V-cycle AND F-cycle multimesh methods for the biharmonic problem using the Hsieh-Clough-Tocher element

Abstract

This paper presents an orderly study of multigrid methods for fourth-order differential equations using the finiteelement as Hsieh-Clough-Tocher (HCT) for its discretization. To which is chosen as a model problem to biarmonicproblem of reinforced concrete slab, showing its variational form, the existence and uniqueness of its solution. It isintended that the solution of the problem is regular enough, to do the search is expands in Sobolev spaces of frac-tional index where it is already known that the problem has an elliptical biarmonic regularlyα∈〈12,1]. A study ofthe multigrid methods is using the additive theory for show the convergence of V-cycle and F-cycle methods. Theyadapt the multigrid methods for the discretization of the element HCT and shows that are convergent multigridmethods for partial differential equations of fourth order. It numerical results show the convergence of multi mesh,verifying the theoretical analysis. The results of different multigrid algebraic algorithms are compared, results of thenumerical solution to the problem of reinforced concrete slab are shown, using the finite element multigrid HCT andV-cycle, W-cycle and F-cycle in the solution the resulting linear system, finally some suggestions for improving them.