Finite element approximation of the Isaacs equation

Abner J. Salgado, Wujun Zhang

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

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 (h|log h|)1/2. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.

Original languageEnglish (US)
Pages (from-to)351-374
Number of pages24
JournalESAIM: Mathematical Modelling and Numerical Analysis
Volume53
Issue number2
DOIs
StatePublished - Mar 1 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Modeling and Simulation
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Discrete maximum principle
  • Finite elements
  • Fully nonlinear equations

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