Least-squares for second-order elliptic problems

In this paper, we introduce and analyze two least-squares methods for second-order elliptic differential equations with mixed boundary conditions. These methods extend to problems which involve oblique derivative boundary conditions as well as nonsymmetric and indefinite problems as long as the original problem has a unique solution. With the methods to be developed, Neumann and oblique boundary conditions are imposed weakly and thus avoid compatibility conditions on the finite 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 first least-squares method involves a discrete, computable H-1-norm of the residual and stabilization terms consisting of the jumps at the interelement boundaries and a weighted elementwise L-2-norm of the residual over the finite elements. This method is developed without the introduction of additional problem variables. The second method involves the use of the flux as an additional unknown. Although this method is similar to the least-squares method for first-order systems introduced in [7], it differs in that discontinuous finite elements are allowed. It is also more general in that it extends to the oblique boundary problem.