Sep 23, 2014
Jan Papež, Charles University in Prague, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic Joint work with Zdenek Strakoš and Mattia Tani In the finite element method (FEM) used for discretization of partial differential equations, the finite dimensional subspaces are typically generated using locally supported basis functions. The resulting algebraic system matrix is then sparse and the sparsity is presented as an advantage of the FEM. On the other hand, when iterative methods are applied for solving the resulting algebraic problem, an algebraic preconditioning is needed in order to assure fast convergence. In the contribution we first briefly recall, following [1, Chapter 8] (see also the references given there), that any algebraic preconditioning can be interpreted as transformation of the discretization basis and, at the same time, transformation of the inner product in the given Hilbert space. This underlines that discretization and preconditioning are tightly coupled. Second, we present the idea of interpreting the algebraic error as a transformation of the discretization basis. We elaborate on [2, 3] where the transformation of the search basis only is considered and the test functions are unchanged. We discuss the general case with the transformation of both the search and the test basis functions.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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