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A. R. MITCHELL: SOME BIOGRAPHICAL AND MATHEMATICAL NOTES

 

D. F. GRIFFITHS, J. D. LAMBERT, G. A. WATSON

Department of Mathematics and Computer Science, University of Dundee, Dundee DD1 4HN, Scotland
E-mail: dfg@maths.dundee.ac.uk, gawatson@maths.dundee.ac.uk

and

G. FAIRWEATHER

Department of Mathematical and Computer Sciences Colorado School of Mines, Golden,Colorado 80401-1887, USA. E-mail: gfairwea@mines.edu

 

The Period up to 1967

Andrew Ronald Mitchell was born on 22 June 1921. He went to school at Morgan Academy, Dundee, and in 1938 went on to read mathematics at the old University College, Dundee (at the time a college of the University of St Andrews) where E. T. Copson held the Chair of Mathematics; Ron graduated with First Class Honours in 1942. Partly due to the war, student numbers were low, and Ron was the only Honours student in mathematics. On graduating, he was called up and sent to the wartime Ministry of Aircraft Production in London, where he remained until after the end of the war. His duties included the interrogation of captured Luftwaffe pilots, in an attempt to get information about their aircraft: some years later he met one of them at a conference. Ron had shown some prowess as a soccer player, and during this period he turned out a few times for Chelsea.

In October 1946, Ron decided to take some time out to do a PhD, and returned to Dundee to see if this might be possible. There was no available supervisor in University College, but he made contact with D. E. Rutherford, who was then a Lecturer in Mathematics and Applied Mathematics at St Andrews University. Lecturing staff were in short supply at that time (they were badly paid even in those days), and Dan Rutherford agreed to act as supervisor in return for Ron taking an Assistant Lectureship for the duration of his PhD.

Although Dan Rutherford was responsible for the Applied Mathematics part of the Mathematics Department in St Andrews, his main research interest was in Lattice Theory, so the supervision must have been fairly nominal, particularly as Ron's thesis was concerned with relaxation methods in compressible flow. Deciding that University life was not so bad after all, Ron stayed in St Andrews after being awarded his PhD in 1950 and was appointed Lecturer, later Senior Lecturer and eventually Reader. Some of Dan Rutherford's later work concerning the eigenvalues of certain banded matrices (having constant diagonals, apart from contributions from boundaries) was almost certainly influenced by Ron's interest in relaxation methods. They did some joint work in this area and, as well as writing two joint papers, actually discovered an early form of SOR before the famous paper of David Young, although this was never published.

During much of this time, Ron continued his active interest in football. No doubt looking to supplement his salary, he signed as a part-time professional footballer with a number of Scottish clubs. During the period 1946-1955, he played with St Johnstone, East Fife, Brechin City and Berwick Rangers in that order. With typical modesty, Ron insists that his playing days at the last-named club were extended beyond their sell-by date because of his vital role as interpreter; five of the Berwick side came from Glasgow and five from Newcastle.

Ron had developed an interest in Numerical Analysis, initially as a means of tackling fluid dynamics problems using Southwell's relaxation methods. Jack Lambert was a member of the Senior Honours class in 1953-54 when Ron taught an Honours special topic in Numerical Analysis. This was the first time Numerical Analysis had been taught in St Andrews.

Ron's first PhD student was Jim Murray, who started in 1953 on a topic in boundary layer fluid dynamics. In these days the Air Ministry published a list of their top ten problems in fluids. Number 6 at that time was flow into a pitot tube: was the speed of flow which was registered the correct speed of the aircraft? This was Jim Murray's PhD problem. As is well known, he went on to an illustrious career, which included an FRS and the Chair of Mathematical Biology at Oxford. Ron's second PhD student was J. Y. Thompson who started in 1954 working on numerical aspects of fluid dynamics. Following the award of his PhD, he went on to a Lectureship in Applied Mathematics in Liverpool, where he married a widowed medical doctor with a large number of children, retrained into the medical profession, and was lost to Mathematics.

In 1959, Ron married Ann, and took up a one year post of Senior Research Fellow in the Mathematics Department at California Institute of Technology. Jack Lambert was appointed as a Lecturer at St Andrews in the same year, and became Ron's third PhD student, working for the degree part-time. He worked on an idea of Ron's of incorporating higher derivatives into methods for ODE's-apparently one of the few times Ron strayed away from PDE's to ODE's. In 1963, Ron and Jack Lambert jointly took on the PhD supervision of Graeme Fairweather and Brian Shaw, but this arrangement resolved itself into 2 pairings with Ron and Graeme Fairweather working on PDE's and the other two on ODE's. In September 1964, the group attended a Workshop in Perugia in Italy on Alcune Questioni di Analisi Numerica. Travel was by car, boat and train, and was a very complicated process. There Ron met Vlastimil Ptak, who visited St Andrews the following year; also present were Peter Wynn, Walter Gautschi and F. L. Bauer. A local newspaper published a picture of the St Andrews contingent listening intently to one of the lectures.

A joint Mitchell/Fairweather paper, published in Numerische Mathematik in 1964, was the first in a series on high order alternating direction finite difference methods for elliptic PDE's. It caused surprise in some quarters, where it had been believed that such higher order methods did not exist.

St Andrews got its first computer in 1964, an IBM 1620 with 64K (or was it 32K?) memory. The machine was capable of solving Laplace's equation in a cube using an optimal alternating direction finite difference method with a 5 ×5×5 mesh in 15 minutes-on a good day. In a square, a 20 ×20 mesh could be tackled. The computer was housed in the Observatory, over a mile from the Department of Mathematics, and hands-on access was provided for an hour each morning and afternoon, with no exceptions, even when the printer ribbon wrapped itself around the type bar, a frequent occurrence. Batch jobs could be run at other times. The method published in the Numerische Mathematik paper was not completely reliable, although it did well on problems with homogeneous Dirichlet boundary conditions on at least two sides of the square. Contrary to Ron's belief that there was an error in the program, it turned out that there was a problem with the handling of the boundary conditions in high-order methods. An elegant way round the problem was obtained by Ron and Graeme Fairweather in 1966, and was published in SINUM the following year. This paper also described how to deal with problems in L-shaped regions. Earlier joint work by the same authors involved the use of a difference scheme based on the Schwarz alternating procedure, published in the Computer Journal in 1966: this paper may have been the first to give numerical results obtained using a domain decomposition method.

Graeme Fairweather had completed his PhD in 1965, and the following year went to Rice University in Texas, a visit set up as a consequence of correspondence between Ron and Jim Douglas Jr. Sandy Gourlay started a PhD with Ron in 1964, as did Pat Keast, followed one year later by John Morris. A good discussion of the work done by Ron and his students around this time is given by Lapidus and Pinder in their book Numerical Solution of Partial Differential Equations in Science and Enginering published by Wiley in 1982.

Throughout his time at the University of St Andrews, Ron continued to live in Dundee. On a typical day, he would catch the 8.02 train from Dundee to St Andrews which got him into his office around 8.40. At 9 he met his students, and he then had a lecture at 10. At 11 he would have coffee in the Staff Club in the Younger Hall, and after possibly seeing his students again he would gather his things together to catch the train back to Dundee at 12.40. An overlong discussion could result in a rush for the train, and a jogging party to the station which would leave the students hanging on to the railings to recover. Ron kept himself extremely fit, and at that time was one of the best squash players in the University. Only a very special event, such as an important visitor, would keep Ron in St Andrews for an afternoon. Seminars were comparatively rare in those days and the relatively large group of research students at St Andrews supervised by Ron and Jack Lambert was likely to turn up anywhere: they became known (to others) as the ``all purpose colloquium audience''. Trips were made as far as Newcastle for a one-hour seminar.

Mike Osborne came to Edinburgh in July 1963 as Assistant Director (to Sidney Michaelson) of the University Computer Unit and he was joined by Donald Kershaw, together with some students including, in October 1964, Alistair Watson. ``Computer Unit'' was, in fact, a bit of a misnomer, as the Unit had no computer, and Atlas Autocode programmes were sent to the Manchester Atlas, but that is another story. John Todd and Olga Tausky were visiting Arthur Erdelyi in Edinburgh, and E. T. Copson (who had moved from Dundee to St Andrews) invited John Todd over to give a talk. Mike Osborne made a fourth in Arthur Erdelyi's car and there was a lively conversation in which the driver was an active participant. One result was that a particular signpost ``St Andrews 10 miles'' was passed several times, and not always in the correct direction.

After the talk, Ron and Mike Osborne swapped concerns about the need for more interaction, and this led to the idea of a ``do it ourselves'' seminar. Ron claimed a friendship with the warden of a particular Hall of Residence, and undertook to see if he could get a good rate (ie student rate) after the June 1965 examinations but before end of term. His persuasive powers were equal to the task, and so the idea was pushed forward, and Ron and Jack Lambert agreed to run the meeting. The Department of Mathematics was very happy to be involved on the basis that if the meeting made a profit, it belonged to the Department, and if it made a loss it came out of the pockets of the organisers. The main organizational arrangements were made by Jack Lambert, and no conference has ever had its financial estimates done more meticulously.

Particular encouragement was given to participation by students (remembered rather differently by the students as a form of coercion), and a surprisingly good response was obtained. In fact the Edinburgh and St Andrews contingents made up only around half those attending, and John Mason from Oxford, Ken Wright from Newcastle, Will McLewin from Manchester and Garry Tee from Lancaster appear to have travelled furthest. If there were any records of the meeting, they seem not to have survived; the best estimates are that there were about 30 attendees and a program extending over two days. Ron's group talked about their work on ADI and high accuracy discretizations.

This model has been used in many other places, but its origins are recorded here in some detail because it is now recognised as the first ``Dundee'' meeting. Emboldened by the success of this meeting, a second one was held in St Andrews from 26-30 June 1967, attracted 85 participants, and established the biennial pattern.

Around 1965-66, Ron went to evening classes in Dundee to learn Russian. Having long since lost his School Leaving Certificate, he experienced some difficulty in persuading the organisers that he had an appropriate level of general education to allow him entry to the course; apparently a PhD was not an acceptable alternative. During Graeme Fairweather's thesis work, it had been realised that some Russians, in particular Samarskii, Andreyev and D'Yakonov were also working on high order difference methods for PDEs. Indeed a method, essentially that of the 1964 Numerische Mathematik paper, had been published in Russian at about the same time, and D'Yakonov had also discovered the loss of accuracy referred to earlier. A knowledge of Russian not only allowed Ron to keep up with the Russian literature as soon as it appeared, but was invaluable when he attended the ICM Meeting in Moscow in 1966. There he met D'Yakonov and, as a result, the latter visited Dundee in the late sixties. In Moscow, Ron was able to indulge his love of soccer: he played for The Rest of the World against the USSR in a soccer match which was organised in the stadium of Moscow Dynamo. The home team, who had been in training for several weeks, won 5-2.

The work of Mitchell/Fairweather lay somewhere between the classical ADI approach of Douglas, Peaceman, Rachford and Gunn, and that of D'Yakonov. The former would not handle the loss of accuracy at the boundary, while the latter would, but was cumbersome. A byproduct was that people in the West became much more aware of the activity in the USSR concerning split operator techniques.

The Period from 1967 onwards

In 1965 D. S. Jones was appointed to the Ivory Chair of Mathematics in Queen's College (formerly University College), Dundee. With a level of priority which was atypical of a classical Applied Mathematician in those days, he decided to build up Numerical Analysis, which he was far-sighted enough to see as a growth area. For example he started an MSc course in Numerical Analysis in 1965. There was a numerical analyst already in Dundee, R. P. Pearce, a Senior Lecturer, who had collaborated with Ron and Jack Lambert while they were at St Andrews. However, he left to fill a Chair at the University of Reading at the end of the 1966/67 academic year. Meantime, Douglas Jones obtained funds to establish a Chair of Numerical Analysis in Dundee, and Ron was appointed in 1967, the year in which Queen's College Dundee formally severed its links with St Andrews and became the University of Dundee. The same year, Jack Lambert, who had moved to Aberdeen in 1965, joined Ron in Dundee as a Senior Lecturer, and Sandy Gourlay came from St Andrews as a Lecturer.

Ron continued to attract research students and, with funding from NCR and the Ministry of Defence obtained largely through the efforts of Douglas Jones, other numerical analysts were appointed to post-doctoral positions. John Morris came from St Andrews to a Research Fellowship, and Sean McKee took up a similar position after completing a PhD with Ron in 1970. Other Research Fellows who came to Dundee from elsewhere at that time were Nancy Nichols and Alistair Watson. Sandy Gourlay left in 1970 to join IBM, and Alistair Watson was appointed to the vacant post. David Griffiths joined the Mathematics Department as a Lecturer in the same year, and he and Ron have worked closely ever since.

The 3rd Biennial Dundee Conference had by now been held in 1969, this time actually in Dundee. It attracted 148 participants. In the following year, Ron obtained substantial Science Research Council funding for a Numerical Analysis Year lasting from September 1970 to September 1971. This was an important and high profile period which went a long way to putting Dundee on the Numerical Analysis map. The year began with a Symposium on the Theory of Numerical Analysis from 15-23 September, 1970 with speakers Gene Golub, Vidar Thomée, Gene Wachspress and Olof Widlund. There was a Conference on the Applications of Numerical Analysis from 23-26 March 1971 with 177 participants, a Conference on Numerical Methods for Nonlinear Optimisation from 28 June-1 July, 1971 with 198 participants, a Seminar on Ritz-Galerkin and the Finite Element Method from 8-9 July, 1971 and finally a Conference on the Numerical Solution of ODEs from 5-6 August, 1971.

In addition to those already mentioned, about 34 other numerical analysts of international repute visited Dundee during the Numerical Analysis Year as Senior Visiting Fellows, some for short periods and others for longer periods up to the full year. Available records list those as C. Bardos, F. L. Bauer, R. Bellman, G. D. Birkhoff, J. Bramble, H. Brunner, J. C. Butcher, L. Collatz, G. Dahlquist, P. J. Davis, J. Douglas Jr., C. W. Gear, W. Gragg, J. L. Greenstadt, P. Henrici, A. S. Householder, T. E. Hull, E. Isaacson, R. E. Kalaba, H. B. Keller, H. O. Kreiss, P. Lascaux, J. L. Lions, M. R. Osborne, M. J. D. Powell, V. Ptak, J. R. Rice, R. D. Richtmyer, J. B. Rosen, I. J. Schoenberg, H. J. Stetter, G. Strang, R. Temam, R. S. Varga.

As those who have spent time with Ron know, unusual and remarkable things are always likely to happen. During his visit, Vidar Thomée was persuaded by Ron to investigate his ancestry by inquiring at a local Tartan shop which claimed to be able to find a tartan associated with any given surname: in this case it turned out to be MacDonald. Ron also managed to persuade Mike Osborne and Gene Wachspress to attend a Burns' supper in Dundee resplendent in kilts.

Although Ron and Jack Lambert were back as colleagues, they had an understanding that they would pursue their own interests in an attempt to keep the Numerical Analysis base wide. A major contribution to the widening of that base was the appointment in 1973 of Roger Fletcher, already a leading figure in optimization, to a Senior SRC Fellowship; he moved to a permanent appointment in 1976. Also in 1973, a Conference on The Numerical Solution of Differential Equations was held from 3-6 July 1973, and attracted 234 participants. The 1975 Conference was on general Numerical Analysis, and this pattern continues to this day, reflecting the wider Numerical Analysis base referred to above. Further information about all the Dundee meetings can be found on the World Wide Web, starting from the home page of the Department of Mathematics and Computer Science: /. Many eminent numerical analysts have had long associations with Dundee, and have strongly supported the Dundee conferences. Two of those, Lothar Collatz and Gene Golub, have been awarded Honorary Degrees by the University of Dundee. Gene Golub, in addition, presented the inaugural A. R. Mitchell lecture at the 1991 meeting which celebrated Ron's 70th birthday. Two other staunch supporters of the Dundee conferences have been Mike Powell and Bill Morton, who gave the A. R. Mitchell lectures in 1993 and 1995, respectively.

The MSc course in Numerical Analysis has been successful in attracting many good students, and once again Ron must take much of the credit for this. Many students went on to do PhD's with Ron, for example Dick Wait who was the first to work with Ron in the newly emerging field of finite elements, and who continued on to a Research Fellowship and further collaboration, including a book. Among others who took the MSc course and have gone on to carve out academic careers for themselves are Mehi Al-Baali, Peter Alfeld, Ken Brodlie, Dugald Duncan, Julian Hall, Per Skafte Hansen, Chus Sanz-Serna, Sven Sigurdsson, Ian Stewart and Rob Womersley.

Ron's interests changed in the late 1960's to finite elements, a move allegedly instigated by Dick Wait who turned up at Ron's doorstep and announced that he would like to do a PhD in the area. This was virgin territory for numerical analysts and Ron did much pioneering work during the next five years with George Phillips, Gene Wachspress, Bob Barnhill and his students Dick Wait, Robin McLeod and Jim Marshall, work which focussed mainly on the treatment of boundaries-the approximation of curved boundaries and the exact matching of boundary data using blending interpolants (see the Addendum).

In the booklet commemorating Lothar Collatz and published by the University of Hamburg, a lecture given by Ron in 1973 is fondly recalled in which he was dealing with a finite element with one curved side. A crude reproduction of Ron's diagram is shown below where he was explaining how to approximate the curved side (solid line) by a ``parabolic arc'' (dashed curve). There were some gasps from the audience which caused him to explain-``Of course a parabola does not behave in this way, except it is a Scottish parabola!''

parabola

The next change of direction occurred as a consequence of a lecture given by Olec Zienkiewicz at the second MAFELAP conference organized by John Whiteman at Brunel University in 1975. In this talk Olec Zienkiewicz described instabilities they had experienced in converting their successful finite element codes for structural problems into codes for solving the Navier-Stokes and related equations in fluid dynamics. Finite difference practitioners had known for many years that the instability could be overcome by the use of ``upwind differencing'' and Ron was immediately intrigued to know how this type of stabilization could be applied to the finite element situation.

On his return to Dundee, he and David Griffiths attacked this problem with some gusto over the next few weeks and the end result was upwind-biased test functions, and what is now known as the Petrov-Galerkin finite element method (this term was coined, we believe, in a joint paper Ron wrote with Bob Anderssen). Ian Christie, who was on the MSc course at that time, then developed the ideas further in both his MSc and PhD dissertations.

There followed several fruitful years working on convection-diffusion problems until, through his interest in diffusion and dispersion effects and his collaboration with Brian Sleeman, he became interested in nonlinear effects in the early 1980's. Some of the problems arose from Mathematical Biology, on which ``Mano'' Manoranjan did much of his PhD work, but Ron was also interested in solitons, particularly those arising from the Korteweg-de Vries and Schrödinger equations. He was instrumental in bringing the subject of spurious solutions to the fore. Nonlinearity continues to be his abiding passion as he currently wrestles with the Korteweg-de Vries-Burgers equation.

As in most of the areas in which Ron has worked, he has had the uncanny knack of alighting on fundamental issues which, through his many papers and conference talks, have drawn others to the subject. He has a long and illustrious list of publications, and a complete bibliography is given as Appendix 1. Equally if not more impressive is the list of his PhD students, given in Appendix 2. One of his great strengths is the way he has been able to motivate and encourage his students, and this is borne out by the many who have gone on to great things; he has a truly outstanding talent for getting the best out of research students and for instilling self-confidence in them.

Acknowledgement

This article is based on information which has been collected from a number of people. In particular Mike Osborne provided material for Section 1 concerning the origins of the 1965 meeting at St Andrews.