Is catharsis a necessary element of "tragedy"? If so, is there catharsis in Macbeth?

Is catharsis necessary to tragedy? It depends who you ask. Some people would say that tragedy creates such tension for its audience that there must be some moment of emotional release; the audience must experience a purging of these emotions. Others would argue that catharsis allows the audience to become too complacent: Bertold Brecht purposely wrote plays that left intense emotions intact rather than resolving them because he hoped it would compel his audience to take action. 

That being said, I do not think catharsis is necessary to tragedy, only to the audience's emotional satisfaction. To that end, I believe Macbeth offers catharsis. When Macduff kills the tyrant and presents his head to Malcolm, the new, rightful king of Scotland, all of the anxiety built up throughout the play is released and order is restored.