This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order ellip-tic problems of the form-Au=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.

Department of Mathematics,University of Nebraska at Omaha,Omaha,NE 68182,USA

[1]Mahboub Baccouch-.Convergence and Superconvergence of the Local Discontinuous Galerkin Method for Semilinear Second-Order Elliptic Problems on Cartesian Grids)[J].应用数学与计算数学学报,2022(02):437-476

Superconvergence,Semilinear,ellip,hp+1,superconvergent,p+2

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