Volume 53, Number 2, March–April 2019
|Page(s)||351 - 374|
|Published online||16 April 2019|
Finite element approximation of the Isaacs equation
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
2 Department of Mathematics, Rutgers University, Piscataway, NJ, USA
* Corresponding author: firstname.lastname@example.org
Accepted: 28 October 2018
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.
Mathematics Subject Classification: 65N12 / 65N15 / 65N30 / 35J60 / 35D40 / 35Q91
Key words: Fully nonlinear equations / discrete maximum principle / finite elements
© EDP Sciences, SMAI 2019
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