Numerical Analysis of Stochastic Elliptic Variational Inequalities of the First Kind

This paper presents a numerical approach to the stochastic obstacle problem using the stochastic Galerkin (SG) method. Due to the low regularity of the solution, linear finite elements are employed in both the physical and random variable spaces. Properties of random fields and variational inequalities of the first kind are employed to establish the well-posedness of the problem. Finite element spaces are introduced to construct suitable approximation subspaces, and a comprehensive SG formulation is proposed to solve the stochastic obstacle problem. Well-posedness of the discrete formulation is shown and an optimal error estimate for the numerical solution in the $H^1$-norm is derived. Numerical experiments validate the effectiveness of the SG method, showing that both the expectation error and second moment error converge at a rate of $O(h)$ in the $H^1$-norm, consistent with theoretical predictions.