Viewing the Finite Element Residual as a Lagrange Multiplier for Discretization

Instead of writing the Galerkin approximation as seeking the stationary point of some functional over a finite-dimensional subspace of the original function space, we write it as the initial, infinite-dimensional variational problem with an explicit constraint that models the fact that we are actually searching in a subspace. By formulating this as a constrained variational problem using a Lagrangian functional, we are led to introduce a Lagrange multiplier for the discreteness constraint. This multiplier turns out to be the residual of the approximation, shedding some light on the basic interpretation of the residual in Galerkin methods. Since Lagrange multipliers indicate the first order response of a functional to perturbations in the constraint, we consider applications of this relationship to mesh refinement strategies and error estimation for finite element methods. After considering an introductory example for the Laplace equation, the fully nonlinear case is treated, where the accurate computation of some functional of the solution is required.