General setting

Recall that, for a given differential operator \( \mathcal{A} \) that is uniformly elliptic on a bounded open domain \( \Omega \subset \mathbb{R}^d\), we formulate the classical eigenvalue problem. This means, we want to find \( v \) and \( \lambda \) such that \[ \mathcal{A} v = \lambda v, \text{ in } \Omega, \ \ v=0, \text{ on } \partial \Omega. \]

Consider the variational formulation of the previous problem. I.e., we seek \( (v,\lambda) \in H\times \mathbb{C} \) such that \[ a(v,w) = \lambda (v,w), \ \ \forall v \in H, \ \lVert v \rVert = 1. \]

Define \( V:=H\times \mathbb{C} \). Then, we can summarize both constrains into the following expression \[ A(v,\lambda; w, \chi) := -a(v,w) + \lambda(v,w) + \overline{\chi}(\lVert v \rVert^2 - 1),\] for all \( (w,\chi) \in \mathbb{C}\).

Optimizing our goal

We still aim to optimize the quantity \[ j_\omega(u) := \frac{1}{2\imath \lVert u\rVert_\omega^2} \int_{\partial \omega} (\partial_n \overline{u})u - (\partial_n u)\overline{u} \, \mathrm{d}s. \] For general purposes, we use \( J : V \to \mathbb{R} \). We might need to study the posibility of regularizing \( j_\omega\).

Euler-Lagrange approach

Defining a Lagrangian and taking into account the Euler-Lagrange equations, we obtain two sets of equations. The primal problem is defined by our primal problem which is defined by the constrain and is equivalent to , i.e., \[ A(v,\lambda; w, \chi) := -a(v,w) + \lambda(v,w) + \overline{\chi}(\lVert v \rVert^2 - 1) = 0, \] for all \( (w,\chi) \in \mathbb{C}\).

The dual problem is defined by the second optimality condition. \[ \dot{A}(v,\lambda; \hat{v}, \hat{\lambda}, w, \chi) = \dot{J}(v,\lambda; \hat{v}, \hat{\lambda}). \] where \[ \dot{A}(v,\lambda; \hat{v}, \hat{\lambda}, w, \chi) = -a(\hat{v},w) + \lambda(\hat{v},w) + \hat{\lambda}(v,w) + 2\chi\Re(v,\hat{v}).\]

Big hammer: a posteriori error representation

For a general constrained problem of the form \( A(u;f) = F(f) \) and a functional \(J\), the error given by the Galerkin approximation of the saddle-point appoximation is given by \[ J(u) - J(u_h) = \frac{1}{2}\rho(u_h;z - \tilde{z_h}) + \frac{1}{2}\rho^\star(z_h;u - \tilde{u_h}) + R, \] where \( R\) is a third-order residual.

Morals of the technique

For some carefully selected \( J\), the residual term vanishes. We can construct different error estimators after breaking down the weak forms, and applying several Cauchy-Schwarz inequalities. I hope to show you some other examples of this technique!