Lecture Harm Askes, University of Twente

Many problems in science and engineering can be described with a system of coupled or uncoupled partial differential equations (PDEs). Typically, these PDEs are of second-order in space, meaning that they contain up to second-order spatial derivatives of the relevant state variables. The consequences for numerical solution methods is that C0-continuous approximations, such as those provided by standard finite element shape functions, are sufficient. However, the rising interest in describing multi-scale and/or multi-physics problems has resulted in PDEs that may be more complicated. An illustrative example is a class of so-called generalised continua in which higher-order spatial derivatives appear in the PDEs.

In this talk, we will explore solution methods for PDEs that are of higher-order than two. In particular, we will investigate possibilities to rewrite such higher-order PDEs into a series of lower-order PDEs, with a view to using standard numerical technology to solve these. In certain cases, it is possible to write the higher-order PDE as a series of uncoupled lower-order PDEs, whereas in other cases the lower-order PDEs are coupled - the latter category will be shown to possess an intrinsic multi-scale character.

Model PDEs covered will include gradient-enriched elasticity, gradient-enriched piezomagnetics, and a relatively unknown plate theory from structural mechanics.