Abstract: Elastic membrane contact problems and elasto-plastic torsion problems are typical variational inequality problems, and the accurate and efficient analyses of these problems, especially in a self-adaptive manner to satisfy the user-preset error tolerance, are still of great challenges. This paper reports a recently developed approach to the self-adaptive finite element (FE) analysis of one-dimensional (1D) variational inequality problems. To obtain the FE solution rapidly based on variational inequalities, the present paper proposed two techniques, i.e. bisection bounding and C-check technique, which greatly accelerate the convergence rate of the conventional relaxation iteration, and in average, it suffices to implement only 4 to 5 steps of linear FE analysis. With the converged FE solution obtained, the super-convergent solution by EEP (Element Energy Projection) method, also developed by the first author’s research team, is calculated to estimate the errors and then to guide a mesh refinement with final results being virtually ‘numerically exact’, i.e. satisfying the error tolerance by the maximum norm. The numerical examples presented show that the proposed approach is efficient, accurate and reliable.