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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.
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This content will become publicly available on May 4, 2026
Dirichlet problems in perforated domains
In this paper we establish W1,p estimates for solutions uε to Laplace’s equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, C1 domain Ωε,η in Rd. The bounding constants depend explicitly on two small parameters ε and η, where ε represents the scale of the minimal distance between holes, and η denotes the ratio between the size of the holes and ε. The proof relies on a large-scale Lp estimate for ∇uε, whose proof is divided into two parts. In the first part, we show that as ε,ηapproach zero, harmonic functions in Ωε,η may be approximated by solutions of an intermediate problem for a Schr¨odinger operator in Ω. In the second part, a real-variable method is employed to establish the large-scale Lp estimate for ∇uε by using the approximation at scales above ε. The results are shown to be sharp except in one particular case d≥3 and p= d or d′.
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- Award ID(s):
- 2153585
- PAR ID:
- 10589033
- Publisher / Repository:
- Taylor and Francis
- Date Published:
- Journal Name:
- Communications in Partial Differential Equations
- Volume:
- 50
- Issue:
- 5
- ISSN:
- 0360-5302
- Page Range / eLocation ID:
- 686 to 722
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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