Recent advances in the DtN FE Method

The Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fie lds of application. Its name comes from the fact that it involves the nonlo cal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems gov erned by the Helmholtz equation or by the equations of time-harmonic elasti city. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the author's group and by others. This a rticle is a state-of-the-art review of the method. In particular, it concen trates on two major recent advances: (a) the extension of the DtN finite el ement method to nonlinear elliptic and hyperbolic problems; (b) procedures for localizing the nonlocal DtN map, which lead to a family of finite eleme nt schemes with local artificial boundary conditions. Possible future resea rch directions and additional extensions are also discussed.