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Volume 3, Issue 1
Effects of Basis Selection and H-Refinement on Error Estimator Reliability and Solution Efficiency for High-Order Methods in Three Space Dimensions

Peter K. Moore

Int. J. Numer. Anal. Mod., 3 (2006), pp. 21-51.

Published online: 2006-03

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  • Abstract

Designing effective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires the selection of a finite element basis, a posteriori error estimator and refinement strategy. Estimator accuracy may depend on the basis chosen, which in turn, may lead to unreliability or inefficiency via under- or over-refinement, respectively. The basis may also have an impact on the size and condition of the matrices that arise from discretization, and thus, on algorithm effectiveness. Herein, the interaction between these three components is studied in the context of an $h$-refinement procedure. The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems. The results demonstrate that popular choices such as the tensor-product basis or the modified Szabό-Babuška basis have significant shortcomings but that promising alternatives exist.

  • Keywords

a posteriori error estimation, adaptivity, high-order finite element basis.

  • AMS Subject Headings

65N15, 65N30, 65N50

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-3-21, author = {Moore , Peter K.}, title = {Effects of Basis Selection and H-Refinement on Error Estimator Reliability and Solution Efficiency for High-Order Methods in Three Space Dimensions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2006}, volume = {3}, number = {1}, pages = {21--51}, abstract = {

Designing effective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires the selection of a finite element basis, a posteriori error estimator and refinement strategy. Estimator accuracy may depend on the basis chosen, which in turn, may lead to unreliability or inefficiency via under- or over-refinement, respectively. The basis may also have an impact on the size and condition of the matrices that arise from discretization, and thus, on algorithm effectiveness. Herein, the interaction between these three components is studied in the context of an $h$-refinement procedure. The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems. The results demonstrate that popular choices such as the tensor-product basis or the modified Szabό-Babuška basis have significant shortcomings but that promising alternatives exist.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/888.html} }
TY - JOUR T1 - Effects of Basis Selection and H-Refinement on Error Estimator Reliability and Solution Efficiency for High-Order Methods in Three Space Dimensions AU - Moore , Peter K. JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 21 EP - 51 PY - 2006 DA - 2006/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/888.html KW - a posteriori error estimation, adaptivity, high-order finite element basis. AB -

Designing effective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires the selection of a finite element basis, a posteriori error estimator and refinement strategy. Estimator accuracy may depend on the basis chosen, which in turn, may lead to unreliability or inefficiency via under- or over-refinement, respectively. The basis may also have an impact on the size and condition of the matrices that arise from discretization, and thus, on algorithm effectiveness. Herein, the interaction between these three components is studied in the context of an $h$-refinement procedure. The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems. The results demonstrate that popular choices such as the tensor-product basis or the modified Szabό-Babuška basis have significant shortcomings but that promising alternatives exist.

Moore , Peter K.. (2006). Effects of Basis Selection and H-Refinement on Error Estimator Reliability and Solution Efficiency for High-Order Methods in Three Space Dimensions. International Journal of Numerical Analysis and Modeling. 3 (1). 21-51. doi:
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