A new aspect for choosing collocation points for solving biharmonic equations

This paper considers the Dirichlet problem for the biharmonic equation. Finite volume and multi-grid methods have been proposed for solving this equation but in this paper, a new idea will be investigated. This equation is reduced to a system of integral equations (SIEs). Collocation methods for solving some SIEs with weakly singular kernels are rather complicated so that a poor choice of a set of collocation points may lead to instability. Therefore, the best special grids highly concentrated near the possible singularity points for one of the kernels is investigated. Also, the SIEs is reduced to a convex functional, and a minimal solution of this functional gives the minimum condition number for the SIEs. A parameter of collocation points is defined such that it is related to minimal solution of the functional. On the other hand, the speed of convergence rate depends on selecting the parameter of collocation points and vice versa. Finally, we test our method using several numerical examples and demonstrate its efficiency.