On a two-level Newton-type procedure applied for solving non-linear elasticity problems

Abstract

The present work is devoted to the damped Newton method applied for solving a class of non‐linear elasticity problems. Following the approach suggested in earlier related publications, we consider a two‐level procedure which involves (i) solving the non‐linear problem on a coarse mesh, (ii) interpolating the coarse‐mesh solution to the fine mesh, (iii) performing non‐linear iterations on the fine mesh. Numerical experiments suggest that in the case when one is interested in the minimization of the L2‐norm of the error rather than in the minimization of the residual norm the coarse‐mesh solution gives sufficiently accurate approximation to the displacement field on the fine mesh, and only a few (or even just one) of the costly non‐linear iterations on the fine mesh are needed to achieve an acceptable accuracy of the solution (the accuracy which is of the same order as the accuracy of the Galerkin solution on the fine mesh). Copyright © 2000 John Wiley & Sons, Ltd.

A two-grid discretization method for decoupling systems of partial differential equations

“…There are also many other authors who have used this method for many different applications. See, for example, Axelsson et al [2,3,4], Girault and Lions [7], Layton et al [9,10,11], and Utnes [15].…”

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

“…We note that some concrete measurements or explicit expressions on k and µ are given, e.g., in [2,36,38], and k is often considered as constant. With the notations of (4.52) and (4.55), (4.77) and (4.82) imply the analogue of (4.55): (where κ > 0), see more details, e.g., in [7,13,36].…”

Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems

“…The adaptivity of the multilevel approach makes it an efficient realization of Newton's method, which produces convergence to the exact solution u * of the boundary value problem, see [8][9][10][11].…”

“…(We note that in many pratical cases the function k is close to or equal to a constant on corresponding subdomains [9].) Let us consider the weak formulation (9) of our problem:…”

“…The foundation of this model is given in the monograph [29], its further discussion and various numerical approaches are found, e.g. in [9,10,19].…”