RESEARCH IN RESIDENCE — Unconditionally stable conforming space-time methods for the Schrödinger equation

This project focuses on the development of unconditionally stable conforming space—time methods for the linear time‑dependent Schrödinger equation. The main challenge is to overcome the lack of inf‑sup stability in the standard Petrov‑Galerkin formulation for this equation. In fact, the conforming space—time finite element discretization requires a restrictive CFL condition. To address this issue, we propose two approaches. ‑The first one is to augment the sesquilinear form of the standard scheme by adding a carefully chosen inconsistent penalty term to improve the discrete properties of the scheme. By introducing this term, we aim to improve stability without compromising convergence order. ‑The second approach involves the composition of the continuous sesquilinear form with an appropriate transform of the test functions. This modification aims to endow the new sesquilinear form with higher stability properties than its unmodified counterpart.