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In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.

Elliptic equations are widely used in the fields of solid mechanics, material science, and image processing and so on. So it is both theoretically and practically important to investigate numerical methods for such equations. Finite difference method is a general and effective method to solve elliptic equations. In the past three decades, a large number of high order compact (HOC) difference schemes [

Coordinate transformation method [

In this paper, we consider the 2D elliptic equation with variable coefficients as follows:

where

We consider a rectangular physical domain

Then, Equation (1) is transformed into the form as:

where the coefficients a, b, c, d, e, f and the right hand term g are the functions of

where

Then, for Equation (10), a fourth order compact difference scheme can be derived. We will give the derivation process of the fourth order compact scheme in the next section.

Firstly, we divide the computational domain

Then the following difference equation is got if we substitute Equations (11)-(15) into Equation (10):

Equation (16) is actually the second order central difference scheme and

In order to get an HOC scheme, the third order and fourth order derivatives in

Using the central differences, the difference approximation of

Finally, substituting Equations (22)-(25) into Equation (17), combining it with Equation (16) and neglecting high order truncation error term, we have:

where

So Equations (26) with (27)-(37) is the HOC difference scheme based on the coordinate transformation for solving Equation (1) on nonuniform grids. The present HOC scheme can be written in the form of nine-point scheme and the corresponding coefficients of them can be written as follows:

where

From the process of derivation, it is easy to know that the truncation error of this scheme is

In order to solve the linear algebraic systems which are arising from various difference schemes, generally, some iterative methods are used. But the convergence speed of traditional iterative methods is very slow, so appears multigrid method [

Multigrid method is achieved by some circulation algorithms such as V cycle, W cycle or Full Multigrid V cycle (FMV) etc. The whole process has three elements: relaxation operator, projection operator and interpolation operator. The function of relaxation operator (or iteration) is to dump the high frequency components of the errors on the current grid. The function of projection and interpolation operators is to transfer error residuals from finer grids to coarser grids and to return the corrected errors from the coarser grids to the finer grids. The multigrid

Multigrid method has been used to solve various linear elliptic equations such as Poisson equation [

In this paper, we adopt the multigrid V cycle method to solve the linear algebraic system arising from the difference schemes. In order to match the HOC scheme, we choose the full weighting projection operator on uniform grids [^{ }

where ^{ }

Then, for relaxation operator, we use the alternating direction line Gauss-Seidel relaxations to remove the residuals on each coarse grid.

In order to demonstrate the high accuracy and high efficiency of the present method, we use it to solve the following three elliptic problems with Dirichlet boundary conditions. All of the problems have the exact solutions. All computation is started with zero initial guesses and is terminated when the residuals in ^{10}. For each problem, we give the multigrid V cycles (Num), the CPU time in seconds, maximum absolute errors (Error) and convergence rates (Order) about different grid numbers in the tables. The procedure is written in Fortran 77 programming language with double precision arithmetic and run on a Pentium IV/Dual-core/3 GHz private computer with 2 GB memory. The convergence order can be got by the following formula:

where

We consider the following 2D convection diffusion problem [^{ }

where

The boundary conditions are:

The exact solution is:

This problem has a steep solution gradient near

where

For this problem, we set Re = 1000 and choose

In order to illustrate the computational accuracy in the whole domain, with the grid number 64 × 64, we give the figures about the contours of the exact solution (

Next, we consider an elliptic problem as follows:

where

The boundary conditions are:

Grids | Uniform grids | Nonuniform grids | ||||||
---|---|---|---|---|---|---|---|---|

Num | CPU | Error | Order | Num | CPU | Error | Order | |

64 × 64 | 11 | 0.042 | 1.28 (−2) | 11 | 0.078 | 5.05 (−4) | ||

128 × 128 | 11 | 0.266 | 8.44 (−4) | 3.97 | 11 | 0.250 | 3.14 (−5) | 4.05 |

256 × 256 | 10 | 1.016 | 5.32 (−5) | 4.01 | 11 | 1.110 | 1.98 (−6) | 4.01 |

64 × 64 | 11 | 0.063 | 2.72 (+0) | 12 | 0.062 | 2.18 (−3) | ||

128 × 128 | 10 | 0.250 | 3.00 (−2) | 6.58 | 11 | 0.265 | 1.48 (−4) | 3.92 |

256 × 256 | 10 | 1.079 | 2.15 (−3) | 3.82 | 11 | 1.140 | 9.43 (−6) | 3.99 |

64 × 64 | 11 | 0.062 | 7.67 (+1) | 10 | 0.063 | 2.46 (−2) | ||

128 × 128 | 11 | 0.281 | 3.70 (+1) | 1.06 | 10 | 0.266 | 1.50 (−3) | 4.08 |

256 × 256 | 10 | 1.125 | 1.36 (+0) | 4.79 | 10 | 1.110 | 9.41 (−5) | 4.02 |

in which

The exact solution is:

For this problem, there are two boundary layers near

We choose Re = 10, 100 and 1000. From

Grids | Uniform grids | Nonuniform grids | |||||||
---|---|---|---|---|---|---|---|---|---|

Num | CPU | Error | Order | Num | CPU | Error | Order | ||

Re = 10 | |||||||||

64 × 64 | 8 | 0.062 | 6.72 (−5) | 8 | 0.063 | 9.01 (−6) | |||

128 × 128 | 8 | 0.281 | 5.71 (−6) | 3.60 | 8 | 0.296 | 6.28 (−7) | 3.89 | |

256 × 256 | 8 | 1.172 | 4.56 (−7) | 3.67 | 8 | 1.203 | 4.26 (−8) | 3.90 | |

Re = 100 | |||||||||

64 × 64 | 8 | 0.062 | 1.36 (−1) | 10 | 0.062 | 2.18 (−4) | |||

128 × 128 | 8 | 0.281 | 2.10 (−2) | 2.73 | 9 | 0.313 | 1.71 (−5) | 3.71 | |

256 × 256 | 9 | 1.182 | 2.38 (−3) | 3.18 | 9 | 1.296 | 1.22 (−6) | 3.83 | |

Re = 1000 | |||||||||

64 × 64 | 17 | 0.140 | 9.63 (−1) | 22 | 0.188 | 9.03 (−4) | |||

128 × 128 | 17 | 0.625 | 9.36 (−1) | 0.04 | 21 | 0.672 | 2.72 (−4) | 1.75 | |

256 × 256 | 17 | 2.656 | 7.24 (−1) | 0.37 | 21 | 2.891 | 3.87 (−5) | 2.83 | |

We consider the following elliptic equation:

The boundary conditions are:

The exact solution is:

There is a boundary layer near

For this problem, the multigrid V(2,2) cycles are used.

Re = 10,000, it just gets the second order accuracy on the uniform grids, the computed errors are dramatically distorted with the increase of Re, and this gives very poor solution. Especially for Re = 10,000, the solution is very bad and unacceptable. Compared with computed results on the uniform grids, it shows that the fourth order accuracy is achieved for all the Re numbers on the nonuniform grids and the computed results are very accurate. So, it demonstrates that the present transformed HOC scheme is effective for solving the boundary layer problems with nonuniform grids in the physical domain.

The aim of this paper is to build an efficient and high accuracy numerical method for solving 2D elliptic equations with variable coefficients and interior/boundary layers on nonuniform grids. Coordinate transformation

Grids | Uniform grids | Nonuniform grids | |||||||
---|---|---|---|---|---|---|---|---|---|

Num | CPU | Error | Order | Num | CPU | Error | Order | ||

Re = 100 | |||||||||

64 × 64 | 6 | 0.063 | 2.66 (−1) | 8 | 0.078 | 5.67 (−5) | |||

128 × 128 | 6 | 0.289 | 2.14 (−2) | 3.68 | 7 | 0.313 | 3.54 (−6) | 4.05 | |

256 × 256 | 6 | 1.187 | 1.43 (−3) | 3.93 | 7 | 1.359 | 2.20 (−7) | 4.03 | |

Re = 1000 | |||||||||

64 × 64 | 6 | 0.094 | 7.79 (+1) | 9 | 0.110 | 8.42 (−4) | |||

128 × 128 | 6 | 0.391 | 1.91 (+1) | 2.05 | 8 | 0.438 | 4.82 (−5) | 4.17 | |

256 × 256 | 6 | 1.703 | 4.08 (+0) | 2.24 | 6 | 1.609 | 2.86 (−6) | 4.15 | |

Re = 10000 | |||||||||

64 × 64 | 6 | 0.093 | 7.88 (+3) | 8 | 0.094 | 1.17 (−2) | |||

128 × 128 | 6 | 0.406 | 2.00 (+3) | 2.00 | 6 | 0.391 | 5.40 (−4) | 4.49 | |

256 × 256 | 6 | 1.703 | 5.04 (+2) | 2.00 | 5 | 1.515 | 2.59 (−5) | 4.41 | |

method is employed to transfer nonuniform grids in the physical domain, which concentrates clustered grid points inside the interior/boundary layers, to uniform grids in the computational domain. A high order compact difference scheme is derived for the transformed equation to achieve the purpose of simplified calculation on uniform grids. It needs to be pointed out that when the transformation parameter is zero, the present HOC scheme reduces to the HOC difference scheme on uniform grids in the physical domain. So, it fits computation on both uniform and nonuniform grids. In order to accelerate the convergence of the traditional iterative methods and to reduce computational cost, a multigrid method is employed to solve the linear algebraic system which is arising from the difference scheme. Some numerical experiments with interior or boundary layer problems are conducted to demonstrate the performances of the present method. It indicates that a nonuniform grid is necessary for solving 2D elliptic problems with interior or boundary layers. By coordinate transformation, a certain number of grid points are clustered in the interior or boundary layers to guarantee that the HOC scheme for transformed equation obtains very accurate numerical solution with not so fine grids. Otherwise, the HOC scheme produces very poor approximation solution on uniform grids.

The present work was supported by the National Science Foundation of China under Grant 11361045 and 11161036, Fok Ying-Tong Education Foundation of China under Grant 121105.