### Abstract

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

Original language | English |
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Pages (from-to) | 263-288 |

Number of pages | 26 |

Journal | Mathematics of Computation |

Volume | 72 |

Issue number | 241 |

Early online date | 4 Dec 2001 |

DOIs | |

Publication status | Published - 1 Jan 2003 |

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### Keywords

- computation mathematics
- convection-diffusion equations
- oscillations

### Cite this

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*Mathematics of Computation*, vol. 72, no. 241, pp. 263-288. https://doi.org/10.1090/S0025-5718-01-01392-8

**A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation.** / Elman, Howard C.; Ramage, Alison.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

AU - Elman, Howard C.

AU - Ramage, Alison

PY - 2003/1/1

Y1 - 2003/1/1

N2 - It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

AB - It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

KW - computation mathematics

KW - convection-diffusion equations

KW - oscillations

UR - http://www.scopus.com/inward/record.url?scp=85009726896&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-01-01392-8

DO - 10.1090/S0025-5718-01-01392-8

M3 - Article

VL - 72

SP - 263

EP - 288

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 241

ER -