Optimal Local Approximation Spaces for Component-Based Static Condensation Procedures

In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircrafts or ships it is crucial to decrease the number of degrees of freedom on the interfaces, or “ports”, in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method we refer to [I. Babuška and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul., 9 (2011), pp. 373–406]. We introduce a spectral greedy algorithm to generalize the just described procedure to the parameter dependent setting and construct a quasi optimal parameter independent port space. Moreover, it is shown that given a certain tolerance and an upper bound for the ports in the system, the spectral greedy constructs a port space that yields a sc approximation error on a system of arbitrary configuration which is smaller than this tolerance for all parameters in a rich train set. We present our approach for isotropic linear elasticity although the idea is rather general and may be readily applied to any linear coercive problem. Numerical experiments demonstrate the very rapid and exponential convergence both of the eigenvalues and the sc approximation based on spectral modes.