9802367 Babuska The project will address the following topics: 1) A posteriori error estimation in the finite element method. It will focus on the error estimation of the values of engineering interest as error in the solution, the gradients (stresses) in the entire domain as well as subdomains, the values of the functionals, etc. The upper and lower a- posteriori bounds will be addressed. The effectivity and robustness of these estimates will be theoretically and computationally investigated. The linear elliptic equations and problems occurring in engineering practice will be focussed on and also the nonlinear and parabolic ones will be investigated. The results will also be applied in the adaptive procedure. 2) The generalized finite element method. A flexible form of FEM allowing one to use special shape functions will be designed and theoretically analyzed. The classical h- and p- version is a special case. The method will be implemented in two dimensions and possibly later, three dimensions as well. This method can typically be applied for solving problems of heterogeneous material, solutions with boundary layers and oscillatory character. 3) The problem of composites. This is a multi-scale problem with particular focus on composite fibrous materials. The main interest will be on a fiber scale. The study will be both deterministic and stochastic. The stochastic formulation is essential because the position of the fiber scale statistical character. The work will be performed in collaboration with the Aeronautics Institute of Sweden. 4) The p-version of FEM. This work will be focused on resolving various aspects of the p-version in 3 dimensions although first the methodology will be used on the two-dimensional setting. The theory will utilize new functional spaces which are needed for addressing the problems occurring in engineering. The work is a continuation of the results of results obtained in the past. The planned research addresses problems with a high level of importance in the engineering computations. It will focus on the reliable and accurate a-posteriori error estimations and adaptive procedures in the Finite Element Method, which are essential for confidence in the computed data. In this way, various accidents could be avoided. Although the finite element method is a major tool in engineering computations, various important problems are practically unsolvable by the standard approach. Hence a new generalized flexible finite element method will be designed with the goal to increase the effectivity of the method when solving unusually difficult problems. The problem of the composite material with the focus on the fiber scale is one of these types of problems when millions of fibers are present with only statistical character in regard to knowledge of their position. Here new approaches which are very tight to the experimental studies are needed and will be addressed.
