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Abstract The accuracy of a numerical discretization of a mathematical problem depends on the consistency and stability of the discretization method used. This theme, that consistency and stability imply convergence, recurs throughout numerical analysis, and is especially important in the numerical solution of partial differential equations. But the concept of numerical stability can be subtle and elusive. Even simple examples can yield unexpected results, and the development of stable numerical methods remains elusive for important classes of problems. We will survey these ideas through a variety of examples, and describe some modern tools from geometry and topology which are taking their place along side the more classical analytic tools for designing and understanding stable algorithms.
About the Speaker Douglas Arnold is the McKnight Presidential Endowed Professor of Mathematics at the University of Minnesota. He is a research mathematician and educator with a strong interest in mathematics in interdisciplinary research and in the public understanding of the role of mathematics. In 2008, Prof. Arnold serves as the President-elect of SIAM, the Society for Industrial and Applied Mathematics. SIAM is the world's leading professional organization for applied mathematicians and computational scientists. Arnold will serve a two-year term as president of SIAM beginning at the start of 2009, followed by a year as past president.Prof. Arnold's research interests include numerical analysis, partial differential equations, mechanics, and in particular, the interplay between these fields. Much of his work concerns the computer solution of partial differential equations, focusing on the development and understanding of methods for simulating physical phenomena ranging from the deformation of elastic plates and shells to the collision of black holes. Around 2002 he initiated the finite element exterior calculus, a new approach to the stability of finite element methods based on geometric and topological structure underlying the relevant partial differential equations. The development of the finite element exterior calculus is a major direction of his current research work. In 1991 Arnold was awarded the first International Giovanni Sacchi Landriani Prize by the Academy of Arts and Letters of Lombardy Institute in Milan in 1991 for "outstanding contributions to the field of numerical methods for partial differential equations." In 2002 he was a plenary lecturer at the International Congress of Mathematicians in Beijing. In 2008 he was awarded a Guggenhein Fellowship. From 2001 through 2008, Prof. Arnold served as director of the Institute for Mathematics and its Applications IMA). The primary mission of the IMA is to increase the impact of mathematics by fostering interdisciplinary research linking mathematics with important scientific and technological problems from other disciplines and industry. Prof. Arnold received his Ph.D. degree in Mathematics from the University of Chicago in 1979. From 1979 through 1989 he was on the faculty of the University of Maryland. In 1989 he moved to Penn State University where he was appointed Distinguished Professor Mathematics, and where he remained until moving to University of Minnesota and assuming the position of Director at the Institute for Mathematics and its Applications in August 2001. To learn more about Professor Arnold's work, please visit his website. Associated 1-Day Meeting In connection with Professor Arnold's visit, a 1-day meeting on the theme "Numerical Algorithms: Theory and Practice" will be held on January 21, 2009 at the ICMS. The meeting will feature the work of four researchers in numerical mathematics The schedule is given below; each researcher's name is linked to his website.
Links Maxwell Institute for Mathematical Sciences University of Edinburgh School of Mathematics |