This paper studies L 2 norm error estimates for the div least-squares method for which the associated homogeneous least-squares functional is equivalent to the H(div) × H 1 norm for the respective dual and primal variables. Least-squares of this type for the second-order elliptic equations, elasticity, and the Stokes equations are an active area of research, and error estimates in the H(div) × H 1 norm were previously established. In this paper, we establish optimal L 2 norm error estimates for the primal variable under the minimum regularity requirement through a refined duality argument. Introduction.There is substantial interest in the use of least-squares principles for the approximate solution of partial differential equations with applications in both solid and fluid mechanics. One advantage of the least-squares approach is that the finite element spaces for the individual unknowns may be chosen independently, and thus based on simplicity, availability, and optimality, or may be chosen from the physics of the underlying problem. Moreover, the linear systems of algebraic equations resulting from well-posed least-squares discretizations are always self-adjoint and positive definite.Many least-squares methods for scalar elliptic equations, elasticity, and the Stokes equations have been proposed and analyzed. The numerical properties depend on the form of the first-order system and the choice of the least-squares norm. Loosely speaking, there are three types of least-squares methods: the inverse approach, the div approach, and the div-curl approach. The inverse approach employs an inverse norm that is further replaced with either the weighted mesh-dependent norm (see [2]) or the discrete H −1 norm (see [4]) for computational feasibility. The div approach uses the L 2 norm, and the corresponding homogeneous least-squares functional is equivalent to the H(div) × H 1 norm. The homogeneous least-squares functional from the div-curl approach is equivalent to the H(div) ∩ H(curl) norm for some variables.The purpose of this paper is to study the L 2 norm error estimates for the div least-squares method. For the scalar elliptic equations, the div approach based on the flux-pressure formulation has been studied extensively (see, e.g., [3,6,9,10,12,13,14,15,16,17,18]). The pressure and the flux are referred to as the primal and dual variables, respectively. For elasticity and the Stokes equations, we recently proposed and analyzed the div least-squares approach in [7,8] based on the stress-

SUMMARYWe consider an inverse problem of identifying the boundary shape of a domain, where the temperature ÿeld is dominated by natural convection, from temperature measurements on the other boundary. The potential applications of the present investigations are the determination of a phase change isotherm in the Bridgman crystal growth or the thermal tomography which detects aws in materials nondestructively. After mapping the irregular domains into a reference one employing a set of parameters, the inverse problem is formulated as a parameter optimization problem which is solved by a conjugate gradient method. The present method is found to identify the domains reasonably accurately even with noisy temperature measurements.

This study present a short proof concerning the coercivity of a first-order least-squares finite element method for general second-order elliptic problems proposed by Cai, Lazarov, Manteuffel and McCormick (Cai et al. J Numer Anal 31 (1994), 1785-1799). Our proof is based on a priori estimate and the technique can be applied to prove L 2 -norm error estimate for the primary function u. After establishing the coercivity bound from the assumed a priori estimate, we observe that the coercivity bound is actually equivalent to the a priori estimate.

Abstract. In this paper, we consider a least-squares method proposed by Bramble, Lazarov and Pasciak (1998) which can be thought of as a stabilized Galerkin method for noncoercive problems with unique solutions. We modify their method by weakening the strength of the stabilization terms and present various new error estimates. The modified method has all the desirable properties of the original method; indeed, we shall show some theoretical properties that are not known for the original method. At the same time, our numerical experiments show an improvement of the method due to the modification.

No abstract