arrow
Volume 20, Issue 6
A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

Mahboub Baccouch

Int. J. Numer. Anal. Mod., 20 (2023), pp. 772-804.

Published online: 2023-11

Export citation
  • Abstract

In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type $−∆u = f(x, u).$ We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order $p+2$ towards the $p$-degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most $p$ are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two $(p+1)$-degree Radau polynomials in the $x-$ and $y−$ directions. The second error converges to zero with order $p+2$ in the $L^2$-norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the $L^2$-norm under mesh refinement at the optimal rate. The order of convergence is proved to be $p+2.$ We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.

  • AMS Subject Headings

65N12, 65N15, 65N30, 65N50

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{IJNAM-20-772, author = {Baccouch , Mahboub}, title = {A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2023}, volume = {20}, number = {6}, pages = {772--804}, abstract = {

In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type $−∆u = f(x, u).$ We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order $p+2$ towards the $p$-degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most $p$ are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two $(p+1)$-degree Radau polynomials in the $x-$ and $y−$ directions. The second error converges to zero with order $p+2$ in the $L^2$-norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the $L^2$-norm under mesh refinement at the optimal rate. The order of convergence is proved to be $p+2.$ We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2023-1034}, url = {http://global-sci.org/intro/article_detail/ijnam/22141.html} }
TY - JOUR T1 - A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids AU - Baccouch , Mahboub JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 772 EP - 804 PY - 2023 DA - 2023/11 SN - 20 DO - http://doi.org/10.4208/ijnam2023-1034 UR - https://global-sci.org/intro/article_detail/ijnam/22141.html KW - local discontinuous Galerkin method, semilinear elliptic problems, a posteriori error estimators, superconvergence, Radau polynomial. AB -

In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type $−∆u = f(x, u).$ We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order $p+2$ towards the $p$-degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most $p$ are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two $(p+1)$-degree Radau polynomials in the $x-$ and $y−$ directions. The second error converges to zero with order $p+2$ in the $L^2$-norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the $L^2$-norm under mesh refinement at the optimal rate. The order of convergence is proved to be $p+2.$ We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.

Mahboub Baccouch. (2023). A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids. International Journal of Numerical Analysis and Modeling. 20 (6). 772-804. doi:10.4208/ijnam2023-1034
Copy to clipboard
The citation has been copied to your clipboard