Application of the Zenger Correction to an Elliptic PDE with Dirac Source Term

In this work, a numerical solution technique for an elliptic Dirac problem is investigated. One reason to study this issue is that this partial differential equation (PDE) is closely related to a mathematical model for flow processes in vascularized tissue. In order to circumvent the numerical difficulties associated with a Dirac source term a regularization technique is used, which is known as Zenger correction. Using the Zenger correction, it can be shown that a Finite Difference discretization for a standard Dirac problem converges with optimal order in regions excluding the singularity of the solution. Besides a Dirac source term, a coupling term occurring in the right hand side of the equation has to be handled. For a numerical treatment of this coupling term an iterative solution scheme is proposed. Thereby, in each iteration step a standard elliptic Dirac problem is solved. The performance of the newly developed numerical solver is illustrated by means of a series of numerical tests.