On the finite element-least squares approximation to higher order elliptic systems
The paper deals with the least square approximation of the biharmonic equation with homogeneous Dirichlet boundary conditions. The problem is transformed into the first order system for variables \(\phi_ 0,u_ 0,\phi_ 1,u_ 1\), where \(\phi_ 0\) is the solution of the original biharmonic problem, \(u_ 0=\nabla \phi_ 0\), \(\phi_ 1=\Delta \phi_ 0\), \(u_ 1=\text{grad} \Delta \phi_ 0\) (the generalization to the \(2m^{th}\) order case is given in the last section). The first order system is then solved by means of the least squares method over suitable finite element spaces. Error estimates in appropriate norms for all above listed variables are derived. The theory presented in the paper holds under the standard regularity assumptions (data as well as the domain are smooth).
scientific article; zbMATH DE number 3963138
scientific article