What is the ruling of Islamic law on a husband marrying a second wife, knowing that the first wife desires for the matter to be completed despite her fear of discord (fitna) and people's talk?

It is permissible for your friend's husband to marry a second, third, or fourth wife, as long as he has the ability to practice polygyny and knows that he can fulfill the rights due to his wives. He may even be rewarded for this if his intention is to protect himself and those he marries from sin. In the Hadith: "Whoever among you can benefit his brother, let him do so," and "Marry the affectionate and fertile, for I will boast of your numbers before the nations." Your friend will also be rewarded if she agrees with her husband and accepts this, in obedience to the Sharia, as she would be cooperating in righteousness. Allah Almighty said: "And cooperate in righteousness and piety." She should not pay attention to the sayings and customs of society as long as the Sharia has permitted it.