Multilevel preconditioning for the finite volume method (Q2894513)

The authors consider the preconditioning of linear systems which result from the finite volume method (FVM) for elliptic boundary value problems. First, they precondition the operator induced by the FVM bilinear form, then they precondition the corresponding finite volume matrix. With the help of the interpolation operator from the trial space to the test space of the FVM and the operator induced by the FVM bilinear form, they show that both wavelet preconditioners and multilevel preconditioners designed originally for the finite element method (FEM) of a boundary value problem can be used to precondition the FVM of the same boundary value problem. The uniform boundedness of the condition numbers of the preconditioned operator and the preconditioned matrix is shown for both methods.NEWLINENEWLINEIn the case of two-dimensional elliptic equations with discontinuous coefficients the authors prove that the FVM bilinear form using linear elements for the trial space for solving the equation can be expressed as the linear FEM bilinear form for solving the same equation with a small perturbation. Numerical examples are presented.