A study of incorporating the multigrid method into the three-dimensional finite element discretization

a modular setting and application

Hwai Ping Cheng, Gour Tsyh Yeh, Jinchao Xu, Ming Hsu Li, Robert Carsel

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h = Ih 2h Ah I2h h and b2h = Ih 2h bh, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEM WATER and 3DLEWASTE,to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting.

Original language English (US) 499-526 28 International Journal for Numerical Methods in Engineering 41 3 https://doi.org/10.1002/(SICI)1097-0207(19980215)41:3<499::AID-NME295>3.0.CO;2-3 Published - Jan 1 1998

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Multigrid Method
Finite Element Discretization
Grid
Three-dimensional
Matrix Equation
Boundary value problems
Linear Matrix Equations
Computer simulation
Computational Results
Finite Difference
Coding
Discretization
Boundary Value Problem
Finite Element
Numerical Simulation
Necessary
Approximation
Model
Demonstrate

All Science Journal Classification (ASJC) codes

• Numerical Analysis
• Engineering(all)
• Applied Mathematics

Cite this

title = "A study of incorporating the multigrid method into the three-dimensional finite element discretization: a modular setting and application",
abstract = "Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h = Ih 2h Ah I2h h and b2h = Ih 2h bh, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEM WATER and 3DLEWASTE,to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting.",
author = "Cheng, {Hwai Ping} and Yeh, {Gour Tsyh} and Jinchao Xu and Li, {Ming Hsu} and Robert Carsel",
year = "1998",
month = "1",
day = "1",
doi = "10.1002/(SICI)1097-0207(19980215)41:3<499::AID-NME295>3.0.CO;2-3",
language = "English (US)",
volume = "41",
pages = "499--526",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "3",

}

A study of incorporating the multigrid method into the three-dimensional finite element discretization : a modular setting and application. / Cheng, Hwai Ping; Yeh, Gour Tsyh; Xu, Jinchao; Li, Ming Hsu; Carsel, Robert.

In: International Journal for Numerical Methods in Engineering, Vol. 41, No. 3, 01.01.1998, p. 499-526.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A study of incorporating the multigrid method into the three-dimensional finite element discretization

T2 - a modular setting and application

AU - Cheng, Hwai Ping

AU - Yeh, Gour Tsyh

AU - Xu, Jinchao

AU - Li, Ming Hsu

AU - Carsel, Robert

PY - 1998/1/1

Y1 - 1998/1/1

N2 - Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h = Ih 2h Ah I2h h and b2h = Ih 2h bh, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEM WATER and 3DLEWASTE,to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting.

AB - Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h = Ih 2h Ah I2h h and b2h = Ih 2h bh, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEM WATER and 3DLEWASTE,to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting.

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U2 - 10.1002/(SICI)1097-0207(19980215)41:3<499::AID-NME295>3.0.CO;2-3

DO - 10.1002/(SICI)1097-0207(19980215)41:3<499::AID-NME295>3.0.CO;2-3

M3 - Article

VL - 41

SP - 499

EP - 526

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 3

ER -