$L^2(H_{\gamma }^1)$ 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 $L^2(H_{\gamma }^1(\mathrm{\Omega }))$ to the viscosity solution without assuming uniform parabolicity of the HJB operator.