Asymptotic Mesh Independence of Newton–Galerkin Methods via a Refined Mysovskii Theorem
Peter Deuflhard, Florian Alexandru Potra · SIAM Journal on Numerical Analysis · 1992
The paper presents a theoretical characterization of the often observed asymptotic mesh independence of Newton’s method, which means that Newton’s method applied to discretized operator equations behaves essentially the same for all sufficiently fine discretizations. Unlike previous theoretical approaches, the theory presented here does not need any uniform Lipschitz assumptions, which may be hard to verify in realistic problems or, in some cases, may not hold at all. Moreover, this approach leads to a significantly simpler presentation. The new results are obtained by means of a new, refined Newton–Mysovskii theorem in affine invariant formulation. This theorem will be of interest in a wider context, because it gives both existence and uniqueness of the solution and quadratic convergence for sufficiently good starting points. Attention is restricted to Galerkin approximations even though similar results should hold for finite difference methods—but corresponding proofs would certainly be more technical. As illustrative examples, adaptive one-dimensional collocation methods and two-dimensional multilevel finite element methods are discussed.