Monotone Iterations for Elliptic Variational Inequalities

A wide range of free boundary problems occurring in engineering and industry can be rewritten as a minimization problem for a strictly convex, piecewise smooth but non–differentiable energy functional. The fast solution of related discretized problems is a very delicate question, because usual Newton techniques cannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex min- imization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the conver- gence speed. We present a general convergence theory and discuss several applications.

[1]  C. M. Elliott,et al.  Weak and variational methods for moving boundary problems , 1982 .

[2]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[3]  J. Rodrigues.  Obstacle Problems in Mathematical Physics , 1987 .

[4]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[5]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[6]  P. Deuflhard.  A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting , 1974 .

[7]  P. Deuflhard,et al.  Global Inexact Newton Multilevel FEM for Nonlinear Elliptic Problems , 1998 .

[8]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[9]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities II , 1996 .

[10]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[11]  J. Crank.  Free and moving boundary problems , 1984 .

[12]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities I , 1994 .

[13]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[14]  A. Friedman.  Variational principles and free-boundary problems , 1982 .

[15]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[16]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[17]  R. Kornhuber.  Adaptive monotone multigrid methods for nonlinear variational problems , 1997 .