Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids

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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 languageEnglish (US)
Pages (from-to)437-476
Number of pages40
JournalCommunications on Applied Mathematics and Computation
Volume4
Issue number2
DOIs
StatePublished - 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

  • Applied Mathematics
  • Computational Mathematics

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