A subdomain Galerkin/Least squares method for first-order elliptic systems in the plane

A finite element method for the approximation of solutions of linear, first-order elliptic systems in two-dimensional domains is considered. The method differs from previous techniques in that the discretization is effected prior to the application of a least squares method. Optimal error estimates in ${\text{\bf H}}^1 (\Omega )$ and ${\text{\bf L}}^2 (\Omega )$-norms are derived for piecewise linear finite element trial spaces. Numerical examples are provided that illustrate the theoretical convergence rates and that indicate that the method is easy to implement and is computationally efficient.