First, second and third order fractional step methods for the three-field viscoelastic flow problem

Abstract

In this paper, three different fractional step methods are designed for the three-field viscoelastic flow problem, whose variables are velocity, pressure and elastic stress. The starting point of our methods is the same as for classical pressure segregation algorithms used in the Newtonian incompressible Navier–Stokes problem. These methods can be understood as an inexact $LU$ block factorization of the original system matrix of the fully discrete problem and are designed at the pure algebraic level. The final schemes allow one to solve the problem in a fully decoupled form, where each equation (for velocity, pressure and elastic stress) is solved separately. The first order scheme is obtained from a straightforward segregation of pressure and elastic stress in the momentum equation, whereas the key point for the second order scheme is a first order extrapolation of these variables. The third order fractional step method relies on Yosida's scheme. Referring to the spatial discretization, either the Galerkin method or a stabilized finite element formulation can be used. We describe the fractional step methods first assuming the former, and then we explain the modifications introduced by the stabilized formulation we employ and that has been proposed in a previous work. This discretization in space shows very good stability, permitting in particular the use of equal interpolation for all variables.