Analysis of least-squares approximations to second-order elliptic problems. I. Finite element method

An L² least-squares finite element method for second-order elliptic problems having non-symmetric diffusion coefficient matrix in two- and three-dimensional bounded domains is proposed and analyzed. The main result is the coercivity estimate of the bilinear form associated with the least-squares functional, which is established by using more direct techniques than that in Ref. [7]. It is proved that the method is not subject to the Ladyzhenskaya-Babuška-Brezzi condition and that the finite element approximation yields a symmetric positive definite linear system with condition number O(h^-2). Optimal error estimates in the H¹(Ω) × H(div; Ω) norm are derived. An equivalent a posteriori error estimator in the above norm is described. Some concluding remarks are also given.