An adaptive method for the numerical solution of a 2D Monge-Ampère equation

Parabolic fully nonlinear equations may be found in various applications,for instance in optimal portfolio management strategy. We focus here on a canonical parabolic Monge-Ampère equationin two space dimensions. A numerical method has beeninvestigated in[1]. The goal is toextend themethodology by coupling atime steppingsemi-implicit methodthat relies on a conservative formulation of the Monge-Ampère equationwith mesh adaptation.The parabolic Monge-Ampère equation can be expressed as astronglynonlinear, heat-type, parabolic equation, where the nonlinear diffusion function is expressed asa function ofthe cofactor matrix of the Hessianmatrixof the solution. We linearize this diffusion operator and advocatea semi-implicit time-stepping algorithm.In particular, we use the time-evolutive equation to reach a stationarysolutioncorresponding to a solutionof the elliptic Monge-Ampère equation. A loworder, piecewise linear,finite element method is used for spacediscretization, together with a mixed approach for the approximation of the second derivatives. The error is bounded above by an error indicator plus an extra term that can be disregarded in special cases. A mesh adaptivity strategy based on these estimates is then implemented within thetime-stepping algorithmfor the nonlinear equation. Numerical experiments exhibit appropriate convergence orders and arobust behavior. Adaptive mesh refinement proves to be efficient and accurate to tackle test cases with singularities. Inparticular, we consider equations with exact solutionswith singularitieson the boundary of the domain, or with right-hand sides involving Dirac functions