The Least-Squares Spectral Element Method (LSSEM)

For equations with self-adjoint and positive definite operators, the Galerkin formulation results in the same system of equations as in the Rayleigh-Ritz formulation. The Galerkin method does not exhibit the best approximation property. In practice, solution to convection dominated transport problems by the Galerkin method are often corrupted by spurious oscillations or wiggles. The classical Galerkin method also behaves poorly for high-speed compressible flows and shallow water wave problems. Notorious difficulties arise even for the solution of elliptic problems by the Galerkin mixed method. One example is incompressible, irrotational flow problems governed by the Laplace or Poisson equations of the potential.
Another well-known example is viscous flow problems governed by the incompressible Navier-Stokes equations. It is not an exaggeration to say that during the past three decades much of research in fluid mechanics was devoted to the modification and improvement of the Galerkin method. The achievement of mathematical analysis of the Galerkin method is remarkable; however, the application of the Galerkin method in CFD and computational electro-magnetics (CEM) has not produced entirely satisfactory results. So, for non-self-adjoint systems, the application of LSSEM might be the right direction to go. The least-squares principle can offer Universality, Effciency, Robustness, Optimality and General-purpose coding.
In this talk I will first discuss the preliminaries of LSSEM and then its applications in solving parabolic and hyperbolic problems.