By Douglas N. Arnold, Gregery T. Buzzard, and Bradley J. Lucier
Jim Douglas Jr., the Compere and Marcella Loveless Distinguished Professor Emeritus of Computational Mathematics at Purdue University, passed away on April 27, 2016, after a brief illness. Jim was a highly-regarded applied mathematician whose work influenced the entire spectrum of research in partial differential equations, from purely-theoretical results to very practical applications in the oil industry. He is particularly well-known for his key contributions to the numerical solution of partial differential equations. Jim’s early work on alternating direction methods for elliptic and parabolic problems brought many large-scale problems into the realm of practical feasibility, and continues to serve as the foundation for new approaches to optimization and related problems. He was a prolific author and dedicated mentor to dozens of graduate students and postdocs at Rice University, the University of Chicago, and Purdue University. Jim was a SIAM Fellow and an inaugural AMS Fellow, as well as the recipient of numerous awards for his work in the petroleum industry.
Jim began his career at Humble Oil, later a part of the ExxonMobil Corporation, where he worked alongside Henry Rachford and Don Peaceman. Their research focused on numerical simulation of the flow of fluids, such as oil or natural gas. Of course, the standards for numerical simulation at that time were different from what they are now. As Peaceman describes in “A Personal Retrospection of Reservoir Simulation,” the computing devices used in 1955 had extremely limited storage: they held only 864 words!
The simulations of Jim and his colleagues required the solution of linear systems arising from the approximation of second derivatives in two directions. Gaussian elimination demands about \(N^3\) intermediate storage locations on an \(N \times N\) grid, so for \(N\) of any reasonable size, their machine did not have enough storage. This led Jim, Peaceman, and Rachford to develop the so-called Alternating Direction Implicit (ADI) method, which could solve finite-difference approximations to parabolic partial differential equations.
As the name implies, ADI is meant to solve auxiliary one-dimensional problems alternately in each direction; each auxiliary problem requires only about \(N\) words of intermediate storage, which can be reused. This huge saving in storage brought many problems within computational reach, so the ADI method received much attention. A later realization revealed that this basic technique could be applied to the sum of two nonlinear, nonsmooth operators of the right form (monotone operators). The Douglas-Rachford variant of this method found significance in convex analysis, monotone operators, and most recently in the field of big data. According to the citation database Scopus, in 2015 alone, 230 entries contained the term “Douglas-Rachford,” 128 in computer science and 25 in decision sciences.
In 1957, Jim returned to Rice University as an assistant professor of mathematics. He was promoted to full professor in 1961 and named the W.L. Moody Professor in 1964.
In August 1963, Richard Courant chose Jim to attend a meeting on partial differential equations in Novosibirsk, USSR. He was one of 23 American mathematicians to do so, and thus participated in the first large US-USSR mathematics meeting, which was organized with support from the US National Academy of Sciences and the Academy of Sciences of the USSR.
Jim moved to the University of Chicago in 1967, where he turned his attention to the mathematical understanding of the finite element method for partial differential equations. He conducted much of this work with Todd Dupont, who was first Jim’s student at Rice and then a colleague at Chicago.
In 1987, Jim became both director of the Center for Applied Mathematics and Purdue’s Compere and Marcella Loveless Distinguished Professor of Computational Mathematics, positions he held until his retirement in 2003.
During his distinguished career, Jim wrote more than 200 papers with over 70 co-authors. In addition to his work on ADI, Jim made many other lasting contributions. Among the most important are his pioneering work on interior penalty methods, which grew into the huge field of discontinuous Galerkin methods for elliptic and parabolic problems, the Brezzi-Douglas-Marini mixed finite element methods for second order elliptic problems, and the use of characteristic time-stepping for convection diffusion problems. In recognition of his impact on many fronts, Jim was named both a SIAM and AMS Fellow. He was also the recipient of the Cedric K. Ferguson Medal from the Society of Petroleum Engineers, as well as the Robert Earll McConnell Award from the American Institute of Mining, Metallurgical, and Petroleum Engineers, and a Commemorative Medal from Charles University in Prague.
Jim was a wonderful mentor for young people. He served as an advisor to graduate students and helped shape many professional mathematicians in their early careers. Many of his students and post-doctoral associates are AMS and SIAM Fellows, and leaders in computational science around the world.
Jim is survived by his wife Graça and his sons Jimmy and Craig (himself a professor of mathematics at the University of Wyoming) with his late wife Mary Lou.
Doug Arnold is McKnight Presidential Professor of Mathematics at the University of Minnesota. He was director of the IMA from 2001 to 2008, president of SIAM from 2009 to 2010, and a student of Jim’s at the University of Chicago. Greg Buzzard is a professor of mathematics at Purdue University, where he serves as department head. Brad Lucier holds a joint appointment as professor of mathematics and professor of computer science at Purdue University. He is a member of the inaugural class of AMS Fellows.