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Purpose – The authors' objective in this paper is to find the numerical solutions of obstacle, unilateral and contact second-order boundary-value problems. Design/methodology/approach – To achieve this, the authors formulate a spatially adaptive grid refinement scheme following Galerkin's finite element method based on a weighted-residual. A residual based a-posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local element balance has been considered as an error assessment criterion. The approach utilizes piece-wise linear approximations utilizing linear Langrange polynomials. Numerical experiments indicate that local errors are large in regions where the gradients are large. Findings – A comparison of the spatially adaptive grid refinement with that of uniform meshing for second order obstacle boundary value problems confirms the superiority of the scheme without increasing the number of unknown coefficients. Originality/value – The authors believe the work has merit not only in terms of the approach but also of the problem solved in the paper. [ABSTRACT FROM AUTHOR]