Least-Squares Methods for Mixed-Type Equations

A least-squares method for the numerical solution of a class of second-order mixed-type partial differential equations is analyzed. The method offers a unified computational treatment of both elliptic and hyperbolic parts of a given domain. Moreover, it has the advantage over other methods that its implementation is independent of specific properties of a given problem. Convergence of the approximation is proved in a natural norm assuming smooth solutions. A particular least-squares method is implemented using bicubic spline polynomials and numerical results for test problems are presented.