UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 626 UCD Building, 1250 14th St., Denver

TIME: 2 pm (Refreshments served at 1:45 pm)

DATE: May 3, 1999

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\centerline{\bf
     The 3-field formulation with bubble stabilization:}
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\centerline{\bf error estimates.}
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\centerline{DONATELLA MARINI}
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     We briefly recall the 3-field formulation for domain decomposition
     methods introduced in [1], [2], where we showed that a finite element 
     discretization on compatible grids gives back the global finite element
     solution of the original problem.  In [3] we extended the formulation
     to the case of suitable finite element approximations for the three
     variables on different, incompatible grids, and we proposed a
     stabilization procedure via the use of suitable boundary bubbles.
     Here we prove convergence and error estimates for a discretization by
     piecewise linear finite elements stabilized with boundary bubbles for
     the internal variable $u$, piecewise linear and piecewise constant
     for the bounadry unknonws $\psi$ and $\lambda$ respectively.
     Optimal error bounds are proved in the broken $H^1$ norm  and in $L^2$ 
     for the internal variable $u$, and in suitable weighted $L^2$ norms 
     for the other variables $\lambda$ and $\psi$.
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{\bf References}
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[1] F.~Brezzi and L.~D. Marini, {\it Macro hybrid elements and domain
  decomposition methods}, in Optimization et Controle, J.~{Desideri et al.},
  ed., Toulouse, 1993, C\'EPADU\`Es-Editions, pp.~89--96.
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[2] F.~Brezzi and L.~D. Marini,{\it A three-field domain decomposition method}, 
in Domain Decomposition Methods in Science and
  Engineering, A.~{Quarteroni et al.}, ed., Providence, 1994, AMS, Series CONM
  157, pp.~27--34.
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[3] F.~Brezzi, L.~P. Franca, L.~D. Marini, and A.~Russo, {\it Stabilization
  techniques for domain decomposition methods with non-matching grids}.
In Domain Decomposition Methods in Science and
  Engineering, ($9th$ International Conference,
Bergen, Norway, 1996) P.~{Bi{\o}rstad et al.}, ed., 
Domain Decomposition Press, Bergen, 1998, pp.~1--11.
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[4] C.~Baiocchi, F.~Brezzi, and L.~D. Marini, {\it Stabilization of Galerkin
  methods and applications to domain decomposition}, in Future Tendencies in
  Computer Science, Control and Applied Mathematics, A.~{Bensoussan et al.},
  ed., vol.~653, Lecture Notes in Computer Science, Springer-Verlag, 1992,
  pp.~345--355.
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