Numerical solution of under-determined 2D laplace equation with internal information

Abstract

This chapter provides a numerical treatment concerning an underdetermined problem of the Laplace equation in two spatial dimensions. The Dirichlet and Neumann data are respectively imposed on two parts of the boundary of the domain. Besides, the values of the unknown function are specified at n distinct internal points. This new problem is regarded as a boundary inverse problem because the proper boundary conditions are to be identified for the rest of the boundary. The treatment is based on the direct variational method. A functional is minimized by the method of the steepest descent. The minimization problem is transformed into iterative primary and dual boundary value problems of the Laplace equation. In simple numerical examples, this chapter compares numerical solution containing internal information with the numerical solution not containing internal information. It is concluded that adding internal information can significantly improve the numerical solution.