A Posteriori Modeling Error Estimates in Continuum Mechanics

• Summer Term 2015
• 2 hours per week
• Date: Wednesday, 9.15-11.00
• Room: A02, MPI for Mathematics in the Sciences

In continuum mechanics, many models of different accuracy are available for a single physical situation: For example, the behavior of a fluid may be described by particle models treating all molecules individually, by the Boltzmann equation, or by the Navier-Stokes equation. The latter more simple models arise from the former more complex models by formal arguments which rely on additional modeling assumptions.
It is often a challenging task in applied analysis to justify such model simplifications rigorously; likewise, in applications it is often a difficult problem to decide which model simplifications are possible in a given situation without introducing an unacceptably large error in the solution.
In this lecture, we shall present a comparably recent approach to modeling error estimation: A posteriori modeling error estimates use the information provided by the solution to the simplified model in order to obtain significantly improved bounds for the model simplification error. In certain situations, they are even the only known way of estimating the modeling error rigorously. The contents of the lecture may include:

• A posteriori error estimation strategies for the numerical error for PDEs
• A posteriori estimates for the modeling error in periodic homogenization of elliptic PDEs
• Bounds on the modeling error for (dimensionally reduced) plate models, as compared to full three-dimensional elasticity
• Estimates for the modeling error caused by approximating a slightly compressible fluid as perfectly incompressible

Prerequisites: basic knowledge of PDEs and Sobolev spaces