In this paper, we present and analyze a posteriori error estimates for the ultra-weak local discontinuous Galerkin (UWLDG) method applied to nonlinear fourth-order boundary-value problems for ordinary differential equations of the form \(-u^{(4)}=f(x,u)\) . Building upon the superconvergence results established in Baccouch (Numer Algor 92(4):1983–2023, 2023), we demonstrate the convergence of the UWLDG solution, in the \(L^2\) -norm, towards a special p-degree interpolating polynomial when piecewise polynomials of degree at most \(p\ge 2\) are employed. The convergence order is proven to be \(p+2\) . Additionally, we decompose the UWLDG error on each element into two components. The dominant component is proportional to a special \((p+1)\) -degree polynomial, represented as a linear combination of Legendre polynomials with degrees \(p-1\) , p, and \(p+1\) . The second component converges to zero with an order of \(p+2\) in the \(L^2\) -norm. These findings enable the construction of computationally efficient a posteriori error estimates for the UWLDG method. These estimates are obtained by solving a local problem on each element without imposing boundary conditions. Furthermore, we establish that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the \(L^2\) -norm as the mesh is refined, with a convergence order of \(p+2\) . In addition, we prove that the global effectivity index converges to unity at a rate of \(\mathcal {O}(h)\) . Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Numerical results are provided to illustrate the reliability and efficiency of the proposed error estimator.
