Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization

Abstract

Abstract The main goal of this article is two fold: (i) To discuss a methodology for the numerical solution of the Dirichlet problem for a Pucci equation in dimension two. (ii) Use the ensuing algorithms to investigate the homogenization properties of the solutions when a coefficient in the Pucci equation oscillates periodically or randomly in space. The solution methodology relies on the combination of a least-squares formulation of the Pucci equation in an appropriate Hilbert space with operator-splitting techniques and mixed finite element approximations. The results of numerical experiments suggest second order accuracy when globally continuous piecewise affine space approximations are used; they also show that the solution of the problem under consideration can be reduced to a sequence of discrete Poisson–Dirichlet problems coupled with one-dimensional optimization problems (one per grid point).