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**Contact**

Fleurianne Bertrand

TU Chemnitz

Fakultät für Mathematik

Reichenhainer Straße 41

09126 Chemnitz

Germany

**TU ChemnitzNumerical analysis and Optimisation**

**with Prof. Daniele Boffi, founded by KAUST.**

We aim at reducing the complexity of the simulations of large, complex systems described in terms of partial differential equations in the spirit of Reduced Order Methods (ROMs).

As a first step, we considered a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition, see our recent preprint here.

**with Prof. Caroline Birk and Prof. Christian Meyer, founded by Mercator Research Center Ruhr (MERCUR).**

Rapid temperature changes may induce thermal stresses, which lead to the initiation and acceleration of damage and fracture processes in structures. The aim of this project is to develop efficient simulation methods for the prediction of thermally-induced crack propagation processes. To this end, a global model based on the scaled boundary finite element method is coupled with a local thermoelastic two-field damage model. Another aim of this project is to develop appropriate error estimators in order to reduce the number of degrees of freedom.

See also our recent publication on the convergence rate of the SBFEM.

**with Prof. Jörg Schröder and Prof. Gerhard Starke, founded in the DFG priority programm SPP 1748.**

The main objective of this Priority Programme is the development of modern non-conventional discretisation methods, including the mathematical analysis for geometrically as well as physically non-linear problems in the fields of e.g. incompressibility, anisotropies and discontinuities (cracks, contact).

**founded by Universität Duisburg-Essen, programm for excellent young researcher.**

A common way to determine the displacement of a structure under forces, the so-called primal approach, is based on the minimisation of an energy depending on the displacement variable. However, this so-called primal approach does not lead to uniform accuracy for nearly incompressible materials such as common blood vessel walls or the heart muscle, except if the degree of the polynomial approximation is kept very high, leading to a high computational effort.

To address the aforementioned issue, my overall aim is to develop numerical methods with very high and guaranteed accuracy. The key objectives are:

• Guaranteed accuracy: given a tolerance, the true error is less and greater than the tolerance, up to multiplicative constants.

• Robustness: the constants in the error bounds don’t depend on model parameter.

• Accessibility : the computational cost is moderate. The constants in the error bounds depend only on local quantities (i.e. mesh-size, shape of the element) and not on global quantities (such that as Poincaré constant of the whole domain)

Elliptic boundary value problems arising in science and engineering are often posed on domains with curved boundaries. For low-order finite elements it is usually sufficient to use a polygonal approximation of the boundary in order to ensure that the finite element order of convergence is not affected by the inaccurate representation of the boundary. For higher-order finite elements a more accurate representation of the boundary is required which is usually achieved by isoparametric finite elements; i.e., the same finite element space is used once more for the parametrization of a more accurate domain approximation.

Related publications:

Parametric Raviart-Thomas elements for mixed methods on domains with curved surfaces

Bertrand, F. & Starke, G., 2016, In: SIAM Journal on Numerical Analysis. 54, 6, p. 3648-3667 20

First-order system least squares on curved boundaries: Lowest-order Raviart-Thomas elements

Bertrand, F., Münzenmaier, S. & Starke, G., 2014, In: SIAM Journal on Numerical Analysis. 52, 2, p. 880-894

First-order system least squares on curved boundaries: Higher-order Raviart-Thomas elements

Bertrand, F., Münzenmaier, S. & Starke, G., 2014, In: SIAM Journal on Numerical Analysis. 52, 6, p. 3165-3180 16 p.

First-order system least-squares for interface problems

Bertrand, F., 2018, In: SIAM Journal on Numerical Analysis. 56, 3, p. 1711-1730 20 p.