The inverse Poisson problem is a prototype of an indirect measurement problem when the underlying physics can be modeled as a diffusion process. Examples of this type include inverse heat transfer problems, electrocardiography problems as well as certain inverse problems of elasticity.
The model describing how the measured quantity depends on the quantity of interest is called the forward problem. The inverse problem can be described as the inverse of the forward problem. The inverse Poisson problem is ill-conditioned in the sense that a small perturbation in the measured quantity can cause an arbitrarily large change in the quantity of interest. Because of this nature, the solution is obtained as a regularized inverse of the forward problem.
Since even the forward Poisson problem can't be solved exactly, only an approximation of the inverse solution is obtainable. This approximate solution is obtained by discretizing the quantity of interest and using a FE-approximation of the forward problem. Both the discretization and the use of an approximate forward problem cause a certain amount of error. The effect of both error sources are investigated.