Is a non-Islamic divorce valid in Islamic law? Is the wife sinful for cutting off relations with her husband's Muslim, elderly, and lonely ex-wife? Should she try to re-establish contact with her?

Divorce is valid and takes effect if the husband pronounces it, intends it, or writes it down. Court procedures hold no value. The illicit relationship between the divorced couple must be severed, and any attempt to re-establish it is impermissible, unless he can marry her. There is no harm in offering her assistance or advice, and it is preferable for the questioner to undertake this himself.