An iteration method for the mixed formulation of parameter dependent problems related to the Stokes equations (Q1093450)

As a generalization of the Stokes equations, the following problem is considered: given \(\{\) f,g\(\}\in V\times W\) and \(\epsilon\geq 0\), find \(\{u_{\epsilon},\lambda_{\epsilon}\}\in V\times W\) such that \(Au_{\epsilon}+B^*\lambda_{\epsilon}=f,\) \(Bu_{\epsilon}-\epsilon \lambda_{\epsilon}=g,\) where V and W are real Hilbert spaces, A, B and \(B^*\) are linear bounded operators, \(B^*\) being the adjoint of B, A being non-negative. To investigate the problem, an iteration scheme is used with a theorem on its convergence, the penalty method being used as the basic one. Also, a case is considered when A is symmetric. All the obtained results are shown to be applicable to the finite-element method.