Multiscale mollifier technique in poroelasticity with an introduction to thermal aspects

Poroelasticity is part of material research discipline and describes the interaction between solids deformation and the pore pressure.
Therefore, this is anywhere interesting where a porous medium and a fluid come into play and have an effect on each other. This is the case in many applications and we want to focus on geothermics. It is important to consider this aspect in reservoir management since the replacement of the water in the reservoir some kilometers below the Earth's surface has an effect on the surrounding material and vice versa.
The underlying physical processes can be described by partial differential equations, called the quasistatic equations of poroelasticity (QEP).
Our aim is to do a multiscale decomposition of the components given by displacement and pore pressure. This enables us to see underlying structures in the different decomposition scales that cannot be seen in the whole data. We want to detect interfaces and extract more details of the data.
First, we start in a more general setting, that is thermoporoelasticity which relates poroelasticity to thermal effects. After the derivation of fundamental solutions, we reduce the setting to poroelasticity. We construct physically motivated scaling functions with the help of a mollifier regularization of the appropriate fundamental solutions. Here we have a closer look that the scaling functions fulfill the necessary theoretical requirements of an approximate identity.
Further, we present numerical experiments with synthetic data, which show the applicability of our constructed functions.