$L^2(H^1_\gamma)$ Finite Element Convergence for Degenerate Isotropic Hamilton-Jacobi-Bellman Equations

TL;DR

This paper proves the strong convergence of monotone finite element methods in a weighted Sobolev space for fully nonlinear Hamilton-Jacobi-Bellman equations with degenerate, isotropic diffusions, without requiring uniform parabolicity.

Contribution

It establishes the first convergence results in a weighted Sobolev space for degenerate HJB equations using finite element methods, relaxing previous uniform parabolicity assumptions.

Findings

Convergence in weighted Sobolev space $L^2(H^1_eta)$ is achieved.

Finite element solutions approximate viscosity solutions accurately.

Results apply to degenerate, isotropic diffusion cases.

Abstract

In this paper we study the convergence of monotone $P1$ finite element methods for fully nonlinear Hamilton-Jacobi-Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space $L^2(H^1_\gamma(\Omega))$ to the viscosity solution without assuming uniform parabolicity of the HJB operator.