The solution of the Zakai equation provides the complete conditional probability density of the state, given the observations. Numerical solution of this equation by the finite difference method usually leads to large systems of equations which have to be solved at each time step, especially when the dimension of state space is more than two. We propose in this paper, for the first time to our best knowledge, a grid-based four dimensional algorithm to solve the Zakai equation. Our approach is based on an adaptive local grid refinement method and is illustrated with a bearings-only target motion analysis example.