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Volume 19, Issue 6
An Arbitrary-Order Ultra-Weak Discontinuous Galerkin Method for Two-Dimensional Semilinear Second-Order Elliptic Problems on Cartesian Grids

Mahboub Baccouch

Int. J. Numer. Anal. Mod., 19 (2022), pp. 864-886.

Published online: 2022-09

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  • Abstract

In this paper, we present and analyze a new ultra-weak discontinuous Galerkin (UWDG) finite element method for two-dimensional semilinear second-order elliptic problems on Cartesian grids. Unlike the traditional local discontinuous Galerkin (LDG) method, the proposed UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a system of equations. The UWDG scheme is presented in details, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in two-dimensional space. The error estimates of the presented scheme are analyzed. The order of convergence is proved to be $p + 1$ in the $L^2$-norm, when tensor product polynomials of degree at most $p$ and grid size $h$ are used. Several numerical examples are provided to confirm the theoretical results.

  • Keywords

Ultra-weak discontinuous Galerkin method, elliptic problems, convergence, a priori error estimation.

  • AMS Subject Headings

65N12, 65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-864, author = {Baccouch , Mahboub}, title = {An Arbitrary-Order Ultra-Weak Discontinuous Galerkin Method for Two-Dimensional Semilinear Second-Order Elliptic Problems on Cartesian Grids }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {19}, number = {6}, pages = {864--886}, abstract = {

In this paper, we present and analyze a new ultra-weak discontinuous Galerkin (UWDG) finite element method for two-dimensional semilinear second-order elliptic problems on Cartesian grids. Unlike the traditional local discontinuous Galerkin (LDG) method, the proposed UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a system of equations. The UWDG scheme is presented in details, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in two-dimensional space. The error estimates of the presented scheme are analyzed. The order of convergence is proved to be $p + 1$ in the $L^2$-norm, when tensor product polynomials of degree at most $p$ and grid size $h$ are used. Several numerical examples are provided to confirm the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/21037.html} }
TY - JOUR T1 - An Arbitrary-Order Ultra-Weak Discontinuous Galerkin Method for Two-Dimensional Semilinear Second-Order Elliptic Problems on Cartesian Grids AU - Baccouch , Mahboub JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 864 EP - 886 PY - 2022 DA - 2022/09 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/21037.html KW - Ultra-weak discontinuous Galerkin method, elliptic problems, convergence, a priori error estimation. AB -

In this paper, we present and analyze a new ultra-weak discontinuous Galerkin (UWDG) finite element method for two-dimensional semilinear second-order elliptic problems on Cartesian grids. Unlike the traditional local discontinuous Galerkin (LDG) method, the proposed UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a system of equations. The UWDG scheme is presented in details, including the definition of the numerical fluxes, which are necessary to obtain optimal error estimates. The proposed scheme can be made arbitrarily high-order accurate in two-dimensional space. The error estimates of the presented scheme are analyzed. The order of convergence is proved to be $p + 1$ in the $L^2$-norm, when tensor product polynomials of degree at most $p$ and grid size $h$ are used. Several numerical examples are provided to confirm the theoretical results.

Baccouch , Mahboub. (2022). An Arbitrary-Order Ultra-Weak Discontinuous Galerkin Method for Two-Dimensional Semilinear Second-Order Elliptic Problems on Cartesian Grids . International Journal of Numerical Analysis and Modeling. 19 (6). 864-886. doi:
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