A least squares finite element method for the magnetostatic problem in a multiply-connected Lipschitz domain

A new least-squares finite element method is developed for the curl-div magnetostatic problem in Lipschitz and multiply connected domains filled with anisotropic nonhomogeneous materials. In order to deal with possibly low regularity of the solution, local L2 projectors are introduced to standard least-squares formulation and applied to both curl and div operators. Coercivity is established by adding suitable mesh-dependent bilinear terms. As a result, any continuous finite elements (lower-order elements are enriched with suitable bubbles) can be employed. A desirable error bound is obtained: ||u-uh||0 = C ||u-˜u||0, where uh and ˜u are the finite element approximation and the finite element interpolant of the exact solution u, respectively. Numerical tests confirm the theoretical results.