Finite element approximation for the Bessel (p,s)

We study the finite element approximation of the Dirichlet problem for the Bessel (p, s)-Laplacian, divs |∇su| p−2∇su
, 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 6= 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.