Variational and D-N approaches for a magnetostatic Cauchy problem

Abstract

This chapter considers a magnetostatic problem in which two kinds of boundary data are simultaneously imposed on a part of the boundary of a bounded domain. This is a variant of the Cauchy problem of the Laplace equation in which Dirichlet and Neumann data are simultaneously prescribed on a part of the boundary. The problem is recast to a variational problem to identify the proper boundary condition. Minimization of a functional using the steepest descent method leads the variational problem to an iterative process in which two-boundary value problems are involved. These problems are called primary and adjoint problems respectively. After computations by using these two numerical methods, it is concluded that the variational algorithm is stable, and the numerical solution is in good agreement with the exact one. The D–N algorithm is better than the variational algorithm in the cost of computations.