Global Inexact Multilevel FEM for Nonlinear Elliptic Problems

Please always quote using this URN: urn:nbn:de:0297-zib-2430
  • The paper deals with the multilevel solution of {\em elliptic} partial differential equations (PDEs) in a {\em finite element} setting: {\em uniform ellipticity} of the PDE then goes with {\em strict monotonicity} of the derivative of a nonlinear convex functional. A {\em Newton multigrid method} is advocated, wherein {\em linear residuals} are evaluated within the multigrid method for the computation of the Newton corrections. The globalization is performed by some {\em 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 {\em affine conjugate} global convergence theory is given, which covers both the {\em exact} Newton method (neglecting the occurrence of approximation errors) and {\em 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~{\sf NEWTON--KASKADE} code are documented.

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Author:Peter Deuflhard, Martin WeiserORCiD
Document Type:ZIB-Report
Date of first Publication:1996/10/14
Series (Serial Number):ZIB-Report (SC-96-33)
Published in:Appeared in: W. Hackbusch, G. Wittum (eds.) Multigrid Methods. Lecture Notes in Computational Science and Engineering 3. Springer, 1998, pp. 71-89