Partial Differential Equations (PDEs) are widely used as the key mathematical models in the research and the study of a multitude of physical problems since decades. A wide range of problems on scientific and technological applications require the use of complex PDEs in which the classical solving methods are not efficient enough.
In this project, referring mainly to application problems involving the interaction-existence of heterogeneous materials (multiphysics, multidomain), the PDEs used for modeling are elliptic or parabolic type and are characterized by the presence of discontinuous coefficients on the interaction's internal borders of the heterogeneous materials, i.e. the interface points (interfaces). This fact implies the discontinuity of the PDE's solution and / or its derivatives. Examples of such problems, which constitute the challenge of the proposed research, derive from the following two applications of significant importance and complexity from the fields of Medicine and Environmental Engineering:
The way to address and deal with the prementioned problems is a serious challenge for the entire scientific community, given the sensitivity and the increasing attention they receive in modern society. Evidently, the accuracy of the computations and the speed of the calculations are of exceptional importance.
The efficient solution of such problems requires either the adaptation of known analytical or numerical PDEs solving methods, in order to realistically approach the solution of the corresponding problems, or the research and, ultimately, the development of new innovative methods which take into account the specificities and complexities of the problem in a higher level of modelling. At the same time, modern computing systems provide new capabilities in solving complex problems but thorough study is required if they are to be fully exploited.
Upon the completion of the project, the aim is to have a research platform of complex (multiphysics, multidomain) problems developed, in which a large scale of complex simulations that require collaboration and integration of physical concepts, mathematical solving methods, discretization methods, algebraic systems solvers, software and computer architectures, can take place. An essential condition for the creation of this platform is the coordination of the interdisciplinary research activity in order to develop high-precision, fidelity and efficiency methods for the solution of such problems, form the respective software and implement in modern computing systems.