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\iteman{io-port 02220007}
\itemau{Bochev, Pavel; Gunzburger, Max}
\itemti{On least-squares finite element methods for the Poisson equation and their connection to the Dirichlet and Kelvin principles.}
\itemso{SIAM J. Numer. Anal. 43, No. 1, 340-362 (2005).}
\itemab
Summary: 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 $H^1(\Omega)\times H(\Omega,\text{div})$) for all variables and optimal $L^2$ convergence for the scalar variable. However, showing optimal $L^2$ 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 $L^2$ accuracy. We show that optimal $L^2$ 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 $H(\Omega,\text{div})$-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 $L^2$ 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.
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\itemut{least-squares finite element methods; Kelvin principle; Dirichlet principle; BDM spaces; BDDF spaces; RT spaces; mixed methods; Poisson equation}
\itemli{doi:10.1137/S003614290443353X}
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