Numerical Solution of Elliptic Differential Equations by Reduction to the Interface This is the first book that deals systematically with the numerical solution of elliptic partial differential equations by their reduction to the interface via the Schur complement. Inheriting the beneficial features of finite element boundary element and domain decomposition methods our approach permits solving iteratively the Schur complement equation with linear logarithmic cost in the number of the interface degrees of freedom. The book presents the detailed analysis of the efficient data sparse approximation techniques to the nonlocal Poincaré Steklov interface operators associated with the Laplace biharmonic Stokes and Lamé equations. Another attractive topic are the robust preconditioning methods for elliptic equations with highly jumping anisotropic coefficients. A special feature of the book is a unified presentation of the traditional iterative substructuring and multilevel methods combined with modern matrix compression techniques applied to the Schur complement on the interface