**Context of the thesis:**

The aim of this PhD is to propose, to implement and to investigate

efficient numerical methods for models of fuel cells (electrochemical devices converting the chemical

energy of a fuel into electricity). The fuel cell is a highly attractive energy source for the transport

of future since it has a high energy efficiency, produces no greenhouse gas and allows fast fuel

loading. However, many challenges remain to be overcome: the optimal design at energy efficiency,

durability, control, security and cost. Since it is difficult to measure the values of the state variables

inside the fuel cell, numerical simulations turn out to be an indispensable tool both to design and

optimise new devices and to control the dynamics of the existing cells in real time. The PhD

project will be focused on the PEMFC (Proton Exchange Membrane Fuel Cell), which is one of

the most popular types of fluid cells, extensively studied at FEMTO-ST/FCLAB. The models

of PEMFC allowing for efficient numerical simulations have been proposed in ^{5}, for example.

However, the mathematical study of such models is still lacking, especially with regard to the

control of errors due to modeling and to numerical discretization. Finding the right compromise

between precision and computation time is a crucial task and understanding of different sources

of approximation errors is fundamental to fulfill it.

**Challenges and expected contributions:**

The starting point for modeling PEMFC is the

system of partial differential equations (PDEs) combining electrochemistry and gas dynamics.

Physically, it is reasonable to consider these equations on a two-dimensional (2D) domain, but in

practice they are partially reduced to the one-dimensional (1D) setting through some modeling

assumptions in order to allow for efficient numerical discretization. The error of the resulting

numerical simulation comes from two sources: modeling approximations and discretization.

One needs some accurate and computationally cheap error bounds in order to control and minimize the

error. In particular, one should be able to decide whether the error comes mostly from modeling

(one should then go back from 1D to 2D) or from discretization (one should then refine the mesh).

In this PhD project, we would explore the techniques of a posteriori error control, one possibility

could be to develop the appropriate error estimators based on flux and potential reconstructions

^{3}. This approach has proven to be efficient in order to split the sources of error, cf. ^{4} for

example. However, adapting it to our context of modeling error estimation will be a non trivial

task. Moreover, if local mesh refinement is deemed necessary, the implementation constraints

may preclude the use of traditional approach of conforming remeshing. Indeed, the real time

computations are desired which should be typically done on uniform meshes. We propose to overcome this difficulty using the approach of finite element patches ^{1} where a fine mesh is put over a global coarse mesh where needed, thus avoiding the construction of complicated meshes.

The first attempt to automatize this process by a posteriori error control is presented in ^{2} using

the explicit residual estimators. In the present project, we propose to explore the capabilities of

the flux reconstruction estimators to guide the placement of a patch and also the degree of mesh

refinement inside the patch.

The thesis will involve both theoretical investigation of the proposed numerical methods and their implementation to demonstrate their capabilities (without necessarily going to heavily optimized coding directly suitable for real-life real-time applications).

**Conditions:**

The PhD student will be attached to the Mathematics Laboratory (Laboratoire de Mathématiques de Besançon), University of Franche-Comté at Besançon, France. The grant will be about 1400 euros net per month. The position does not involve any teaching by default but a light teaching load may be added (not guaranteed) leading to an increase of the salary. It will be possible to benefit of a housing on the Campus at Besançon.

**Application and deadlines:**

Candidates should send an application containing a CV with a clear indication of academic achievements, a motivation letter, and (if possible) one or more letters of recommendation to:

**Alexei Lozinski (alexei.lozinski@univ-fcomte.fr)**

The successful candidate will have a solid background in numerical methods for PDEs. Skills in

programming (e.g. c++, matlab) and/or modeling of physical phenomena as those mentioned above will be appreciated.

The deadline for application is August 25, 2019.

The PhD is expected to start in October 2019. The funding will be guaranteed for 3 years.

**References**

1] *R. Glowinski, J. He, A. Lozinski, J. Rappaz and J.Wagner, Finite element approximation of multi-scale elliptic problems using patches of elements, Numer. Math. (2005) 101(4):663–687.*

^{2} *M. Duval, A. Lozinski, J.C. Passieux, and M. Salaün, Residual error based adaptive mesh refinement with the non-intrusive patch algorithm, Computer Methods in Applied Mechanics and Engineering Computer Methods (2018) 329:118–143.*

^{3} *A. Ern and M. Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal.(2015) 53:1058–1081*

^{4} *A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput.(2013) 35(4):A1761–A1791.*

^{5} *D. Zhou, F. Gao, E. Breaz, A. Ravey, and A. Miraoui, Tridiagonal Matrix Algorithm for Real-Time*

*Simulation of a Two-Dimensional PEM Fuel Cell Model, IEEE Transactions on Industrial Electronics (2018) 65(9):7106–7118.*