Characterizing mesh independent quadratic convergence of Newton's method for a class of elliptic problems (Q2902719)

Newton's iterative method in combination with finite element discretization constitutes a standard approach in the numerical treatment of nonlinear elliptic partial differential equations, mainly due to its super linear local convergence up to quadratic rate. In addition, Newton's method verifies the classical mesh independence principle (MIP): the number of required iterations for some tolerance remains essentially the same, i.e., the discrete iterations exhibit the same convergence behavior as the mesh is refined, and this common convergence is quadratic. This property, however, has been proved only for semilinear problems.NEWLINENEWLINEIn the paper under review, a general class of second order elliptic boundary value problems is considered and it is shown that mesh uniform quadratic estimates cannot be produced unless the principal part of the involved operator is linear. In fact, a weaker property than the MIP is considered: the so-called mesh independence principle for quadratic convergence (MIPQC). Instead of demanding the same quadratic convergence for all meshes, it is only required that the quadratic convergence rate is uniformly bounded as the mesh is refined. Even in this case it only holds if and only if the principal part of the operator is linear (i.e., the elliptic equation is semilinear). This negative result is not due to the lack of smoothness or the space where the analysis takes place, but it is inherent for the class of the considered problems.