Optimal finite element error estimates and Newton convergence for a quasilinear elliptic problem with mixed boundary conditions

Abstract

The article studies finite element approximations of a quasilinear elliptic heat-conduction problem with inhomogeneous mixed boundary conditions. The conductivity tensor is matrix-valued, anisotropic, possibly nonsymmetric, and dependent on both position and temperature, rendering the problem nonlinear, nonmonotone, and nonpotential. The nonlinear algebraic system arising from the Galerkin discretization is solved using Newton's method, with a posteriori guarantees provided by a computable Newton-Kantorovich criterion and a mesh-dependent stopping rule that ensures that the algebraic error is asymptotically negligible relative to the discretization error. Since the discrete solution need not be unique, we prove that every discrete solution satisfies the optimal convergence rates. The analysis combines mixed-boundary elliptic regularity with an Aubin-Nitsche duality argument adapted to the quasilinear setting. Numerical experiments in two and three dimensions confirm the predicted convergence rates and demonstrate the feasibility of the proposed criterion.