A Class of Stable, Efficient Navier-Stokes Solvers

We study a class of numerical schemes for Navier-Stokes equations (NSE) or Stokes equations (SE) for incompressible fluids in a bounded domain with given boundary value of velocity. The incompressibility constraint and non-slip boundary condition have made this problem very challenging. Their treatment by finite element method leads to the well-known inf-sup compatibility condition. Their treatment by finite difference method leads to the very popular projection method, which suffers from low resolution near the boundary. In [LLP], the authors propose an unconstrained formulation of NSE or SE….

Contents

1 Introduction
1.1 Steady state equations
1.2 Time dependent equations
1.2.1 C0 and C1 FE schemes for time-dependent problem
1.2.2 Another C0 finite element scheme for the time-dependent problem
1.3 Outline of the thesis
2 The unconstrained formulation and pressure estimates
2.1 Steady-state NSE
2.2 Time-dependent NSE
2.3 Pressure estimates
3 Steady-state Stokes equations
3.1 Semi-discrete iterative solver for steady-state Stokes equations
3.2 FE solver for steady-state SE which solves (u; p) together
3.3 Iterative FE solver for steady-state SE
3.4 Numerics for steady-state computation
4 Time dependent problems: Part I
4.1 Stability for C1 FE scheme (1.18),(1.20)
4.1.1 CFL condition for convection diffusion equation
4.2 Error estimates for C1 FE scheme (1.18),(1.20)
4.2.1 Approximation error
4.2.2 Discretization error for pressure
4.2.3 Discretization error for velocity
4.3 Practical issues
4.3.1 Divergence damping
4.3.2 Obtuse corners and C1 finite elements
4.3.3 Non-homogeneous boundary conditions
4.3.4 Higher-order time integration
4.3.5 How to solve for the pressure
4.4 Numerical results
4.4.1 Test problems and parameters
4.4.2 Numerical results for C1 FE schemes like (1.18),(1.20)
4.4.3 Numerical results for C0 FE schemes like (1.18),(1.19)
5 Time dependent problems: Part II
5.1 Another reformulation of NSE
5.1.1 Stability and error estimates of the semi-discrete scheme
5.1.2 How large we can take for coefficient of the divergence damping?
5.2 C0 finite element schemes (1.25) and (5.1)
5.2.1 Proof of Theorem 7
5.2.2 Proof of Theorem 8
5.3 Numerics of FE schemes (1.25) and (5.1)
6 Conclusions
A Projection methods
A.1 First order projection methods
A.1.1 Chorin [Ch]; Temam [Te]
A.1.2 van Kan [vK]; Bell, Colella and Glaz [BCG]
A.1.3 Orszag, Israeli and Deville [OID]; Timmermans, Minev and Van De Vosse [TMV]; Kim and Moin [KM]; Petersson [Pe];
Brown, Cortez and Minion [BCM]; Guermond and Shen [GS2]
A.1.4 Johnston and Liu [JL], E and Liu [EL], Guermond and Shen [GS1]
A.2 Second order projection methods
A.2.1 van Kan [vK]; Bell, Colella and Glaz [BCG]
A.2.2 Orszag, Israeli and Deville [OID]; Timmermans, Minev and Van De Vosse [TMV]; Kim and Moin [KM]; Petersson [Pe];
Brown, Cortez and Minion [BCM]; Guermond and Shen [GS2]
A.2.3 Johnston and Liu [JL], E and Liu [EL], Guermond and Shen [GS1]
A.3 Analysis of projection methods
A.3.1 Chorin [Ch]; Temam [Te]
A.3.2 van Kan [vK]; Bell, Colella and Glaz [BCG]
A.3.3 Orszag, Israeli and Deville [OID]; Timmermans, Minev and Van De Vosse [TMV]
A.3.4 Johnston and Liu [JL], E and Liu [EL], Guermond and Shen [GS1]
A.4 Normal modes analysis
A.4.1 normal modes analysis of 1st order numerical scheme
Bibliography

Author: Liu, Jie

Source: University of Maryland

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