Convergence of the Neumann series in BEM for the Neumann problem of the Stokes system
The boundary value problem to the Stokes system is considered in a bounded domain \(\Omega\subset\mathbb{R}^m\) with connected Lipschitz boundary \(\Gamma\) \[ \begin{cases} \Delta u-\nabla p=0,\quad \text{div}\,u=0&\text{in}\;\Omega,\\ \mathbb{T}(u,p)n=g &\text{on}\;\partial\Omega=\Gamma. \end{cases} \] Here \(u\) is the velocity field, \(p\) is the pressure, \(\mathbb{T}(u,p)\) is the stress tensor, \[ \mathbb{T}(u,p)=[\nabla u +(\nabla u)^T]-pI, \] \(n\) is the unit outward normal on \(\Gamma\). The author formulates the condition to the vector \(g\) which is necessary and sufficient for the existence of a weak solution to the problem. The problem is reduced to the operator (integral) equation in the space \(H^{-1/2}(\Gamma)\). The solution of this equation might be found by successive approximations. The numerical realization of successive approximations is discussed in the paper.
scientific article; zbMATH DE number 5988271
scientific article