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

作　　者：Mahboub Baccouch 

机构地区：[1]Department of Mathematics,University of Nebraska at Omaha,Omaha,NE 68182,USA

出　　处：《Communications on Applied Mathematics and Computation》2022年第2期437-476,共40页应用数学与计算数学学报（英文）

基　　金：This research was supported by the NASA Nebraska Space Grant(Federal Grant/Award Number 80NSSC20M0112).

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 

分 类 号：O17[理学—数学]