Numerical Solution of Elliptic Differential Equations by Reduction to the Interface

This is the first book that deals with the numerical solution of elliptic partial differential equations by their reduction to the interface. Inheriting the beneficial features of finite element, boundary element and domain decomposition methods, the new approach exhibits linear-logarithmic complexity in the number of the interface degrees of freedom and it is well suited for parallel computations. Of particular interest to the reader will be the data-sparse approximation techniques to the fundamental Poincaré-Steklov operators for the Laplace, biharmonic, Stokes and Lamé equations as well as the robust preconditioning methods in the case of highly jumping anisotropic coefficients. A special feature of the book is a unified presentation of the traditional finite element, domain decomposition and multilevel methods combined with modern techniques for a data-sparse approximation to the nonlocal operators.