Volume 12, Issue 3
Fast Solvers for the Symmetric Ipdg Discretization of Second Order Elliptic Problems

Liuqiang Zhong, Eric T. Chung & Chunmei Liu

DOI:

Int. J. Numer. Anal. Mod., 12 (2015), pp. 455-475

Published online: 2015-12

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  • Abstract

In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a linear conforming space and a space containing high frequency components. To derive such decomposition, we introduce a novel interpolation operator which projects piecewise polynomials of arbitrary order to continuous piecewise linear functions. We prove that this operator is stable which allows us to derive the required space decomposition easily. Moreover, we prove that both the condition number of the preconditioned system and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results.

  • Keywords

Discontinuous Galerkin methods iterative method preconditioner

  • AMS Subject Headings

65F08 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-12-455, author = {}, title = {Fast Solvers for the Symmetric Ipdg Discretization of Second Order Elliptic Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {3}, pages = {455--475}, abstract = {In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a linear conforming space and a space containing high frequency components. To derive such decomposition, we introduce a novel interpolation operator which projects piecewise polynomials of arbitrary order to continuous piecewise linear functions. We prove that this operator is stable which allows us to derive the required space decomposition easily. Moreover, we prove that both the condition number of the preconditioned system and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results.}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/498.html} }
TY - JOUR T1 - Fast Solvers for the Symmetric Ipdg Discretization of Second Order Elliptic Problems JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 455 EP - 475 PY - 2015 DA - 2015/12 SN - 12 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/ijnam/498.html KW - Discontinuous Galerkin methods KW - iterative method KW - preconditioner AB - In this paper, we develop and analyze a preconditioning technique and an iterative solver for the linear systems resulting from the discretization of second order elliptic problems by the symmetric interior penalty discontinuous Galerkin methods. The main ingredient of our approach is a stable decomposition of the piecewise polynomial discontinuous finite element space of arbitrary order into a linear conforming space and a space containing high frequency components. To derive such decomposition, we introduce a novel interpolation operator which projects piecewise polynomials of arbitrary order to continuous piecewise linear functions. We prove that this operator is stable which allows us to derive the required space decomposition easily. Moreover, we prove that both the condition number of the preconditioned system and the convergent rate of the iterative method are independent of the mesh size. Numerical experiments are also shown to confirm these theoretical results.
Liuqiang Zhong, Eric T. Chung & Chunmei Liu. (1970). Fast Solvers for the Symmetric Ipdg Discretization of Second Order Elliptic Problems. International Journal of Numerical Analysis and Modeling. 12 (3). 455-475. doi:
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