Abstract
In this paper, we propose an optimal least-squares finite element method for 2D and 3D elliptic problems and discuss its advantages over the mixed Galerkin method and the usual least-squares finite element method. In the usual least-squares finite element method, the second-order equation, −▽·(▽ u) + u = f, is recast as a first-order system, −▽· p + u = f, ▽u - p = 0 . Our error analysis and numerical experiments show that, in this usual least-squares finite element method, the rate of convergence for flux p is one-order lower than optimal. In order to get an optimal least-squares method, the irrotationality ▽ x p = 0 should be included in the first-order system.
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