On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations

We study the convergence of monotone $P1$ finite element methods on unstructured meshes for fully nonlinear Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under nondegeneracy assumptions, strong $L^2$ convergence of the gradients.

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