"The fact that we live at the bottom of a deep gravity well, on the surface of a gas covered planet going around a nuclear fireball 90 million miles away and think this to be normal is obviously some indication of how skewed our perspective tends to be." ― Douglas Adams
First recorded use of the term "finite element method" by Ray Clough, to describe the methods of Courant, Hrenikoff and Zienkiewicz, among others. See also here.
Molecular dynamics was invented independently by Aneesur Rahman.
Cooley and Tukey re-invent the Fast Fourier transform (voted one of the top 10 algorithms of the 20th century), an algorithm first discovered by Gauss.
Birch and Swinnerton-Dyer conjecture formulated through investigations on a computer.
Grobner bases and Buchberger's algorithm invented for algebra
Frenchman Verlet (re)discovers a numerical integration algorithm, (first used in 1791 by Delambre, by Cowell and Crommelin in 1909, and by Carl Fredrik Störmer in 1907, hence the alternative names Störmer's method or the Verlet-Störmer method) for dynamics.
Risch invents algorithm for symbolic integration.
1970s
Computer algebra replicates and extends the work of Delaunay in lunar theory.