In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.

中文翻译：

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笛卡尔网格上非线性二阶椭圆问题的局部不连续Galerkin方法的最优误差估计

在本文中，我们研究了局部间断Galerkin（LDG），用于所述类型的两维二阶非线性椭圆问题的方法Ü _XX  +  Ù _YY = ˚F（X，ÿ，ü，ü _X，ü _ÿ），在一个矩形区域Ω用的边界上经典边界条件Ω。建立了解和近似于其梯度的辅助变量的收敛性质。更具体地说，我们使用对偶性参数来证明LDG解与L中的精确解之间的误差当使用度最大为p的张量积多项式时， ²范数达到最佳（p  +1）阶收敛。此外，我们证明了LDG解的梯度朝着精确解的Gauss-Radau投影的梯度具有p  + 1阶的超闭合。结果示于上使用的程度张量积多项式笛卡尔网格二维空间有效p  ≥1 ，以及用于二者混合狄利克雷-诺伊曼和周期性边界条件。初步的数值实验表明我们的理论发现是最优的。