Finite-element approximation in quantum field theory

In this talk the author describes a research program that has been underway for several years. The objective is to find a fast and accurate scheme for solving quantum field theory on a lattice which does not involve a Monte Carlo algorithm. They use an alternative strategy based on the method of finite elements. They are able to formulate fully consistent quantum-mechanical systems directly on a lattice in terms of operator difference equations. One of the many advantages of doing this is that the fermion doubling problem is completely eliminated. The method for solving operator difference equations is discussed for various elementary quantum-mechanical and field-theoretic models including the Schwinger and Sine-Gordon models. Good numerical results are obtained