Fleurianne Bertrand

In solid mechanics, the quantity expressing the internal forces is the stress. Surely, the stresses can be computed from the primal approach in a post-processing but this approach leads to stresses, which divergence is not squarely integrable. In other words, they are not H(div)-conforming and surface or interface traction forces cannot be evaluated. Moreover, momentum cannot be exactly conserved. As this quantities are driving quantities the standard methods are not accurate enough in many challenging applications and stress-based method are known to be a game changer to obtain guaranteed accuracy. However, in solid mechanics, the symmetric gradient arising in the corresponding partial differential equation is not an easy condition to impose at the discrete level. The conforming discretisation of those stresses, the so-called Arnold-Winter approach, is highly sophisticated and increase the computational cost enormously.
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)