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A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for one- dimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is obtained by solving a matrix-free problem. An application in elasticity coupling the two operators is presented. We consider a beam subject to a combination of tensile and bending loads, where the main goal is the stress critical point determination for an intramedullary nail.This research was financed by FEDER funds through Programa Operacional Factores de Competitividade-COMPETE and by Portuguese Funds through FCT-Fundacao para a Ciencia e a Tecnologia, within the projects PEst-C/MAT/UI0013/2014 and PTDC/MAT/121185/2010 via a grant for research initiation (BII) awarded by the Centre of Mathematics of the School of Science at the University of Minho, Portugal. G.J. Machado and S. Clain were financed by Programa Pessoa Cooperacao Transnacional Portugal/Franca Fundacao para a Ciencia e a Tecnologia with the reference 130631467126259

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation