An eclectic view on numerical methods for PDEs:Presentation of the special issue "advanced Numerical Methods: Recent Developments, Analysis and Applications"

The design and analysis of numerical methods for partial differential equations (PDEs) is a field of constant growth. It seems that the more we develop numerical analysis for PDEs, the more we face questions and problems that require additional developments – maybe a consequence of an increased knowledge that gives the tools to tackle questions previously inaccessible and of the growing place of scientific computing in Physics and Engineering. The aspects currently taken into account when dealing with numerical schemes for PDEs include: the need for geometrical flexibility, as the meshes available for certain applications may comprise cells with complicated shapes, possibly degenerate faces or bad aspect-ratios; the capacity to handle complex models, with non-linearities, couplings, strong heterogeneities, weakly regular data or even singularities; the respect of physical properties of the model, such as conservation of energy or mass, and proper asymptotic behaviour. The above mentioned challenges are ubiquitous in practical problems arising from Engineering, Life Sciences, and Industry