A method for solving biharmonic-type equation with a singularly occuring small parameter
An iterative algorithm to solve the boundary value problem \(- \varepsilon^ 2 \Delta^ 2 u+\Delta u=f\) in a domain \(\Omega\subset\mathbb{R}^ m\), \(u|_{\partial\Omega} ={{\partial u} \over {\partial n}}\Bigl|_{\partial \Omega}=0\) is proposed. For this purpose, the decomposition into two Poisson problems \[ - \varepsilon^ 2 \Delta u^{k+1}+ u^{k+1}=v^ k, \quad {{\partial u^{k+1}} \over {\partial n}} \biggl|_{\partial\Omega}=0, \quad \Delta v^ k=f, \quad v^ k|_{\partial\Omega} =v^ k_ 0, \] is employed with a special sequence \(v^ k_ 0\) to satisfy the boundary condition \(u|_{\partial\Omega}=0\) as \(k\to \infty\). The convergence is proved only if \(\Omega\) is a ball or parallelepiped. Some numerical tests are presented to demonstrate the computational efficiency of the algorithm.
scientific article; zbMATH DE number 220262