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Paper IPM / M / 8110  


Abstract:  
This paper considers the Dirichlet problem for the biharmonic
equation. Finite volume and multigrid 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 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.
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