## Project Description

The project will be focused on the a posteriori error control and adaptivity of time- dependent, two-level Shrödinger systems with conical crossings.

• Mathematically, any generic two-level Shrödinger system will exhibit conical crossings.

• Shrödinger systems with crossings can describe many chemical systems, and the most interesting phenomena, such as chemical reactions, are related exactly with the conical crossing.

• Finally, not only there are no exact solutions for such systems, but in general very little is known even for the qualitative behaviour of the solutions.

For these reasons, it is of fundamental importance to have mathematical tools that will lead us to accurate approximations without the knowledge of the exact solution of the system. This can be achieved through the construction of solvers with a posteriori error estimates.

An a posteriori error estimate can simply be described as follows: If u is the exact solution of the problem and U a computed approximation of it, we are interested in estimates of the form:

(1)

u − U

≤ E(U),

for some norm natural for the problem. In (1) the estimator E(U) is a computable quantity depending only on the approximate solution U and the known data of the problem. For meaningful computations we also require E(U) to reduce as fast as the error

u − U

, i.e. to give a reasonable estimation of the actual error (even though in general it will not be sharp). Having at hand E(U) we can make it smaller than a given tolerance Tol, E(U) ≤ Tol, and thus from (1), we will also have

u − U

≤ Tol. In this way, the error control through a posteriori error estimates not only leads to reliable computations, but also provides mathematical guarantees on how accurate the approximate solution is. One of the aims of this project is to obtain estimates of the form (1) for time-dependent, two-level Schrödinger systems with conical crossings.

Another great advantage of the a posteriori error control is that it may lead to the construction of an adaptive algorithm, i.e., a numerical algorithm in which non-uniform grids (variable spatial mesh sizes and time-steps) are allowed. This is very important for the reduction of the computational cost, i.e. to allocate the degrees of freedom of the computation where they are most needed. Additionally, for problems with singularities, such as problems with conical crossings, it is very difficult to numerically study the singular area of the solution without adaptivity. Since the singular area is usually the most interesting part of the solution mathematically, but also from the point of view of applications, adaptivity is crucial.

A related question is “How can we use E(U) in (1) to construct an adaptive algorithm?” An efficient solver E(U) can be written in the form of a sum of local in space and time estimators. This gives an understanding where the error is coming from; in particular, using this sum of local estimators, the regions where the solution exhibits singular behaviour can be detected. The idea is then to add grid points wherever the error is big with the aim to reduce it, and to remove points from areas where nothing interesting seems to happen, i.e., the error is small. In this way, the computational cost is substantially reduced, and numerical computations close to the singular points become feasible. Another main aim of the project is the proposition of efficient adaptive algorithms or time-dependent, two-level Schrödinger systems with conical crossings. The proposition of the adaptive algorithms will be based on rigorous a posteriori error control.

## Funding Information

This is a self-funded project only.

## Eligibility Requirements

Applicants must have obtained, or expect to obtain, a first or 2.1 UK honours degree, or equivalent for degrees obtained outside the UK in a relevant discipline.

English language requirements can be found here.

## Application Process

To apply, follow the steps below:

Step 1: Email Dr Irene Kyza ([email protected]) to (1) send a copy of your CV and (2) discuss your potential application and any practicalities (e.g. suitable start date).

Step 2: After discussion with Dr Kyza, formal applications can be made via UCAS Postgraduate:

Apply for the Doctor of Philosophy (PhD) degree in Mathematics using this link.

In the ‘provider questions’ section:

– Write the lead supervisor’s name and give brief details of your previous contact with them in the ‘previous contact with the University of Dundee’ box.

## References

[1] G. Akrivis, Ch. Makridakis, R.H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp. 75 (2006) 511–531.

[2] A. Athanassoulis, Th. Katsaounis, I. Kyza, Regularized semiclassical limits: linear flows with infinite Lyapunov exponents, Comm. Math. Sci. 14 (2016) 1821–1858.

[3] S. Jin, P. Qi, Z. Zhang, An Eulerian surface hopping method for the Schr ̈odinger equation with conical crossings, Multiscale Model. Simul. 9 (2011) 258–281.

[3] Th. Katsaounis, I. Kyza, A posteriori error control and adaptivity for Crank-Nicolson finite element approximations for the linear Schro ̈dinger equation, Numer. Math. 129 (2015) 55–90.

[4] Th. Katsaounis, I. Kyza, A posteriori error analysis for evolution nonlinear Schro ̈dinger equations up to

the critical exponent, SIAM J. Numer. Anal. 56 (2018) 1405–1434.