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Volume 14, Issue 3
An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems

Changjuan Zhang, Zhilin Li & Xingye Yue

Numer. Math. Theor. Meth. Appl., 14 (2021), pp. 773-796.

Published online: 2021-06

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  • Abstract

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.

  • AMS Subject Headings

65N06, 65N50

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-14-773, author = {Zhang , ChangjuanLi , Zhilin and Yue , Xingye}, title = {An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2021}, volume = {14}, number = {3}, pages = {773--796}, abstract = {

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2020-0112}, url = {http://global-sci.org/intro/article_detail/nmtma/19197.html} }
TY - JOUR T1 - An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems AU - Zhang , Changjuan AU - Li , Zhilin AU - Yue , Xingye JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 773 EP - 796 PY - 2021 DA - 2021/06 SN - 14 DO - http://doi.org/10.4208/nmtma.OA-2020-0112 UR - https://global-sci.org/intro/article_detail/nmtma/19197.html KW - 3D elliptic interface problem, augmented IIM, fast Poisson solver, directional least squares interpolation. AB -

A new fast algorithm based on the augmented immersed interface method and a fast Poisson solver is proposed to solve three dimensional elliptic interface problems with a piecewise constant but discontinuous coefficient. In the new approach, an augmented variable along the interface, often the jump in the normal derivative along the interface is introduced so that a fast Poisson solver can be utilized. Thus, the solution of the Poisson equation depends on the augmented variable which should be chosen such that the original flux jump condition is satisfied. The discretization of the flux jump condition is done by a weighted least squares interpolation using the solution at the grid points, the jump conditions, and the governing PDEs in a neighborhood of control points on the interface. The interpolation scheme is the key to the success of the augmented IIM particularly. In this paper, the key new idea is to select interpolation points along the normal direction in line with the flux jump condition. Numerical experiments show that the method maintains second order accuracy of the solution and can reduce the CPU time by 20-50%. The number of the GMRES iterations is independent of the mesh size.

Changjuan Zhang, Zhilin Li & Xingye Yue. (2021). An Acceleration Technique for the Augmented IIM for 3D Elliptic Interface Problems. Numerical Mathematics: Theory, Methods and Applications. 14 (3). 773-796. doi:10.4208/nmtma.OA-2020-0112
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