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Volume 16, Issue 1
Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

Jihong Xiao, Zimo Zhu & Xiaoping Xie

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 79-110.

Published online: 2023-01

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  • Abstract

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k (k ≥ 1)$ for the stress approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We show the existence and uniqueness of the semi-discrete and fully discrete solutions, and derive optimal a priori error estimates. Numerical examples are provided to support the theoretical analysis.

  • AMS Subject Headings

35Q74, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-79, author = {Xiao , JihongZhu , Zimo and Xie , Xiaoping}, title = {Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {1}, pages = {79--110}, abstract = {

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k (k ≥ 1)$ for the stress approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We show the existence and uniqueness of the semi-discrete and fully discrete solutions, and derive optimal a priori error estimates. Numerical examples are provided to support the theoretical analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0024}, url = {http://global-sci.org/intro/article_detail/nmtma/21344.html} }
TY - JOUR T1 - Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model AU - Xiao , Jihong AU - Zhu , Zimo AU - Xie , Xiaoping JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 79 EP - 110 PY - 2023 DA - 2023/01 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0024 UR - https://global-sci.org/intro/article_detail/nmtma/21344.html KW - Quasistatic Maxwell viscoelastic model, weak Galerkin method, semi-discrete scheme, fully discrete scheme, error estimate. AB -

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k (k ≥ 1)$ for the stress approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We show the existence and uniqueness of the semi-discrete and fully discrete solutions, and derive optimal a priori error estimates. Numerical examples are provided to support the theoretical analysis.

Xiao , JihongZhu , Zimo and Xie , Xiaoping. (2023). Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model. Numerical Mathematics: Theory, Methods and Applications. 16 (1). 79-110. doi:10.4208/nmtma.OA-2022-0024
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