On Least-Squares Finite Element Methods for the Poisson Equation and Their Connection to the Dirichlet and Kelvin Principles

Least-squares finite element methods for first-order formulations of the Poisson equation are not subject to the inf-sup condition and lead to stable solutions even when all variables are approximated by equal-order continuous finite element spaces. For such elements, one can also prove optimal convergence in the "energy" norm (equivalent to a norm on $H^{1}(\Omega)\times H(\Omega,\mbox{\rm div})$) for all variables and optimal L2 convergence for the scalar variable. However, showing optimal L2 convergence for the flux has proven to be impossible without adding the redundant curl equation to the first-order system of partial differential equations. In fact, numerical evidence strongly suggests that nodal continuous flux approximations do not posses optimal L2 accuracy. In this paper, we show that optimal L2 error rates for the flux can be achieved without the curl constraint, provided that one uses the div-conforming family of Brezzi--Douglas--Marini or Brezzi--Douglas--Duran--Fortin elements. Then, we proceed to reveal an interesting connection between a least-squares finite element method involving $H(\Omega,\mbox{\rm div})$-conforming flux approximations and mixed finite element methods based on the classical Dirichlet and Kelvin principles. We show that such least-squares finite element methods can be obtained by approximating, through an L2 projection, the Hodge operator that connects the Kelvin and Dirichlet principles. Our principal conclusion is that when implemented in this way, a least-squares finite element method combines the best computational properties of finite element methods based on each of the classical principles.