This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type \(u^{(4)}=f(x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime })\), x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables 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 polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L² norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal.

本文涉及类型为（（u ^ {（4）} = f（x，u，u ^ {）的非线性四阶边值问题的局部不连续Galerkin（LDG）有限元方法的收敛性和超收敛性\素}中，u ^ {\素\素}中，u ^ {\素\素\素}）\） ，X ∈[一，b ]与在端点古典边界条件。建立了解和所有三个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，当次数为p的多项式时，LDG解和L ²范数中精确解之间的误差达到最优（p +1）阶收敛。被使用。我们还证明，LDG的解决方案，并在精确解的高斯- 10期Radau突起之间的误差的衍生物大号²范数与顺序超收敛p + 1。此外，一个（2 p + 1）阶超收敛为在准均匀网格下，还可以获得网格节点处的数值通量以及单元平均值的误差。最后，我们证明了LDG解对特定解的精确投影具有p + 2阶的超收敛性。在本文提出的误差分析是有效的p ≥1，数值实验表明，我们的理论研究结果是最优的。