Mixed Finite Element Solution for a Magnetostatic Inverse Problem

Abstract

The purpose of this paper is to present a numerical method of solution for a two-dimensional magnetostatic problem. The normal component of the magnetic flux density and the tangential component of the magnetic field with errors are simultaneously imposed on a part of the boundary of a bounded domain of the problem. The problem can be regarded as a boundary value inverse problem, because the proper boundary condition is to be identified for the rest of the boundary. The treatment is based on the method of least squares, and the steepest descent method minimizes an objective functional with a regularization term. The direct variational method paraphrases the inverse problem to the primary and the adjoint boundary value problems. The mixed finite element method using the edge element and the conventional finite element method are applied to the numerical solutions of the boundary value problems. Based on numerical computations, it is concluded that an estimated solution for the boundary data without errors is in agreement with an exact one, and the regularization term yields good approximate solutions for the boundary data with errors.