Geodesic
A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation
International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) no. 3, pp. 529-537.

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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.
Keywords: finite volume method, polynomial reconstruction operator, harmonic operator, biharmonic operator, high order method 
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Costa, R.; Machado, G. J.; Clain, S. A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation. International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) no. 3, pp. 529-537. http://geodesic.mathdoc.fr/item/IJAMCS_2015_25_3_a6/

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