Finite Element Approximation of Hamilton-Jacobi-Bellman equations with nonlinear mixed boundary conditions

Abstract

We show strong uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretised via a lower Dini derivative. In time the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard's algorithm. Keywords: Finite element method, Hamilton-Jacobi-Bellman equation, Mixed boundary conditions, Fully nonlinear equation, Viscosity solution