Volume 12, Issue 3
Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: Weak Galerkin Method

Xiu Ye & Shangyou Zhang

East Asian J. Appl. Math., 12 (2022), pp. 590-598.

Published online: 2022-04

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  • Abstract

A simple stabilizer free weak Galerkin (SFWG) finite element method for a one-dimensional second order elliptic problem is introduced. In this method, the weak function is formed by a discontinuous $k$-th order polynomial with additional unknowns defined on vertex points, whereas its weak derivative is approximated by a polynomial of degree $k+1.$ The superconvergence of order two for the SFWG finite element solution is established. It is shown that the elementwise lifted $P_{k+2}$ solution of the $P_k$ SFWG one converges at the optimal order. Numerical results confirm the theory.

  • Keywords

Finite element, weak Galerkin, stabilizer free.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-12-590, author = {Xiu and Ye and and 22868 and and Xiu Ye and Shangyou and Zhang and and 22869 and and Shangyou Zhang}, title = {Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: Weak Galerkin Method}, journal = {East Asian Journal on Applied Mathematics}, year = {2022}, volume = {12}, number = {3}, pages = {590--598}, abstract = {

A simple stabilizer free weak Galerkin (SFWG) finite element method for a one-dimensional second order elliptic problem is introduced. In this method, the weak function is formed by a discontinuous $k$-th order polynomial with additional unknowns defined on vertex points, whereas its weak derivative is approximated by a polynomial of degree $k+1.$ The superconvergence of order two for the SFWG finite element solution is established. It is shown that the elementwise lifted $P_{k+2}$ solution of the $P_k$ SFWG one converges at the optimal order. Numerical results confirm the theory.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.030921.141121 }, url = {http://global-sci.org/intro/article_detail/eajam/20408.html} }
TY - JOUR T1 - Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: Weak Galerkin Method AU - Ye , Xiu AU - Zhang , Shangyou JO - East Asian Journal on Applied Mathematics VL - 3 SP - 590 EP - 598 PY - 2022 DA - 2022/04 SN - 12 DO - http://doi.org/10.4208/eajam.030921.141121 UR - https://global-sci.org/intro/article_detail/eajam/20408.html KW - Finite element, weak Galerkin, stabilizer free. AB -

A simple stabilizer free weak Galerkin (SFWG) finite element method for a one-dimensional second order elliptic problem is introduced. In this method, the weak function is formed by a discontinuous $k$-th order polynomial with additional unknowns defined on vertex points, whereas its weak derivative is approximated by a polynomial of degree $k+1.$ The superconvergence of order two for the SFWG finite element solution is established. It is shown that the elementwise lifted $P_{k+2}$ solution of the $P_k$ SFWG one converges at the optimal order. Numerical results confirm the theory.

Xiu Ye & Shangyou Zhang. (2022). Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: Weak Galerkin Method. East Asian Journal on Applied Mathematics. 12 (3). 590-598. doi:10.4208/eajam.030921.141121
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