<p>Finite-volume discretization of a quasilinear elliptic problem with state-dependent diffusion induces non-normality in the Jacobian. The derivative of the diffusion coefficient with respect to the state variable is multiplied by the gradient of the state, and the resulting product acts as an effective drift that generates an advection-like skew-symmetric component. This asymmetry reduces the numerical stability of both direct and iterative linear solvers. The present work proposes a preconditioning procedure organized in three stages. The first stage diagnoses the source of asymmetry through the antisymmetric part operator, defined as the difference between a linear operator and its formal adjoint. The second stage employs a commutator-based local indicator to drive r-adaptive mesh relocation via a weighted graph Laplacian. The third stage introduces an edge-wise logarithmic ratio indicator that admits a closed-form equivalence with the sum of a geometric volume ratio and a one-dimensional line integral of the drift-to-diffusion ratio. A discrete Helmholtz–Hodge decomposition splits the indicator into an irrotational component, absorbed by a diagonal scaling, and a solenoidal component, identified with the skew-symmetric component of the rescaled Jacobian and subtracted from the Jacobian within an inexact Newton method. A field-of-values argument bounds the spectrum of the preconditioned operator within a complex disk centered at unity. Numerical verification by the method of manufactured solutions on one- and two-dimensional problems with discontinuous diffusion confirms the absorption of non-normality and the spectral clustering predicted by the field-of-values bound.</p>