Tomáš Vejchodský

Grants
- Guaranteed bounds of eigenvalues and eigenfunctions of differential operators (Neuron Impuls 24/2016)
  from 01/01/2017 to 31/12/2019 main investigator
- We will propose new a posteriori error estimates for eigenvalue problems of symmetric elliptic partial differential operators. We will prove their reliability and local efficiency. We will use them in the adaptive finite element method for reliable error estimates of the size of the error in eigenvalues and eigenfunctions.
- Advanced methods for flow-field analysis (GA14-02067S)
  from 01/01/2014 to 31/12/2016 investigator
- 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.
- Stochastic and deterministic modelling of biological and biochemical phenomena with applications to circadian rhythms and pattern formation (StochDetBioModel(328008))
  from 01/03/2013 to 31/08/2014 main investigator
- Marie Curie Intra European Fellowship for Tomáš Vejchodsky at the University of Oxford. Grant Agreement Number: PIEF-GA-2012-328008.
- The finite element method for higher dimensional problems (IAA100190803)
  from 01/01/2008 to 31/12/2012 investigator
- 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.
- Advanced algorithms for solution of coupled problems in electromagnetism (102/07/0496)
  from 01/01/2007 to 31/12/2009 main investigator
- The project resulted in internationally recognized results in the development and implementation of advanced algorithms for numerical modeling of coupled problems in the field of heavy current electrical engineering and electrotechnics. These tasks are characterized by an interaction of several physical fields. Characterization of such interactions is essential for reliable and economical design. Members of the research team focused primarily on advanced finite element method of higher order accuracy (hp-FEM) and the selected method of integral and integro-differential equations. The obtained were published in Dolezel, I., Karban, P. Solin, P.: Integral Methods in Low-Frequency Electromagnetics. Wiley, Hoboken, NJ, UA (2009), 388 pages.
- Methods of higher order of accuracy for solution of multi-physics coupled problems (IAA100760702)
  from 01/01/2007 to 31/12/2011 main investigator
- The design of efficient numerical methods for computer simulations of large nonlinear and associated transient problems belongs among the most recent topics in the sphere of technical and scientific computing. Examples include processing solid and liquid metals by electromagnetic field, problems of thermoelasticity and termoplasticity, fluid interaction with solid structures and others. The difficulty of coupled problems stems from the fact that various components of solutions exhibit specific characters, such as boundary layers in fluids or singularities in electromagnetic fields. Efficient and accurate solution of these problems requires the representation of various components by geometrically different meshes. From the mathematical point of view, various solution components belong to different Hilbert spaces and, therefore, their approximations require various types of finite elements. For each solution component we use the modern hp-adaptive version of the finite element method (hp-FEM), which is known for its exponential convergence.
- Mesh adaptivity for numerical solution of parabolic partial differential equations (P201/04/P021)
  from 01/01/2003 to 31/12/2006 main investigator
- We analyzed adaptive methods for numerical solution of partial differential equations. We concentrated on the hp-version of the finite element method (hp-FEM) and on the problem of hp-adaptivity. One of the studied aspects were the a posteriori error estimates. We developed a new guaranteed error estimate, which enables to compute an approximate solution with guaranteed accuracy. We also optimized the hp-FEM basis functions in order to improve the conditioning properties of the resulting matrices. Another aspect of the project was the analysis of the discrete maximum principles. We developed a simple conditions that guarantee the nonnegativity of the hp-FEM solutions. Within the project we also participated on the development of the hp-FEM software project Hermes.