Finite Element Methods for Isotropic Isaacs Equations with Viscosity and Strong Dirichlet Boundary Conditions

TL;DR

This paper develops and analyzes a monotone finite element method for solving fully non-linear Isaacs equations with isotropic diffusion, ensuring convergence to viscosity solutions even with complex boundary conditions.

Contribution

It introduces a novel finite element scheme that handles boundary singularities and consistency violations, with proven convergence and efficient solvability.

Findings

Uniform convergence to viscosity solutions demonstrated

Method effectively manages boundary singularities

Finite dimensional systems are well-posed and efficiently solvable

Abstract

We study monotone P1 finite element methods on unstructured meshes for fully non-linear, degenerately parabolic Isaacs equations with isotropic diffusions arising from stochastic game theory and optimal control and show uniform convergence to the viscosity solution. Elliptic projections are used to manage singular behaviour at the boundary and to treat a violation of the consistency conditions from the framework by Barles and Souganidis by the numerical operators. Boundary conditions may be imposed in the viscosity or in the strong sense, or in a combination thereof. The presented monotone numerical method has well-posed finite dimensional systems, which can be solved efficiently with Howard's method.