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On a generalized \(L_2\) projection and some related stability estimates in Sobolev spaces. (English) Zbl 0997.65120
The paper deals with the stability estimate needed in hybrid finite and boundary element methods, especially in hybrid coupled domain decomposition methods including mortar finite elements. This stability estimate is equivalent to the stability of a generalized \(L_2\) projection in certain Sobolev spaces. The construction of the dual mesh is described. The stability estimate \[ {1\over c_s} \|u_h\|_{H^s(\Gamma)}\leq \sup_{0\neq W_h\in W_h} {\langle u_h,w_h\rangle\over \|w_h\|_{\widetilde H^{-s}(\Gamma)}}\text{ for all }u_h\in V_h\subset H^s(\Gamma),\;c_s> 0, \] is crucial for the stability analysis of hybrid coupled domain decomposition methods. (Here, \(\Gamma\) is an open or closed Lipschitz’ manifold in \(\mathbb{R}^n\), \(n= 2,3\) and \(V_h\subset H^s(\Gamma)\) with \(s\in (0,1)\) is given finite element space, \(W_h\subset (H^s(\Gamma))^*\).)
Main result: Using piecewise linear trial spaces and appropriate piecewise constant test spaces, the stability of the generalized \(L_2\) projection is proved assuming some mesh conditions locally.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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