Structure-preserving finite elements and the convergence of augmented Lagrangian methods

Problems with physical constraints, such as the incompressibility constraint for mass conservation in fluids or Gauss's laws for electric and magnetic fields, result in generalized saddle point systems. So-called structure-preserving finite elements respect the constraints pointwise, resulting in more physically accurate solutions that are typically robust with respect to some problem parameters. However, constructing these finite elements may involve complicated spaces for the Lagrange multiplier variables. Augmented Lagrangian methods (ALMs) provide one process to compute the solution without the need for an explicit basis for the Lagrange multiplier space. In this talk, we present new convergence estimates for a standard ALM method, sometimes called the iterated penalty method, applied to structure-preserving discretizations of linear saddle point systems.