Abstract
This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin (LDG) method for two-dimensional semilinear second-order elliptic problems of the form - Δ u= f(x, y, u) on Cartesian grids. By introducing special Gauss-Radau projections and using duality arguments, we obtain, under some suitable choice of numerical fluxes, the optimal convergence order in L2-norm of O(hp+1) for the LDG solution and its gradient, when tensor product polynomials of degree at most p and grid size h are used. Moreover, we prove that the LDG solutions are superconvergent with an order p+ 2 toward particular Gauss-Radau projections of the exact solutions. Finally, we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves (p+ 1) -th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results.
Original language | English (US) |
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Pages (from-to) | 437-476 |
Number of pages | 40 |
Journal | Communications on Applied Mathematics and Computation |
Volume | 4 |
Issue number | 2 |
DOIs | |
State | Published - Jun 2022 |
Keywords
- A priori error estimation
- Gauss-Radau projections
- Local discontinuous Galerkin method
- Optimal superconvergence
- Semilinear second-order elliptic boundary-value problems
- Supercloseness
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics