On some nonlocal, nonlinear diffusion problems

TL;DR

The paper analyzes nonlocal, nonlinear elliptic diffusion problems where coefficients depend on a nonlocal quantity $\ell(u)$ and reformulates the computation as a fixed-point problem in $\mathbb{R}$ via $\mu=\ell(u_{\mu})$. It establishes existence (and, in the Lipschitz case with $\lambda=0$, conditional uniqueness) and develops a Galerkin discretization with a fixed-point iteration, accompanied by a rigorous error analysis based on Céa’s lemma and Aubin–Nitsche arguments. The authors provide detailed convergence results for the fixed-point iterations, including a comprehensive numerical study in 1D that exhibits convergent, bounded-divergent, and unbounded-divergent regimes, and demonstrate linear convergence in the spatial discretization together with quadratic superconvergence for the fixed-point error. The work yields practical, provably convergent algorithms for nonlocal diffusion, with explicit error bounds and insights into stable and unstable parametric regimes, complemented by an appendix on a modified fixed-point scheme.

Abstract

This note is devoted to some nonlocal, nonlinear elliptic problems with an emphasis on the computation of the solution of such problems, reducing it in particular to a fixed point argument in R. Errors estimates and numerical experiments are provided.

Figures (8)

• Figure 1: Illustration of the left- and right-hand side of the fixed point equation $x=G\left( x\right)$ under the assumptions stated in (\ref{['mc17']}).
• Figure 2:
• Figure 3:
• Figure 4: Convergence of the fixed point iteration for a mesh with $h=2^{-13}$. Left: The iterations $\mu_{h,n}$ are converging rapidly to the discrete fixed point $\mu_{h}$ which is very close to $\mu _{0}=1$ for this fine mesh. Right: The semilog plot of the error illustrates the exponential convergence of the fixed point iteration.
• Figure 5: Divergent fixed point iteration for the parameter choice (\ref{['stas_div_bound']}) and starting guess $\mu_{h,0}:=\nu_{1}$ such that the iterates alternate between two values.

Theorems & Definitions (21)

• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• Corollary 2.5
• Example 2.6
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Theorem 3.4