Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

2016 ◽  
Vol 42 (6) ◽  
pp. 1363-1380 ◽  
Author(s):  
Xiaochuan Tian ◽  
Qiang Du ◽  
Max Gunzburger
Keyword(s):  
Fractional Laplacian ◽  
Nonlocal Diffusion ◽  
Bounded Domains ◽  
Diffusion Problems

ON A RESCALED NONLOCAL DIFFUSION PROBLEM WITH NEUMANN BOUNDARY CONDITIONS

2021 ◽  
Vol 10 (8) ◽  
pp. 3013-3022
Author(s):  
C.A. Gomez ◽  
J.A. Caicedo
Keyword(s):  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Neumann Boundary ◽  
Diffusion Problem ◽  
Neumann Boundary Conditions ◽  
Nonlocal Diffusion ◽  
Banach Fixed Point Theorem ◽  
Positive Continuous Function ◽  
Normalization Constant ◽  
Diffusion Problems

In this work, we consider the rescaled nonlocal diffusion problem with Neumann Boundary Conditions \[ \begin{cases} u_t^{\epsilon}(x,t)=\displaystyle\frac{1}{\epsilon^2} \int_{\Omega}J_{\epsilon}(x-y)(u^\epsilon(y,t)-u^\epsilon(x,t))dy\\ \qquad \qquad+\displaystyle\frac{1}{\epsilon}\int_{\partial \Omega}G_\epsilon(x-y)g(y,t)dS_y,\\ u^\epsilon(x,0)=u_0(x), \end{cases} \] where $\Omega\subset\mathbb{R}^{N}$ is a bounded, connected and smooth domain, $g$ a positive continuous function, $J_\epsilon(z)=C_1\frac{1}{\epsilon^N}J(\frac{z}{\epsilon}), G_\epsilon(x)=C_1\frac{1}{\epsilon^N}G(\frac{x}{\epsilon}),$ $J$ and $G$ well defined kernels, $C_1$ a normalization constant. The solutions of this model have been used without prove to approximate the solutions of a family of nonlocal diffusion problems to solutions of the respective analogous local problem. We prove existence and uniqueness of the solutions for this problem by using the Banach Fixed Point Theorem. Finally, some conclusions are given.

scholarly journals Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

2020 ◽  
Vol 150 (5) ◽  
pp. 2682-2718 ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Antonio J. Fernández
Keyword(s):  
Bounded Domain ◽  
Lebesgue Space ◽  
Fractional Laplacian ◽  
Existence Result ◽  
Existence Results ◽  
Real Parameter ◽  
Nonlocal Diffusion ◽  
Smooth Bounded Domain ◽  
Image Position ◽  
Regularity Results

AbstractLet$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form $$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$ where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, $\mu \in L^{\infty}$ and $\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by $$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$ Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

scholarly journals Reduced-order modeling for nonlocal diffusion problems

2016 ◽  
Vol 83 (3) ◽  
pp. 307-327 ◽  
Author(s):  
David R. Witman ◽  
Max Gunzburger ◽  
Janet Peterson
Keyword(s):  
Reduced Order Modeling ◽  
Nonlocal Diffusion ◽  
Reduced Order ◽  
Diffusion Problems

Initial-boundary value problem for a fractional type degenerate heat equation

2018 ◽  
Vol 28 (06) ◽  
pp. 1199-1231
Author(s):  
Gerardo Huaroto ◽  
Wladimir Neves
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Diffusion ◽  
Laplacian Operator ◽  
Initial Boundary Value ◽  
Fractional Laplacian Operator ◽  
Fractional Type

In this paper, we study a fractional type degenerate heat equation posed in bounded domains. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition. The nonlocal diffusion effect relies on an inverse of the [Formula: see text]-fractional Laplacian operator, and the solvability is proved for any [Formula: see text].

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