The study of partial differential equations (PDE's) is a fundamental subject area of mathematics which links important strands of pure mathematics to applied and computational mathematics. Indeed PDE's are ubiquitous in almost all of the applications of mathematics where they provide a natural mathematical description of phenomena in the physical, natural and social sciences.
Partial differential equations and their solutions exhibit rich and complex structures. Unfortunately, closed analytical expressions for their solutions can be found only in very special circumstances, and these are mostly of limited theoretical and practical interest. Thus, scientists and mathematicians have been naturally led to seeking techniques for the approximation of solutions. Indeed, the advent of digital computers has stimulated the incarnation of Computational Mathematics, much of which is concerned with the construction and the mathematical analysis of numerical algorithms for the approximate solution of partial differential equations.
The present programme focuses on some of the most exciting and promising mathematical ideas in these fields, and those branches of PDE theory that provide a source of physically relevant and mathematically hard problems to stimulate future development. The main themes of the programme are:
- Cognitive algorithms: adaptivity, feed-back and a posteriori error control;
- Hierarchical modelling: multi-scale mathematical models and algorithms;
- Nonlinear degenerate PDEs and problems with non-smooth solutiopns and with interfaces and line singularities: fourth-order degenerate parabolic equations, fully nonlinear second-order equations, free boundary problems for unknown codimension 1 interfaces, and equations of motion for line and point defects.
The aim of this programme is to bring together experts in the numerical, applied and computational analysis of PDE's and their applications and to address several major topics of research of significant mathematical, scientific and computational importance.
Concurrent Newton Institute Programme: