Abstract
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.
Original language | English (US) |
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Pages (from-to) | 201-224 |
Number of pages | 24 |
Journal | Applied Numerical Mathematics |
Volume | 162 |
DOIs | |
State | Published - Apr 2021 |
Keywords
- A posteriori error estimator
- Adaptive mesh refinement
- Elliptic boundary-value problems
- Local discontinuous Galerkin method
- Superconvergence
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics