In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. On a two-dimensional rectangular grid Using the finite difference numerical method to discretize the 2-dimensional Poisson equation (assuming a uniform spatial discretization, ) on an grid gives the following formula: where and . The preferred arrangement of the solution vector is to use natural ordering which, prior to removing boundary elements, would look like: This will result in an linear system: where is the identity matrix, and , also , is given by: and is defined by For each equation, the columns of correspond to a block of components in : while the columns of to the left and right of each correspond to other blocks of components within : and respectively. From the above, it can be inferred that there are block columns of in . It is important to note that prescribed values of (usually lying on the boundary) would have their corresponding elements removed from and . For the common case that all the nodes on the boundary are set, we have and , and the system would have the dimensions , where and would have dimensions . Example For a 3×3 ( and ) grid with all the boundary nodes prescribed, the system would look like: with and As can be seen, the boundary 's are brought to the right-hand-side of the equation. The entire system is while and are and given by: and Methods of solution Because is block tridiagonal and sparse, many methods of solution have been developed to optimally solve this linear system for . Among the methods are a generalized Thomas algorithm with a resulting computational complexity of , cyclic reduction, successive overrelaxation that has a complexity of , and Fast Fourier transforms which is . An optimal solution can also be computed using multigrid methods. Applications In computational fluid dynamics, for the solution of an incompressible flow problem, the incompressibility condition acts as a constraint for the pressure. There is no explicit form available for pressure in this case due to a strong coupling of the velocity and pressure fields.