Objectives:
The investigation aims at developing advanced methods for flow-field analysis particularly for transitional and turbulent flows. The work deals with decomposition techniques, mainly with decomposition of motion and vorticity. The vortex-identification criterion under development at present is associated with the corotation of line segments near a point and with the so-called residual vorticity obtained after the 'removal' of local shearing motion. The new quantity - the average corotation - is a vector, thus it provides a good starting point for developing both region- and line-type vortex-identification methods, especially predictor-corrector type schemes. Further, the project aims to analyze the strain-rate skeleton so as to draw a more complete picture of the flow. For testing purposes, large-scale numerical experiments based on the solution of the Navier-Stokes equations (NSE) will be performed for selected 3D flow problems using finite element method on parallel supercomputers. Some qualitative properties of the solutions to the NSE will be studied and described in detail.
The proposed project is a free continuation of the grant Finite element method for three-dimensional problems IAA1019201, which terminated in 2006 and which was fulfilled with excellent results. The main goal of the new project will be a thorough mathematical and numerical analysis of the finite element method for solving partial differential equations in higher dimensional spaces. The necessity of solving such problems arises, e.g., in theory of relativity, statistical and particle physics, financial mathematics. In particular, we would like to deal with generation of simplicial meshes of polytopic domains. Further, we will investigate the existence and uniqueness of continuous and approximate solutions of problems that are often nonlinear. A special emphasize will be laid also on a priori and a posteriori error estimates, superconvergence, discrete maximum principle, stability of numerical schemes, etc.