### Abstract

We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P_{2}/P_{1}- and P_{1}+/P_{1}-finite elements optimal error estimates are proved in ℓ^{∞}(H^{1})×ℓ^{2}(L^{2}) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.

Original language | English |
---|---|

Pages (from-to) | 39-67 |

Number of pages | 29 |

Journal | Mathematics of Computation |

Volume | 87 |

Issue number | 309 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Fingerprint

### Keywords

- Exact computation
- Finite element method
- Lagrange-Galerkin scheme
- Navier-Stokes equations

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*87*(309), 39-67. https://doi.org/10.1090/mcom/3222

**An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates.** / Tabata, Masahisa; Uchiumi, Shinya.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 87, no. 309, pp. 39-67. https://doi.org/10.1090/mcom/3222

}

TY - JOUR

T1 - An exactly computable Lagrange-Galerkin scheme for the Navier-Stokes equations and its error estimates

AU - Tabata, Masahisa

AU - Uchiumi, Shinya

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P2/P1- and P1+/P1-finite elements optimal error estimates are proved in ℓ∞(H1)×ℓ2(L2) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.

AB - We present a Lagrange-Galerkin scheme, which is computable exactly, for the Navier-Stokes equations and show its error estimates. In the Lagrange-Galerkin method we have to deal with the integration of composite functions, where it is difficult to get the exact value. In real computations, numerical quadrature is usually applied to the integration to obtain approximate values, that is, the scheme is not computable exactly. It is known that the error caused from the approximation may destroy the stability result that is proved under the exact integration. Here we introduce a locally linearized velocity and the backward Euler method in solving ordinary differential equations in the position of the fluid particle. Then, the scheme becomes computable exactly, and we show the stability and convergence for this scheme. For the P2/P1- and P1+/P1-finite elements optimal error estimates are proved in ℓ∞(H1)×ℓ2(L2) norm for the velocity and pressure. We present some numerical results, which reflect these estimates and also show robust stability for high Reynolds numbers in the cavity flow problem.

KW - Exact computation

KW - Finite element method

KW - Lagrange-Galerkin scheme

KW - Navier-Stokes equations

UR - http://www.scopus.com/inward/record.url?scp=85038959574&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038959574&partnerID=8YFLogxK

U2 - 10.1090/mcom/3222

DO - 10.1090/mcom/3222

M3 - Article

VL - 87

SP - 39

EP - 67

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 309

ER -