Galerkin schemes and inverse boundary value problems in reflexive Banach spaces

We develop the Galerkin method for a recent version of the Lax–Milgram theorem. The generation of the corresponding finite-dimensional subspaces for concrete boundary value problems leads us to consider certain biorthogonal systems in the reflexive Banach spaces in question. In addition, we present an application to the numerical solution of inverse problems involving certain elliptic boundary value problems.