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Research


Multiscale problems
Multiscale problems are some of the greatest challenges in computational mathematics today. In all branches of the engineering sciences we encounter problems with features on several different scales. A typical example is simulations in a heterogenous media where material data such as module of elasticity, conductivity or permeability, varies in space over several different scales. In order to solve these problems we develop and analyze efficient generalized finite element methods.
Multiphysics problems
Engineering applications that involves different kinds of physics (multi-physics) are very computationally challenging. As computers get more and more powerful, complicated coupled systems of non-linear partial differential equations can be solved. As a result of this development much more complex physical phenomena can be analyzed by computational means. Reliability and efficiency becomes crucial when solving these problems. Reliability comes down to a need for bounds of the error between the computed approximation and the exact solution, and efficiency is achieved by designing adaptive algorithms that distributes the computational effort where it is most needed.
Uncertainty quantification
In engineering applications it is very common that the data is given by experimental measurements or even that data is missing in some regions. It is natural to model this lack of exact information using a probabilistic representation of the data. We develop efficient methods for forward sensitvity analysis of partial differential equations with uncertainty in the data. We present error estimates taking into account both the numerical and the statistical error e.g. when approximating the cumulative distribution function of a quantity of interest using numerical techniques.
Error analysis for finite element methods
Error estimation in numerical simulation is a very important field of research since it shows how reliable the computation is. A priori error bounds with rates show the convergence behaviour of the proposed method. A posteriori bounds gives computable bounds of the error and can be used to drive adaptivity. The solution is typically improved by refining the computational mesh or decreasing the time steps, but it can also be improved by improving the representation of the geometry, the data, or the numerical quadrature used in the finite element method.



Supervision


Postdocs
Siyang Wang, Chalmers University of Technology, 2017-
Efthimios Karatzas, University of Gothenburg, 2016-2017, now at University of Trieste
Tony Stillfjord, University of Gothenburg, 2015-2017, now at Max Planck institute in Magdeburg
Patrick Henning, Uppsala University, 2013-2014, now at Royal Institute of Technology

Ph.D. candidates
Gustav Kettil, Chalmers/Fraunhofer AEM Graduate Programme, (lic), 2014-
Anna Persson, Chalmers University of Technology, (lic), 2013-
Fredrik Hellman, Uppsala University, (lic), (phd), 2012-2017, now at High performance consulting
Daniel Elfverson, Uppsala University, (lic), (phd), 2011-2015, now at Umeå University

M.Sc. students
Per Ljung, Chalmers University of Technology, 2017
Tim Keil, University of Gothenburg, (msc), 2017, now at University of Munster
Robert Forslund, Chalmers University of Technology, (msc), 2015, now at Chalmers/Arcam
Daniel Elfverson, Uppsala University, (msc), 2010, now at Umeå University
Klas Pettersson, Uppsala University, (msc), 2010, now at Narvik University College
Yang Zeng, Uppsala University, (msc), 2009

B.Sc. students
Viktor Bäck, Uppsala University, (bsc), 2012



External funding sources


The Swedish Research Council, 2012-2019
The Swedish Foundation for Strategic Research, 2014-2019
The Centre for Interdisciplinary Mathematics at Uppsala University, 2012-2017
The Göran Gustafsson Foundation, 2008, 2012-2014


Co-authors


Daniel Elfverson, Umeå University, Sweden
Donald Estep, Colorado State University, USA
Fritjof Fagerlund, Uppsala University, Sweden
Emmanuil Georgoulis, University of Athens, Greece
Victor Ginting, University of Wyoming, USA
Fredrik Hellman, Uppsala University, Sweden
Patrick Henning, Royal Institute of Technology, Sweden
Michael Holst, University of California at San Diego, USA
Max Jensen, University of Sussex, UK
Mats G. Larson, Umeå University, Sweden
Auli Niemi, Uppsala University, Sweden
Anna Persson, Chalmers University of Technology, Sweden
Daniel Peterseim, University of Bonn, Germany
Michael Presho, University of Texas, USA
Tony Stillfjord, University of Gothenburg, Sweden
Robert Söderlund
Simon Tavener, Colorado State University, USA

Erdös number: 4, e.g. through: Erdös - Faber - Manteuffel - Holst - Målqvist


Links


Department of Information Technology at Uppsala University
Mathematisches Forschungsinstitut Oberwolfach
My place in the global academic tree
Department of Mathematics at UCSD
Department of Mathematics at CSU
ENUMATH 2009 conference