TY - JOUR
T1 - Optimal superconvergence and asymptotically exact a posteriori error estimator for the local discontinuous Galerkin method for linear elliptic problems on Cartesian grids
AU - Baccouch, Mahboub
N1 - Funding Information:
This research was supported by NASA Nebraska Space Grant (Federal Grant/Award Number: 80NSSC20M0112 ). The author would like to thank the anonymous reviewers for the valuable comments and suggestions which improved the quality of the paper.
Publisher Copyright:
© 2020 IMACS
PY - 2021/4
Y1 - 2021/4
N2 - The purpose of this paper is twofold: to study the superconvergence properties and to present an efficient and reliable a posteriori error estimator for the local discontinuous Galerkin (LDG) method for linear second-order elliptic problems on Cartesian grids. We prove that the LDG solution is superconvergent towards a particular projection of the exact solution. The order of convergence is proved to be p+2, when tensor product polynomials of degree at most p are used. Then, we prove that the actual error can be split into two parts. The components of the significant part can be given in terms of (p+1)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We prove that the proposed residual-type a posteriori error estimates converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. We provide several numerical examples illustrating the effectiveness of our procedures.
AB - The purpose of this paper is twofold: to study the superconvergence properties and to present an efficient and reliable a posteriori error estimator for the local discontinuous Galerkin (LDG) method for linear second-order elliptic problems on Cartesian grids. We prove that the LDG solution is superconvergent towards a particular projection of the exact solution. The order of convergence is proved to be p+2, when tensor product polynomials of degree at most p are used. Then, we prove that the actual error can be split into two parts. The components of the significant part can be given in terms of (p+1)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We prove that the proposed residual-type a posteriori error estimates converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. We provide several numerical examples illustrating the effectiveness of our procedures.
KW - A posteriori error estimator
KW - Adaptive mesh refinement
KW - Elliptic boundary-value problems
KW - Local discontinuous Galerkin method
KW - Superconvergence
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U2 - 10.1016/j.apnum.2020.12.019
DO - 10.1016/j.apnum.2020.12.019
M3 - Article
AN - SCOPUS:85098660285
SN - 0168-9274
VL - 162
SP - 201
EP - 224
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
ER -