Some aspects of numerical realization of the optimal finite-element method for a biharmonic problem with boundary conditions of the second type (Q1968524)

The variational problem under approximation is connected with the equation \(\Delta^2 u=f\) and rectangular grids. The first iteration of the method is a standard finite element approximation with bipolynomial spline spaces (Hermitian cubic polynomials are used). The authors discuss the possibility to decrease the obtained approximation of the minimal value of the energy functional via passing the new basic functions. For them the authors suggest a system of ordinary differential equations and give analytical expressions. For the new spaces it is necessary to solve a new system on the grid. Convergence of the iterations is not proved but the authors consider the method as ``comparable'' with traditional ones (they and the grids in experiments are not specified).