Let $R$ be a commutative Noetherian ring with unit. Suppose $P$ is a prime ideal that is not maximal. How can we go about finding a normal (reduced) primary decomposition of the power of $P$, say a normal decomposition for $P^2$ or $P^3$?
I was just recently looking into symbolic powers and primary decompositions, and I remember looking at this script. It didn't really help me with my question, but I think that section 8 might be of perfect use to you. You might find more texts by Swanson on the topic, she has done some very nice work in that area.