Convergence of a regularized finite element discretization of the two-dimensional Monge-Ampère equation

12/20/2021

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by   Dietmar Gallistl, et al.

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This paper proposes a regularization of the Monge-Ampère equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution u_ε and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of u_ε to the convex Alexandrov solution u to the Monge-Ampère equation as the regularization parameter ε approaches 0. A mixed finite element method for the approximation of u_ε is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions u.