Our final goal is to derive a posteriori error estimates for the Adaptive Local Basis (ALB) method that is used solving non-linear eigenvalue problems in the framework of Kohn-Sham models in computational chemistry.
The characteristics of the ALB-method is that it constructs local basis functions by diagonalizing the operator locally and then solving the global eigenvalue problem using the Discontinuous Galerkin (DG) technique.
We start with analyzing the a posteriori error estimates for the DG-method applied to linear Schrödinger equations, but where the nature of the basis functions are unknown.

The main challenge is that no inverse estimates, which are commonly used in a posteriori error analyses, are available for generic basis functions. To overcome this, we accept computations on a very fine grid on each element of the DG-method, but they should remain local and independent. We then compute local constants that are subsequently used in the error estimates. Finally, we present numerical examples that illustrate the behavior of the estimates.