Finite element approximation of the Isaacs equation

2015/12/30 by Abner J. Salgado, Salgado, Abner J., Wujun Zhang +1 · 2 citations
Computer Science · Engineering · Mathematics · #35D40 #35J60 #35Q91 #65N12 #65N15 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1512.09091

openalex publication_date 2015/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε, h→0, and ε \gtrsim h^1/2|log h|. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.

Citations

• Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form

Cited by

• Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form
• Numerical analysis of strongly nonlinear PDEs

Related

• Finite Element Methods for Isotropic Isaacs Equations with Viscosity and Strong Dirichlet Boundary Conditions
• To the theory of viscosity solutions for uniformly elliptic Isaacs equations
• On the rate of convergence of finite-difference approximations for elliptic Isaacs equations in smooth domains
• To the theory of viscosity solutions for uniformly parabolic Isaacs equations
• Comparison results for the Stokes equations