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.