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Volume 15, Issue 1
High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids

Yan Yang & Xiaoping Xie

DOI: 10.4208/nmtma.OA-2021-0055

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 227-250.

Published online: 2022-02

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  • Abstract

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H($div) − $P_k (k ≥ 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, and the displacement field is approximated by $C^{−1}− P_{k−1}$ polynomial vectors enriched with higher order terms. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress and displacement in $H$(div) norm and $L^2$ norm, respectively. Numerical results confirm the theoretical analysis.

  • Keywords

Linear elasticity, mixed finite element, mass lumping, error estimate.

  • AMS Subject Headings

65N15, 65N30, 74H15, 74S05

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{NMTMA-15-227, author = {Yang , Yan and Xie , Xiaoping}, title = {High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {1}, pages = {227--250}, abstract = {

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H($div) − $P_k (k ≥ 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, and the displacement field is approximated by $C^{−1}− P_{k−1}$ polynomial vectors enriched with higher order terms. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress and displacement in $H$(div) norm and $L^2$ norm, respectively. Numerical results confirm the theoretical analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0055}, url = {http://global-sci.org/intro/article_detail/nmtma/20228.html} }
TY - JOUR T1 - High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids AU - Yang , Yan AU - Xie , Xiaoping JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 227 EP - 250 PY - 2022 DA - 2022/02 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2021-0055 UR - https://global-sci.org/intro/article_detail/nmtma/20228.html KW - Linear elasticity, mixed finite element, mass lumping, error estimate. AB -

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H($div) − $P_k (k ≥ 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, and the displacement field is approximated by $C^{−1}− P_{k−1}$ polynomial vectors enriched with higher order terms. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress and displacement in $H$(div) norm and $L^2$ norm, respectively. Numerical results confirm the theoretical analysis.

Yang , Yan and Xie , Xiaoping. (2022). High Order Mixed Finite Elements with Mass Lumping for Elasticity on Triangular Grids. Numerical Mathematics: Theory, Methods and Applications. 15 (1). 227-250. doi:10.4208/nmtma.OA-2021-0055
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