Volume 10, Issue 2
The Immersed Finite Element Method for Parabolic Problems Using the Laplace Transformation in Time Discretization

T. Lin & D. Sheen

Int. J. Numer. Anal. Mod., 10 (2013), pp. 298-313

Published online: 2013-10

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  • Abstract
In this paper we are interested in solving parabolic problems with a piecewise constant diffusion coefficient on structured Cartesian meshes. The aim of this paper is to investigate the applicability and convergence behavior of combining two non-conventional but innovative methods: the Laplace transformation method in the discretization of the time variable and the immerse finite element method (IFEM) in the discretization of the space variable. The Laplace transformation in time leads to a set of Helmholtz-like problems independent of each other, which can be solved in highly parallel. The employment of immerse finite elements (IFEs) makes it possible to use a structured mesh, such as a simple Cartesian mesh, for the discretization of the space variable even if the material interface (across which the diffusion coefficient is discontinuous) is non-trivial. Numerical examples presented indicate that the combination of these two methods can perform optimally from the point of view of the degrees of polynomial spaces employed in the IFE spaces.
  • Keywords

Immersed finite element method interface problems Laplace transform parallel algorithm

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@Article{IJNAM-10-298, author = {T. Lin and D. Sheen}, title = {The Immersed Finite Element Method for Parabolic Problems Using the Laplace Transformation in Time Discretization}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {2}, pages = {298--313}, abstract = {In this paper we are interested in solving parabolic problems with a piecewise constant diffusion coefficient on structured Cartesian meshes. The aim of this paper is to investigate the applicability and convergence behavior of combining two non-conventional but innovative methods: the Laplace transformation method in the discretization of the time variable and the immerse finite element method (IFEM) in the discretization of the space variable. The Laplace transformation in time leads to a set of Helmholtz-like problems independent of each other, which can be solved in highly parallel. The employment of immerse finite elements (IFEs) makes it possible to use a structured mesh, such as a simple Cartesian mesh, for the discretization of the space variable even if the material interface (across which the diffusion coefficient is discontinuous) is non-trivial. Numerical examples presented indicate that the combination of these two methods can perform optimally from the point of view of the degrees of polynomial spaces employed in the IFE spaces.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/569.html} }
TY - JOUR T1 - The Immersed Finite Element Method for Parabolic Problems Using the Laplace Transformation in Time Discretization AU - T. Lin & D. Sheen JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 298 EP - 313 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/569.html KW - Immersed finite element method KW - interface problems KW - Laplace transform KW - parallel algorithm AB - In this paper we are interested in solving parabolic problems with a piecewise constant diffusion coefficient on structured Cartesian meshes. The aim of this paper is to investigate the applicability and convergence behavior of combining two non-conventional but innovative methods: the Laplace transformation method in the discretization of the time variable and the immerse finite element method (IFEM) in the discretization of the space variable. The Laplace transformation in time leads to a set of Helmholtz-like problems independent of each other, which can be solved in highly parallel. The employment of immerse finite elements (IFEs) makes it possible to use a structured mesh, such as a simple Cartesian mesh, for the discretization of the space variable even if the material interface (across which the diffusion coefficient is discontinuous) is non-trivial. Numerical examples presented indicate that the combination of these two methods can perform optimally from the point of view of the degrees of polynomial spaces employed in the IFE spaces.
T. Lin & D. Sheen. (1970). The Immersed Finite Element Method for Parabolic Problems Using the Laplace Transformation in Time Discretization. International Journal of Numerical Analysis and Modeling. 10 (2). 298-313. doi:
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