An a posteriori error estimate for a linear-nonlinear transmission problem
in plane elastostatics
Mauricio A. Barrientos, Gabriel N. Gatica and Norbert Heuer
Calcolo 39, 61-86, 2002.
We consider the coupling of dual-mixed finite element and boundary element
methods to solve a linear-nonlinear transmission problem in plane
hyperelasticity with mixed boundary conditions. Besides the displacement
and the stress tensor, we introduce the strain tensor as an additional unknown,
which yields a two-fold saddle point operator equation as the corresponding
variational formulation. We derive a reliable a-posteriori error estimate
that depends on the solution of local Dirichlet problems and on residual terms
on the transmission and Neumann boundaries, which are given in a negative order
Sobolev norm. Our approach does not need the exact Galerkin solution, but any
reasonable approximation of it. In addition, the analysis does not depend on
special finite element or boundary element subspaces. However, for certain
specific subspaces we are able to provide two fully local a-posteriori error
estimates, in which the residual terms are bounded by weighted local
$L^2$-norms. Further, one of the error estimates does not require the explicit
solution of the local problems.