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 - <p style="text-align: justify;">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.</p>