Near and full quasi-optimality of finite element approximations of stationary second-order mean field games

We establish a priori error bounds for monotone stabilized finite element discretizations of stationary second-order mean field games (MFG) on Lipschitz polytopal domains. Under suitable hypotheses, we prove that the approximation is asymptotically nearly quasi-optimal in the $H^1$-norm in the sense that, on sufficiently fine meshes, the error between exact and computed solutions is bounded by the best approximation error of the corresponding finite element space, plus possibly an additional term, due to the stabilization, that is of optimal order with respect to the mesh size. We thereby deduce optimal rates of convergence of the error with respect to the mesh-size for solutions with sufficient regularity. We further show full asymptotic quasi-optimality of the approximation error in the more restricted case of sequences of strictly acute meshes. Our third main contribution is to further show, in the case where the domain is convex, that the convergence rate for the $H^1$-norm error of the value function approximation remains optimal even if the density function only has minimal regularity in $H^1$.