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Monotone two-scale methods for a class of integrodifferential operators and applications

Juan Pablo Borthagaray, Ricardo H. Nochetto, Abner J. Salgado, Céline Torres

TL;DR

The paper develops a monotone two-scale discretization for integrodifferential operators of order $2s$ with $s\in(0,1)$, combining a regularization scale near the origin with a discretization scale on graded meshes to preserve a discrete maximum principle. It establishes pointwise convergence rates for linear and obstacle problems, proves regularity and free-boundary estimates, and derives Hausdorff-distance error bounds for the free boundary. It then extends the framework to a concave fully nonlinear, nonlocal equation, providing a convergent scheme built from a sequence of obstacle-type problems. The approach relies on weighted Hölder regularity, barrier arguments, and carefully graded meshes to handle boundary singularities, delivering robust monotone schemes with provable convergence and meaningful rates.

Abstract

We develop a monotone, two-scale discretization for a class of integrodifferential operators of order $2s$, $s \in (0,1)$. We apply it to develop numerical schemes, and derive pointwise convergence rates, for linear and obstacle problems governed by such operators. As applications of the monotonicity, we provide error estimates for free boundaries and a convergent numerical scheme for a concave fully nonlinear, nonlocal, problem.

Monotone two-scale methods for a class of integrodifferential operators and applications

TL;DR

The paper develops a monotone two-scale discretization for integrodifferential operators of order with , combining a regularization scale near the origin with a discretization scale on graded meshes to preserve a discrete maximum principle. It establishes pointwise convergence rates for linear and obstacle problems, proves regularity and free-boundary estimates, and derives Hausdorff-distance error bounds for the free boundary. It then extends the framework to a concave fully nonlinear, nonlocal equation, providing a convergent scheme built from a sequence of obstacle-type problems. The approach relies on weighted Hölder regularity, barrier arguments, and carefully graded meshes to handle boundary singularities, delivering robust monotone schemes with provable convergence and meaningful rates.

Abstract

We develop a monotone, two-scale discretization for a class of integrodifferential operators of order , . We apply it to develop numerical schemes, and derive pointwise convergence rates, for linear and obstacle problems governed by such operators. As applications of the monotonicity, we provide error estimates for free boundaries and a convergent numerical scheme for a concave fully nonlinear, nonlocal, problem.
Paper Structure (21 sections, 27 theorems, 218 equations, 1 figure)

This paper contains 21 sections, 27 theorems, 218 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Function spaces and their norms
  4. The integrodifferential operator
  5. Two-scale discretization
  6. The regularization scale
  7. The discretization scale
  8. Consistency of interpolation
  9. The linear problem
  10. Regularity
  11. Pointwise error estimates
  12. The obstacle problem
  13. Pointwise error estimates
  14. Regularity of free boundaries
  15. Regular points
  16. ...and 6 more sections

Key Result

Proposition 2

Let $K \colon \mathbb{R}^d \to \mathbb{R}$ be a positive function. Define the operator ${\mathcal{L}}_K$ via Let $w \colon \mathbb{R}^d \to \mathbb{R}$ be such that, in the weak sense, ${\mathcal{L}}_K[w] \geq 0$ almost everywhere in $\Omega$ and $w \geq 0$ in $\Omega^c$. Then, $w \geq 0$ in $\Omega$.

Figures (1)

  • Figure 1: Examples of singular free boundary points where the function $u-\psi$ has strict quadratic growth. We expect quadratic growth in the direction of the red arrows. The case on the right is more degenerate than the case on the left, and worse for the approximation of the free boundary $\Gamma$ (depicted in blue). In both cases we have that $\dim \ker{{\mathbf{A}}} = 1$. A discrete free boundary $\Gamma_{\mathscr{T}}$, which is at a distance ${\mathcal{O}}(\delta(h)^{1/2})$ is depicted in dashed brown.

Theorems & Definitions (71)

  • Definition 1: class ${\mathcal{C}}(\lambda,\Lambda)$
  • Proposition 2: comparison principle
  • proof
  • Theorem 3: interior consistency
  • proof
  • Lemma 4: barrier
  • proof
  • Theorem 5: consistency
  • proof
  • Lemma 6: refined interpolation estimate
  • ...and 61 more