Least Squares for Second Order Elliptic Problems

In this paper we introduce and analyze two least squares methods for second order elliptic di erential equations with mixed boundary conditions These methods extend to problems which involve oblique derivative boundary conditions as well as nonsym metric and inde nite problems as long as the original problem has a unique solution With the methods to be developed Neumann and oblique boundary conditions are im posed weakly and thus avoid compatiblity conditions on the nite element subspaces The resulting least squares approximations are unconditionally stable no conditions on the step size h and will be shown to converge at an optimal rate The rst least squares method involves a discrete computable H norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise L norm of the residual over the nite elements This method is developed without the introduction of additional problem variables The second method involves the use of the ux as an additional unknown Although this method is similar to the least squares method for rst order systems introduced in it di eres in that discontinuous nite elements are allowed It is also more general in that it extends to the oblique boundary problem