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

doi:10.1007/s42967-021-00123-8

摘要/Abstract

摘要： 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 -Δu=f (x, y, u) on Cartesian grids. By introducing special GaussRadau projections and using duality arguments, we obtain, under some suitable choice of numerical fuxes, the optimal convergence order in L²-norm of O(h^p+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.

关键词: Semilinear second-order elliptic boundary-value problems, Local discontinuous Galerkin method, A priori error estimation, Optimal superconvergence, Supercloseness, Gauss-Radau projections

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 -Δu=f (x, y, u) on Cartesian grids. By introducing special GaussRadau projections and using duality arguments, we obtain, under some suitable choice of numerical fuxes, the optimal convergence order in L²-norm of O(h^p+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.

Key words: Semilinear second-order elliptic boundary-value problems, Local discontinuous Galerkin method, A priori error estimation, Optimal superconvergence, Supercloseness, Gauss-Radau projections

中图分类号:

Mahboub Baccouch. Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 437-476.

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