A spectral minimum-residual method for 2D Riesz fractional reaction–diffusion equations with weak endpoint singularities

A penalized Levenberg–Marquardt minimum-residual method is proposed for efficiently solving nonlinear two-dimensional Riesz space-fractional reaction–diffusion equations. We discretize time with the second-order backward differentiation formula and approximate space using fractional Jacobi–weighted spectral bases. A boundary-penalty term is embedded into the residual functional, which yields a unified least-squares framework that simultaneously delivers high interior accuracy and weakly enforces Dirichlet data. A Riesz-adapted approximation space is introduced, and an \(L^2(\Omega)\) -stability bound together with corresponding weighted error estimates is derived. Numerical tests show second-order temporal accuracy and rapid spatial error decay for the tested fractional orders. For boundary-limited low-regularity solutions, suitable fractional Jacobi parameters improve the observed convergence. Additional tests with perturbed Dirichlet data indicate that the boundary-penalized residual formulation reduces sensitivity to boundary perturbations. Compared with representative methods at comparable resolutions, the proposed approach attains smaller global errors while maintaining competitive iteration counts and residual reductions.