FETI-DP domain decomposition methods for elasticity with structural changes: $P$-elasticity

Klawonn, Axel; Neff, Patrizio; Rheinbach, Oliver; Vanis, Stefanie

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 3, pp. 563-602.

We consider linear elliptic systems which arise in coupled elastic continuum mechanical models. In these systems, the strain tensor ε_P := sym (P^-1∇u) is redefined to include a matrix valued inhomogeneity P(x) which cannot be described by a space dependent fourth order elasticity tensor. Such systems arise naturally in geometrically exact plasticity or in problems with eigenstresses. The tensor field P induces a structural change of the elasticity equations. For such a model the FETI-DP method is formulated and a convergence estimate is provided for the special case that P^-T = ∇ψ is a gradient. It is shown that the condition number depends only quadratic-logarithmically on the number of unknowns of each subdomain. The dependence of the constants of the bound on P is highlighted. Numerical examples confirm our theoretical findings. Promising results are also obtained for settings which are not covered by our theoretical estimates.

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