Finite element approximation of Hamilton–Jacobi–Bellman equations with nonlinear mixed boundary conditions

Abstract

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 discretized 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.