  We propose an approximation scheme for a class of semilinear variational
inequalities whose Hamiltonian is convex and coercive. The proposed scheme is a
natural extension of a previous splitting scheme proposed by Liang,
Zariphopoulou and the author for semilinear parabolic PDEs. We establish the
convergence of the scheme and determine the convergence rate by obtaining its
error bounds. The bounds are obtained by Krylov's shaking coefficients
technique and Barles-Jakobsen's optimal switching approximation, in which a key
step is to introduce a variant switching system.
