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)\documentclass[12pt]{minimal}
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\begin{document}$$-\Delta u=f(x,y,u)$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-norm of O(hp+1)\documentclass[12pt]{minimal}
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\begin{document}$${O}(h^{p+1})$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$p+ 2$$\end{document} 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)\documentclass[12pt]{minimal}
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\begin{document}$$(p+1)$$\end{document}-th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results.
