On a generalized \(L_2\) projection and some related stability estimates in Sobolev spaces
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.
scientific article; zbMATH DE number 1742797