  This article deals with the adaptive and approximative computation of the
Lam\'e equations. The equations of linear elasticity are considered as boundary
integral equations and solved in the setting of the boundary element method
(BEM). Using BEM, one is faced with the solution of a system of equations with
a fully populated system matrix, which is in general very costly. Some adaptive
algorithms based on hierarchical matrices and the adaptive cross approximation
are proposed. At first, an adaptive matrix-vector multiplication scheme is
introduced for the efficient treatment of multiplying discretizations with
given data. The strategy, to reach this aim, is to use error estimators and
techniques known from adaptivity. The case of approximating the system matrix
appearing in the linear system of equations with this new type of adaptivity is
also discussed.
