12 research outputs found
Local convergence of the FEM for the integral fractional Laplacian
We provide for first order discretizations of the integral fractional
Laplacian sharp local error estimates on proper subdomains in both the local
-norm and the localized energy norm. Our estimates have the form of a
local best approximation error plus a global error measured in a weaker norm
Finite element methods for fractional-order diffusion problems with optimal convergence order
A convergence result is stated for the numerical solution of space-fractional diffusion problems. For the spatial discretization, an arbitrary family of finite elements can be used combined with the matrix transformation technique. The analysis covers the application of the implicit Euler method for time integration to ensure unconditional stability. The spatial convergence rate does not depend on the fractional power of the Laplacian operator. An efficient numerical implementation is developed avoiding the direct computation of matrix powers
Numerical approximations for a fully fractional Allen-Cahn equation
A finite element scheme for an entirely fractional Allen-Cahn equation with
non-smooth initial data is introduced and analyzed. In the proposed nonlocal
model, the Caputo fractional in-time derivative and the fractional Laplacian
replace the standard local operators. Piecewise linear finite elements and
convolution quadratures are the basic tools involved in the presented numerical
method. Error analysis and implementation issues are addressed together with
the needed results of regularity for the continuous model. Also, the asymptotic
behavior of solutions, for a vanishing fractional parameter and usual
derivative in time, is discussed within the framework of the Gamma-convergence
theory
Weighted analytic regularity for the integral fractional Laplacian in polyhedra
We prove weighted analytic regularity of solutions to the Dirichlet problem
for the integral fractional Laplacian in polytopal three-dimensional domains
and with analytic right-hand side. Employing the Caffarelli-Silvestre extension
allows to localize the problem and to decompose the regularity estimates into
results on vertex, edge, face, vertex-edge, vertex-face, edge-face and
vertex-edge-face neighborhoods of the boundary. Using tangential
differentiability of the extended solutions, a bootstrapping argument based on
Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides
control of higher order derivatives.Comment: arXiv admin note: text overlap with arXiv:2112.0815