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Volume 31, Issue 1
Optimality of Local Multilevel Methods for Adaptive Nonconforming P1 Finite Element Methods

Xuejun Xu, Huangxin Chen & R.H.W. Hoppe

DOI: 10.4208/jcm.1203-m3960

J. Comp. Math., 31 (2013), pp. 22-46.

Published online: 2013-02

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  • Abstract

In this paper, a local multilevel product algorithm and its additive version are considered for linear systems arising from adaptive nonconforming P1 finite element approximations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jacobi or Gauss-Seidel smoothers performed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

  • Keywords

Local multilevel methods, Adaptive nonconforming P1 finite element methods, Convergence analysis, Optimality.

  • AMS Subject Headings

65F10, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-31-22, author = {Xuejun Xu, Huangxin Chen and R.H.W. Hoppe}, title = {Optimality of Local Multilevel Methods for Adaptive Nonconforming P1 Finite Element Methods}, journal = {Journal of Computational Mathematics}, year = {2013}, volume = {31}, number = {1}, pages = {22--46}, abstract = {

In this paper, a local multilevel product algorithm and its additive version are considered for linear systems arising from adaptive nonconforming P1 finite element approximations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jacobi or Gauss-Seidel smoothers performed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1203-m3960}, url = {http://global-sci.org/intro/article_detail/jcm/9719.html} }
TY - JOUR T1 - Optimality of Local Multilevel Methods for Adaptive Nonconforming P1 Finite Element Methods AU - Xuejun Xu, Huangxin Chen & R.H.W. Hoppe JO - Journal of Computational Mathematics VL - 1 SP - 22 EP - 46 PY - 2013 DA - 2013/02 SN - 31 DO - http://doi.org/10.4208/jcm.1203-m3960 UR - https://global-sci.org/intro/article_detail/jcm/9719.html KW - Local multilevel methods, Adaptive nonconforming P1 finite element methods, Convergence analysis, Optimality. AB -

In this paper, a local multilevel product algorithm and its additive version are considered for linear systems arising from adaptive nonconforming P1 finite element approximations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jacobi or Gauss-Seidel smoothers performed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

Xuejun Xu, Huangxin Chen and R.H.W. Hoppe. (2013). Optimality of Local Multilevel Methods for Adaptive Nonconforming P1 Finite Element Methods. Journal of Computational Mathematics. 31 (1). 22-46. doi:10.4208/jcm.1203-m3960
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