A two-parameter defect-correction method for computation of steady-state viscoelastic fluid flow

Abstract

The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge. In this paper a two-parameter defect-correction method for viscoelastic fluid flow is presented and analyzed. In the defect step the Weissenberg number is artificially reduced to solve a stable nonlinear problem. The approximation is then improved in the correction step using a linearized correction iteration. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method.

Computation of viscoelastic fluid flows using continuation methods

“…The numbers of steps and steplength reductions required for each method for computations with a = 1 on mesh M1 are reported in Table 5 temporally unsteady solutions) exist beyond some critical value of λ. The same observation was also made during numerical experiments with a defect-correction method for viscoelasticity [16]. Trebotich, et al [45] also observed this situation and suggested that the wave-like behavior is related to the elastic Mach number.…”

“…The defect-correction method has been applied to steady-state viscoelastic flows [17,16,32] for high Weissenberg number. In their approach, the defect step consisted of a nonlinear iteration in which the Weissenberg number was replaced with an artificially reduced value, and the correction step sought to improve on the approximation found in the defect step.…”

Modified characteristics mixed defect-correction finite element method for the time-dependent Navier–Stokes problems

“…Due to its good efficiency, there are many works devoted to this method, e.g. the convection-diffusion equation, [24] adaptive refinement for the convection-diffusion problems, [25] adaptive defect-correction methods for the viscous incompressible flow, [26] two-parameter defect-correction method for computation of the steady-state viscoelastic fluid flow, [27] variational methods for the elliptic boundary value problems, [28] defect-correction parameter-uniform numerical method for a singularly perturbed convection-diffusion problem, [29] the convection-dominated flow, [30] finite volume local defect-correction method for solving the transport equation, [31] the singular initial value problems, [32] the time-dependent Navier-Stokes equations, [33][34][35] the stationary Navier-Stokes equation, [36] second-order defect-correction scheme, [37] finite element eigenvalues with applications to quantum chemistry, [38] and so on. In [28], a method which makes it possible to apply the idea of iterated defect correction to finite element methods was given.…”

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

“…Because it is highly efficient, there are many works devoted to this method, e.g. the convection-diffusion equation [17], adaptive refinement for the convection-diffusion problems [2], adaptive defect correction methods for the viscous incompressible flow [18], two-parameter defectcorrection method for computation of the steady-state viscoelastic fluid flow [20], variational methods for the elliptic boundary value problems [21], defectcorrection parameter-uniform numerical method for a singularly perturbed convection-diffusion problem [28], the convection-dominated flow [33], finite volume local defect correction method for solving the transport equation [34], the singular initial value problems [35], the time-dependent Navier-Stokes equations [36,37,41], the stationary Navier-Stokes equation [39], second order defect correction scheme [42], finite element eigenvalues with applications to quantum chemistry [46] and so on. In [21], Frank et al give a method which makes it possible to apply the idea of iterated defect-correction to finite element methods.…”