Global Inexact Newton Multilevel FEM for Nonlinear Elliptic Problems

The paper deals with the multilevel solution of elliptic partial differential equations (PDEs) in a finite element setting: uniform ellipticity of the PDE then goes with strict monotonicity of the derivative of a nonlinear convex functional. A Newton multigrid method is advocated, wherein linear residuals are evaluated within the multigrid method for the computation of the Newton corrections. The globalization is performed by some damping of the ordinary Newton corrections. The convergence results and the algorithm may be regarded as an extension of those for local Newton methods presented recently by the authors. An affine conjugate global convergence theory is given, which covers both the exact Newton method (neglecting the occurrence of approximation errors) and inexact Newton–Galerkin methods addressing the crucial issue of accuracy matching between discretization and iteration errors. The obtained theoretical results are directly applied for the construction of adaptive algorithms. Finally, illustrative numerical experiments with a NEWTON–KASKADE code are documented.