Finite element approximation for the Bessel (p,s)-Laplacian
Abstract
We study the finite element approximation of the Dirichlet problem for the Bessel (p,s)-Laplacian, divs (|∇s u|p−2∇s u), built from the Riesz fractional gradient ∇s and posed on the Bessel potential space Xs,p, obtained by complex interpolation between Lp and W1,p. For p = 2 this operator reduces to the fractional Laplacian, while for p ≠ 2 it differs from the fractional p-Laplacian arising from real interpolation. Since a direct Galerkin discretization requires reassembling a dense stiffness matrix at every nonlinear iteration, we propose an augmented Lagrangian formulation based on the projected fractional gradient Bh = Πh ∇s, in which the only dense matrix is assembled once and the nonlinearity decouples elementwise. We analyze the resulting consistency error, establish a priori convergence rates in Xs,p, conditional on a discrete inf-sup condition for the augmented Lagrangian scheme, and present numerical experiments that illustrate the convergence of the method and its ability to deal with degenerate and strongly nonlinear problems.
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BibTeX
@misc{borthagaray2026finite,
title = {{Finite element approximation for the Bessel $(p,s)$-Laplacian}},
author = {Juan Pablo Borthagaray and José Camilo Rueda Niño},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37747},
eprint = {2609.37747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37747},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite element approximation for the Bessel (p,s)-Laplacian
AU - Juan Pablo Borthagaray
AU - José Camilo Rueda Niño
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37747
UR - https://arxiv.org/abs/2609.37747
AB - We study the finite element approximation of the Dirichlet problem for the Bessel (p,s)-Laplacian, div^s (|∇^s u|^p−2∇^s u), built from the Riesz fractional gradient ∇^s and posed on the Bessel potential space X^s,p, obtained by complex interpolation between L^p and W^1,p. For p = 2 this operator reduces to the fractional Laplacian, while for p ≠ 2 it differs from the fractional p-Laplacian arising from real interpolation. Since a direct Galerkin discretization requires reassembling a dense stiffness matrix at every nonlinear iteration, we propose an augmented Lagrangian formulation based on the projected fractional gradient B_h = Π_h ∇^s, in which the only dense matrix is assembled once and the nonlinearity decouples elementwise. We analyze the resulting consistency error, establish a priori convergence rates in X^s,p, conditional on a discrete inf-sup condition for the augmented Lagrangian scheme, and present numerical experiments that illustrate the convergence of the method and its ability to deal with degenerate and strongly nonlinear problems.
ER -