# Douglas Arnold

Principal Investigator (WAVE), Department of Mathematics (UMN), Simons Collaboration on Localization of Waves

Research

Numerical analysis, partial differential equations, mechanics, and in particular, the interplay between these fields

Biography

- 1979 — Ph.D. in Mathematics, University of Chicago
- 1976 — M.S. in Mathematics, University of Chicago
- 1975 — B.A. in Mathematics, Brown University

- 2008 — McKnight Presidential Professor of Mathematics
- 2001–2008 — Director, Institute for Mathematics and its Applications Professor, School of Mathematics

- 1995–2002 — Distinguished Professor, Department of Mathematics
- 1997–2001 — Co-Director, Center for Computational Mathematics and Applications
- 1996–2001 — Associate Director, Inst. for High Performance Computing and Applications
- 1989–1995 — Professor, Department of Mathematics
- 1994–1995 — Acting Department Chair
- 1991–1994 — Associate Chair for Computing

- 1989 — Professor, Department of Mathematics
- 1984–1989 — Associate Professor, Department of Mathematics
- 1979–1984 — Assistant Professor, Dept. of Mathematics and Inst. for Phys. Sci. & Tech.

- American Association for the Advancement of Science
- American Mathematical Society
- Association for Women in Mathematics
- Mathematical Association of America
- Society for Industrial and Applied Mathematics

Douglas N. Arnold is the McKnight Presidential Professor of Mathematics at the University of Minnesota. His research interests are in computational mathematics, centered on the design and analysis of numerical algorithms for partial differential equations and their application to physical problems ranging from the deformation of elastic shells to the collision of black holes. In a plenary lecture at the 2002 International Congress of Mathematicians, Arnold initiated the finite element exterior calculus, an approach to the stability of finite element methods based on geometric and topological structures underlying the relevant PDE, which has grown to be the subject of many hundreds of papers and dozens of PhD theses. His 2006 and 2010 joint papers establishing the foundations for this subject are each among the 30 most cited papers published in mathematics during their publication year (according to MathSciNet). His earlier work on discontinuous Galerkin methods is similarly highly cited, his thesis publication being the fourth most cited paper in mathematics in 1982, and a later joint work on the subject being the single most cited paper in all of mathematics in 2001.

From 2001 through 2008, Professor Arnold served as director of the Institute for Mathematics and its Applications (IMA). In 2009 and 2010, he was the elected President of the Society for Industrial and Applied Mathematics (SIAM), the world’s leading professional organization for applied mathematicians and computational scientists. He serves as Editor-in-Chief or on the editorial boards of numerous journals and is a Fellow of the American Mathematical Society, the Society of Industrial and Applied Mathematics, and the American Association for the Advancement of Science. Among his other honors are a Guggenheim Fellowship, membership in the Norwegian Academy of Science and Letters, the SIAM Prize for Distinguished Service to the Profession, and the J. Tinsley Oden Medal of the US Association for Computational Mechanics. Arnold served on scientific boards including the Board of Mathematical Sciences and Applications of the National Research Council, the U.S. National Committee for Mathematics, the Mathematical and Physical Sciences Advisory Committee of the National Science Foundation, the Program Committees for both the International Congress of Mathematicians and the International Congress of Industrial and Applied Mathematics, and the boards of many research institutes including, currently, ICERM at Brown University and the Fine Institute for Theoretical Physics in Minnesota.

Read more on his University of Minnesota homepage.

Publications & Awards

- J. Tinsley Oden Medal, awarded by the US Association of Computational Mechanics, 2015.
- “Highly Cited Research” designation by Thomson Reuters, 2014.
- SIAM Prize for Distinguished Service to the Profession, 2013.
- Overseas Fellow of Churchill College, Cambridge University, elected 2013.
- AMS Fellow, appointed 2012. AAAS Fellow, elected 2011.
- Foreign member, Norwegian Academy of Science and Letters, elected 2009.
- SIAM Fellow, appointed 2009.
- AMS-MAA Invited Address, Joint Math Meetings, 2009.
- President of the Society for Industrial and Applied Mathematics (SIAM), 2009–2010.
- Guggenheim Fellow, 2008–2009. McKnight Presidential Professor of Mathematics, University of Minnesota, 2008– 2007
- Science and Engineering Visualization Challenge winner (National Science Foundation and Science journal), honorable mention with J. Rogness. International Congress of Mathematicians plenary lecture, 2002.
- “Highly Cited Research” designation by Thomson/ISI, 2001.
- Distinguished Service Award of the Penn State Eberly College of Science Alumni Society, 2000.
- Teresa Cohen Service Award, Penn State Math Department, 1998. George W. Atherton Award for Excellence in Teaching (Penn State’s highest level of recognition for undergraduate education), 1996.
- Distinguished Professor of Mathematics, Penn State University, 1995–2002.
- First International Giovanni Sacchi Landriani Prize, awarded by the Academy of Sciences and Letters, Milan, Italy (Instituto Lombardo), 1991.
- NATO Postdoctoral Research Fellow, 1982–1983.

- D. N. Arnold, Finite Element Exterior Calculus, volume 93 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018.
- D. N. Arnold, P. B. Bochev, R. B. Lehoucq, R. A. Nicolaides, and M. Shashkov, Editors, Compatible Spatial Discretizations, volume 142 of IMA Volumes in Mathematics and its Applications, Springer, Berlin, 2006.

- The Hellan-Herrmann-Johnson method with curved elements, 2019
- Finite Element Exterior Calculus, 2018
- Localization of eigenfunctions via an effective potential, 2017
- Computing spectra without solving eigenvalue problems , 2017
- Finite element exterior calculus for parabolic problems, 2017
- Finite element exterior calculus with lower-order terms, 2016
- Effective confining potential of quantum states in disordered media, 2016
- Finite element differential forms on curvilinear cubic meshes and their approximation properties, 2015
- Mixed methods for elastodynamics with weak symmetry, 2014
- Finite element differential forms on cubical meshes, 2014
- On the consistency of the combinatorial codifferential, 2014
- Finite element exterior calculus: from Hodge theory to numerical stability, 2010
- Finite element exterior calculus, homological techniques and applications, 2006
- Differential complexes and numerical stability, 2002
- Unified analysis of discontinuous Galerkin methods for elliptic problems, 2002
- Mixed finite elements for elasticity, 2002
- Multigrid in H(div) and H(curl), 2000
- A uniformly accurate finite element method for the Reissner-Mindlin plate, 1989
- A stable finite element for the Stokes equations, 1984
- An interior penalty finite element method with discontinuous elements, 1982

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