On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. II: Implementation.

*(English)*Zbl 0712.76036Summary: [For part I, see the foregoing entry (Zbl 0712.76035).]

Ever since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the ‘consistent mass matrix’ has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full ‘velocity-pressure coupling’ is also inherent. The high cost of such (high-quality) FEM calculations led to the development of simpler but ad hoc methods in which the ‘lumped’ mass matrix is employed and the velocity and pressure are uncoupled to the maximum extent possible. Resulting computer codes were less expensive to use but suffered a significant loss of accuracy, caused by lumping the mass when the flow was advection-dominated and accurate transport of ‘information’ was important. In the second part of this paper we re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized.

Ever since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the ‘consistent mass matrix’ has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full ‘velocity-pressure coupling’ is also inherent. The high cost of such (high-quality) FEM calculations led to the development of simpler but ad hoc methods in which the ‘lumped’ mass matrix is employed and the velocity and pressure are uncoupled to the maximum extent possible. Resulting computer codes were less expensive to use but suffered a significant loss of accuracy, caused by lumping the mass when the flow was advection-dominated and accurate transport of ‘information’ was important. In the second part of this paper we re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized.

##### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76M10 | Finite element methods applied to problems in fluid mechanics |

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\textit{P. M. Gresho} and \textit{S. T. Chan}, Int. J. Numer. Methods Fluids 11, No. 5, 621--659 (1990; Zbl 0712.76036)

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