unlawful, as the Prophet, peace and blessings be upon him, said: "No man is alone with a woman but the third of them is Satan," and his saying: "I have not left behind me a trial more harmful to men than women." Meeting a non-mahram girl and exchanging words of love with her is a great tribulation and a delight to Satan, who seeks to lure you both into indecency. So fear Allah and stay away from temptations.