The project aims to develop new techniques for the numerical analysis of partial differential equations in real-world situations. These equations are essential to model and understand phenomena such as oil extraction, carbon sequestration, and groundwater contamination. Due to their complexity, such phenomena can be predicted only by numerical schemes. Confidence in the outcomes of these schemes critically depends on mathematicians to rigorously establish their accuracy. By moving beyond the analysis of numerical schemes in idealised situations, and by drawing on pure mathematics tools, the project's new techniques aim to significantly improve the reliability of the predictions under assumptions that are compatible with field applications.