[Paper Review] Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

Óscar Ciaurri, Luz Roncal+3 more|arXiv (Cornell University)|Aug 31, 2016

Nonlinear Partial Differential Equations25 references6 citations

TL;DR

This paper establishes a rigorous framework for nonlocal discrete diffusion equations driven by fractional powers of the discrete Laplacian on a uniform mesh. It derives a pointwise nonlocal formula for the fractional discrete Laplacian, proves its equivalence to a Dirichlet-to-Neumann operator via a semidiscrete extension problem, and demonstrates convergence of solutions to the continuous fractional Laplacian with explicit error estimates in Hölder spaces as the mesh size $ h \to 0 $.

ABSTRACT

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-Δ_h)^su=f, \] for $u,f:\mathbb{Z}_h o\mathbb{R}$, $0

Motivation & Objective

• To develop a comprehensive theory for nonlocal discrete diffusion equations involving fractional powers of the discrete Laplacian on a uniform mesh.
• To establish a pointwise nonlocal formula for $ (-\Delta_h)^s u $ and characterize it as a Dirichlet-to-Neumann operator via a semidiscrete extension problem.
• To prove regularity estimates in discrete H"older spaces, existence and uniqueness for the discrete Dirichlet problem, and derive discrete Sobolev and Poincar\'e inequalities.
• To analyze the convergence of the discrete fractional Laplacian to the continuous fractional Laplacian as $ h \to 0 $, providing uniform error estimates under minimal regularity assumptions.
• To demonstrate that solutions to the continuous Poisson problem for $ (-\Delta)^s U = F $ can be approximated by solutions to the discrete problem with explicit error bounds.

Proposed method

• Define the fractional discrete Laplacian $ (-\Delta_h)^s u $ using the semigroup method via the solution to the semidiscrete heat equation $ \partial_t w = \Delta_h w $.
• Derive a pointwise nonlocal formula for $ (-\Delta_h)^s u_j $ as a weighted sum over lattice points: $ \sum_{m \neq j} (u_j - u_m) K^h_s(j - m) $, with an explicit kernel $ K^h_s(m) $ involving the Gamma function.
• Establish the equivalence of $ (-\Delta_h)^s $ to a Dirichlet-to-Neumann map for a semidiscrete degenerate elliptic extension problem in $ \mathbb{Z}_h \times (0,\infty) $.
• Prove discrete versions of the Hardy--Littlewood--Sobolev inequality, fractional Sobolev embedding, and Poincar\'e inequality for the discrete setting.
• Use the extension problem to derive Schauder-type estimates in discrete H"older spaces and establish existence and uniqueness for the discrete Dirichlet problem.
• Analyze the limit $ h \to 0 $, showing that solutions to the discrete problem converge uniformly to solutions of the continuous Poisson problem $ (-\Delta)^s U = F $ with error bounds depending on $ h $ and the regularity of $ F $.

Experimental results

Research questions

• RQ1 How can fractional powers of the discrete Laplacian be characterized via a nonlocal pointwise formula on a uniform mesh?
• RQ2 What is the precise relationship between the fractional discrete Laplacian and a semidiscrete extension problem, and how does this yield a Dirichlet-to-Neumann characterization?
• RQ3 What regularity properties, such as Schauder estimates, hold for solutions to the discrete nonlocal Dirichlet problem in discrete H"older spaces?
• RQ4 How do solutions to the discrete fractional Poisson problem converge to solutions of the continuous fractional Poisson problem as the mesh size $ h \to 0 $, and what are the error estimates?
• RQ5 What discrete analogues of classical inequalities—such as Sobolev embedding and Poincar\'e inequality—hold for the fractional discrete Laplacian?

Key findings

• A pointwise nonlocal formula for $ (-\Delta_h)^s u_j $ is derived as $ \sum_{m \neq j} (u_j - u_m) K^h_s(j - m) $, with an explicit kernel $ K^h_s(m) $ involving the Gamma function.
• The fractional discrete Laplacian $ (-\Delta_h)^s $ is shown to be equivalent to the Dirichlet-to-Neumann operator for a semidiscrete extension problem in $ \mathbb{Z}_h \times (0,\infty) $.
• Schauder estimates in discrete H"older spaces are established, showing that solutions to the discrete Dirichlet problem inherit H"older regularity from the data.
• The discrete fractional Sobolev embedding and Poincar\'e inequality are proven, which are essential tools for the analysis of the discrete problem.
• The discrete fractional Laplacian converges to the continuous fractional Laplacian in H"older norms as $ h \to 0 $, with error bounds of order $ h^{\alpha} $ for $ \alpha < \min(1, 2s) $, under minimal regularity assumptions.
• Solutions to the continuous Poisson problem $ (-\Delta)^s U = F $ in $ \mathbb{R} $ are approximated by solutions to the discrete problem with uniform error estimates in terms of $ h $, valid for $ F \in C^{0,\alpha} $.