The div—curl system is an important class of first-order partial differential equations. This system governs, for example, static electromagnetic fields, and incompressible irrotational fluid flows. The div—curl system is also fundamental from a theoretical point of view, since the Stokes equations and the incompressible Navier—Stokes equations written in the first-order velocity—pressure—vorticity formulation, as well as the Maxwell equations consist of two div—curl systems. The three-dimensional div—curl system is traditionally considered as “overdetermined” or “overspecified”, because it has four equations involving only three unknowns. For this reason, it is not easy to solve by using conventional numerical methods. In this chapter, we will prove that the div—curl system is really well determined and strongly elliptic by introducing a dummy variable, and explain that for the well-posedness the div—curl system should have two algebraic boundary conditions. We will also show that the LSFEM is the best choice for numerical solution of the div—curl system.
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