Using the boundary element method to solve mixed boundary value problems, we only consider the unknowns on the boundary, resulting in a reduction of spatial dimension, but also in a non-sparse operator. Consequently, in order to reduce storage, the involved integral operators need to be approximated. This is done by Hierarchical matrices. The standard formulation of the boundary integral equations involves a hypersingular integral operator. In order to avoid the time-consuming computation of this operator, we can also use a mixed approximation, which was introduced by Olaf Steinbach for the Laplace equation. Our aim is the application of this mixed formulation to the Lamé equation from linear elasticity in order to speed up the simulation.