Partition of unity for localization in implicit a posteriori finite element error control for linear elasticity

The partition of unity for localization in adaptive finite element method (FEM) for elliptic partial differential equations has been proposed in Carstensen and Funken (SIAM J. Sci. Comput. 2000; 21: 1465–1484) and is applied therein to the Laplace problem. A direct adaptation to linear elasticity in this paper yields a first estimator L based on patch-oriented local-weighted interface problems. The global Korn inequality with a constant CKorn yields reliability |u − uh| CKorn L for any finite element approximation uh to the exact displacement u. In order to localize this inequality further and so to involve the global constant CKorn directly in the local computations, we deduce a new error estimator L. The latter estimator is based on local-weighted interface problems with rigid body motions (RBM) as a kernel and so leads to effective estimates only if RBM are included in the local FE test functions. Therefore, the excluded first-order FEM has to be enlarged by RBM, which leads to a partition of unit method (PUM) with RBM, called P1 +RBM, or to second-order FEMs, called P2 FEM. For P1 +RBM and P2 FEM (or even higher-order schemes) one obtains the sharper reliability estimate |u − uh| L. Efficiency holds in the strict sense of L |u − uh||. The local-weighted interface problems behind the implicit error estimators L and L are usually not exactly solvable and are rather approximated by some FEM on a refined mesh and/or with a higherorder FEM. The computable approximations ˜ L L and ˜ L L are shown to be reliable in the sense of |u − uh| min{ ˜ L, ˜ L}+osc. The oscillations osc are known functions of the given data and higher-order terms if the data are smooth for first-order FEM. The mathematical proofs are based on weighted Korn inequalities and inverse estimates combined with standard arguments. The numerical experiments for uniform and adapted FEM on benchmarks such as an L-shape problem, Cook’s membrane, or a slit problem validate the theoretical estimates and also concern numerical bounds for CKorn and the locking phenomena. Copyright q 2007 John Wiley & Sons, Ltd