$L^2(H^1_γ)$ Finite Element Convergence for Degenerate Isotropic Hamilton-Jacobi-Bellman Equations

In this paper we study the convergence of monotone $P1$ finite element methods for fully nonlinear Hamilton-Jacobi-Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space $L^2(H^1_γ(Ω))$ to the viscosity solution without assuming uniform parabolicity of the HJB operator.