Vector potential formulation for three-dimensional magnetostatics

Three-dimensional magnetostatics problems, with no a priori assumptions about model geometry or topology, admit two vector potential formulations: the physical formulation which is a consequence of Maxwell’s equations, and an alternate formulation which includes the vector Laplacian formulation as a special case. The lack of uniqueness inherent in the physical formulation is computationally undesirable because numerical discretization may produce singular coefficient matrices. The alternate formulation, on the other hand, can yield a unique solenoidal vector potential and nonsingular finite element matrices. This alternate formulation also satisfies the physical formulation if the physical problem is well posed.