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 themorewe develop numerical analysis for PDEs, themorewe 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 cellswith 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. Such challenges require both the design of new schemes, and the development of a rigorous mathematical analysis of old and new schemes to ensure that they properly handle these questions with innovative toolsmade available by themost advanced research inmathematics – benchmarking is essential to assess the quality of a scheme, but it cannot cover all situations encountered in applications. The analysis of advanced numerical methods and their application to problems arising in the field of Engineering and Computer Science also identifies the key elements that make schemes robust and efficient, and clarifies relationships between the various methods. These themes, among others, were at the core of two main events of the program on Numerical Methods for PDEs, organised at the Institute Henri Poincaré (Paris, France) at the end of 2016 by D. A. Di Pietro, A. Ern and L. Formaggia. The first of these events was an introductory school on current topics in numerical methods for PDEs: virtual element methods [5], hybridizable discontinuous Galerkin methods [9], the gradient discretisation method [11], defective interface conditions [13], mimetic spectral methods [16], low-rank tensor methods [19], reduced basis method [15], adaptive and a posteriori finite element methods [10, 18]. The second event was a conference on “Advanced numerical methods: Recent developments, analysis, and applications”. The whole IHP program included other major events and tackled many more issues around numerical PDEs, we refer to the webpage http://imag.edu.umontpellier.fr/event/ihp-quarter-on-numericalmethods-for-pdes/ for more details.