On the least-squares method Original Research Article

It is shown that the theoretical basis of the general least-squares method is the bounded inverse theorem. This explains why the least-squares finite element method (LSFEM) within one mathematical/computational framework without any special treatment can provide numerical solutions for all types of partial differential equations. Error estimates of LSFEM for general and elliptic first-order systems of partial differential equations are established. As an example, the incompressible Stokes equations are considered. The principal part of the Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems. This knowledge is employed to derive all permissible combinations of non-standard boundary conditions for the Stokes equations and to show that under these conditions the Stokes operator is bounded below in the full H1 sense, and hence the corresponding LSFEM is optimal. The numerical results are also given to support this conclusion.