I read this example sentence in Duden:
Das Gericht muss sich mit dem Fall beschäftigen.
I am wondering whether this is a formula used in court of law when the judge makes such announcements as "The court is adjourned."?
No, this isn't a formula, legal or otherwise. It's simply a free combination of words that means exactly what you would expect. And it doesn't sound like something a judge would say either; it sounds like something a reporter would write about a dispute as it being escalated to a legal dispute, i.e. before it goes to court.
This is just a normal sentence. No formula and no idiomatic expression. It means:
The court has to deal with the case.
Your suggestion "The court is adjourned" is wrong. A court can only adjourn a case that is has opened before. This is not what the sentence above really means. It means:
The court has to open a new case. The court has not dealt with this case before. There already is an dispute, but the court was not involved into this dispute until now. But in the future the court has to deal with it.