Multilevel Preconditioning for Finite Volume Element Methods

We consider precondition of linear systems resulting from the finite volume element method (FVEM) for elliptic boundary value problems. With the help of the interpolation operator from a trial space to a test space and the operator induced by the FVEM bilinear form, we show that both wavelet preconditioners and multilevel preconditioners designed originally for the finite element method for the same boundary value problems can be used to precondition the finite volume element matrices. We prove that such preconditioners ensure that the resulting coefficient matrix has a uniformly bounded condition number. We also present four numerical examples to confirm the theoretical result.