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Least-squares finite element methods for the Poisson equation and their connection to the Dirichlet and Kelvin principles

Max Gunzburger
Mathematics and School of Computational Science, Florida State University


Least-squares finite element methods for first-order formulations ofthe Poisson equation are not subject to the inf-sup condition andlead to stable solutions even when all variables are approximated byequal-order continuous finite element spaces. For such elements, onecan also prove optimal convergence in the ``energy' norm for allvariables and optimal L^2 convergence for the scalar variable.However, showing optimal L^2 convergence for the flux has proven tobe impossible without adding the redundant curl equation to the first-order system of partial differential equations. In fact, numericalevidence strongly suggests that nodal continuous flux approximationsdo not posses optimal L^2 accuracy. In this talk, we show thatoptimal L^2 error rates for the flux can be achieved without the curlconstraint, provided that one uses the div-conforming family ofBrezzi-Douglas-Marini or Brezzi-Douglas-Duran-Fortin elements. Then,we proceed to reveal an interesting connection between a least-squares finite element method involving div-conforming fluxapproximations and mixed finite element methods based on theclassical 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 theKelvin and Dirichlet principles. Our principal conclusion is thatwhen implemented in this way, a least-squares finite element methodcombines the best computational properties of mixed-Galerkin finiteelement methods based on each of the classical principles.

Thursday, May 17, 2007
11:00AM AP&M 5402