
Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers (GA2001074S)
from 01/01/2020
to 31/12/2022 main investigator

Objectives:
The project deals with the numerical solution of several types of partial differential equations (PDEs) describing various practical phenomena and problems. The aim is to develop reliable and efficient numerical methods allowing to obtain approximate solutions of PDEs under the given tolerance using a minimal number of arithmetic operations. The whole process includes the proposals and analysis of discretization schemes together with suitable solvers for the solution of arising algebraic systems, a posteriori error estimation including algebraic errors and adaptive techniques balancing various error contributions. We focus on the use of adaptive higherorder schemes which allow to reduce significantly the number of necessary degrees of freedom required for the achievement of the prescribed accuracy. The adaptive mesh refinement must also take into account the properties of the resulting algebraic systems. The expected outputs of this projects are adaptive reliable and efficient numerical methods for the solution of the considered types of PDEs.



Matematický ústav AV ČR usiluje o HR Award  zavedení profesionálního řízení lidských zdrojů (CZ.02.2.69/0.0/0.0/18_054)
from 01/01/2020
to 31/12/2022 investigator

Matematický ústav je přední českou věřejnou organizací, kterých posláním je vědecký výzkum v oblastech matematiky a jejích aplikací. Pro posílení jeho konkurenceschopnosti v mezinárodním kontextu je klíčové uvést dosavadní strategii řízení a rozvoje lidských zdrojů do souladu s Evropskou chartou pro výzkumné pracovníky, a tím umožnit získání ocenění HR Award. Pro ústav jde o mimořádnou příležitost, jak zkvalitnit a zprofesionalizovat péči o lidské zdroje, které jsou alfou a omegou jeho úspěchu.
Tento projekt je podpořen z operačního programu Výzkum, vývoj a vzdělávání, Výzva č. 02_18_054 pro Rozvoj kapacit pro výzkum a vývoj II v prioritní ose 2 OP, reg. č. CZ.02.2.69/0.0/0.0/18_054/0014664.



Advanced flowfield analysis (GA1809628S)
from 01/01/2018
to 31/12/2020 investigator

Objectives:
The research deals with an advanced flowfield analysis, particularly for transitional and turbulent flows. Local vortex identification and more general classification of flow regions based on the velocity gradient are investigated. The velocity gradient is usually decomposed in strainrate tensor and vorticity tensor, consequently the identification and classification criteria are determined by the inner velocitygradient configuration. The impact of configuration is usually significant though hidden, and will be, including representative data sets, analyzed and described. Largescale 3D numerical experiments based on the solution of the NavierStokes equations (NSE) will be performed with the help of new effective methods (parallel domain decomposition (DD) with adaptive mesh refinement) while these methods will be further developed. Some suitable qualitative properties of the solutions to the NSE will be studied and described in detail, focusing on the regularity criteria with only one or two velocity components and one or several velocitygradient entries.



Doktorská škola pro vzdělávání v oblasti matematických metod a nástrojů v HPC (CZ.02.2.69/0.0/0.0/16_018)
from 01/09/2017
to 30/09/2022 main investigator

Hlavním cílem projektu je ustavení Doktorské školy pro vzdělávání v oblasti matematických metod a nástrojů v HPC integrující doktorská studia MFF UK v Praze, MÚ AV ČR a NSC IT4Innovations VŠBTUO a navazující na jejich širší spolupráci v oblasti výzkumné. Součástí projektu je modernizace a internacionalizace jednoho doktorských programů školy (Výpočetní vědy, VŠBTUO) a vytvoření programu double degree. Projekt navazuje na související projekt ERDF Vzdělávací tréninkové centrum IT4Innovations.



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 flowfield analysis (GA1402067S)
from 01/01/2014
to 31/12/2016 investigator

Objectives:
The investigation aims at developing advanced methods for flowfield analysis particularly for transitional and turbulent flows. The work deals with decomposition techniques, mainly with decomposition of motion and vorticity. The vortexidentification criterion under development at present is associated with the corotation of line segments near a point and with the socalled 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 linetype vortexidentification methods, especially predictorcorrector type schemes. Further, the project aims to analyze the strainrate skeleton so as to draw a more complete picture of the flow. For testing purposes, largescale numerical experiments based on the solution of the NavierStokes 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: PIEFGA2012328008.



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 threedimensional 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 (hpFEM) and the selected method of integral and integrodifferential equations. The obtained were published in Dolezel, I., Karban, P. Solin, P.: Integral Methods in LowFrequency Electromagnetics. Wiley, Hoboken, NJ, UA (2009), 388 pages.



Methods of higher order of accuracy for solution of multiphysics 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 hpadaptive version of the finite element method (hpFEM), 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 hpversion of the finite element method (hpFEM) and on the problem of hpadaptivity. 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 hpFEM 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 hpFEM solutions. Within the project we also participated on the development of the hpFEM software project Hermes.
