A Monotone Discretization for Integral Fractional Laplacian on Bounded Lipschitz Domains: Pointwise Error Estimates under Hölder Regularity

2022/11/23 by Rubing Han, Shuonan Wu · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · doi:10.1137/21m1448239

Abstract

.We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogeneous Dirichlet boundary condition. The method is inspired by a quadrature-based finite difference method of Huang and Oberman, but is defined on unstructured grids in arbitrary dimensions with a more flexible domain for approximating singular integrals. The scale of the singular integral domain not only depends on the local grid size, but also on the distance to the boundary, since the Hölder coefficient of the solution deteriorates as it approaches the boundary. By using a discrete barrier function that also reflects the distance to the boundary, we show optimal pointwise convergence rates in terms of the Hölder regularity of the data on both quasi-uniform and graded grids. Several numerical examples are provided to illustrate the sharpness of the theoretical results.Keywordsmonotone discretizationbounded Lipschitz domainsunstructured gridspointwise error estimateHölder regularityMSC codes35R1165N0665N1265N15

Citations

• Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach
• Direct and inverse error estimates for finite elements with mesh refinements
• Regularity of the obstacle problem for a fractional power of the laplace operator
• A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations
• Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution
• Discretizations of the Spectral Fractional Laplacian on General Domains with Dirichlet, Neumann, and Robin Boundary Conditions

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