Volkswagen Foundation, Project I/72342, Accurate Numerical Solution of Convection-Diffusion Problems

The work in accordance with the project started in the end of 1997 and finished in the beginning of 2001. The final list of russian team includes 7 participants:

V. V. Shaidurov — professor, doktor of physical and mathematical sciences, director of Institute of Computational Modelling of Russian Academy of Sciences; head of Chair on Softwear of Krasnoyarsk State Technical University;
I. V. Kireev — kandidat of physical and mathematical sciences, scientific worker of Institute of Computational Modelling of Russian Academy of Sciences;
E. G. Bykova — kandidat of physical and mathematical sciences, dozent of Krasnoyarsk State Technical University;
L. V. Gilyova — kandidat of physical and mathematical sciences, scientific worker of Institute of Computational Modelling of Russian Academy of Sciences;
E. D. Karepova — kandidat of physical and mathematical sciences, scientific worker of Institute of Computational Modelling of Russian Academy of Sciences;
S. F. Pyataev — diplom. mathematician, scientific worker of Institute of Computational Modelling of Russian Academy of Sciences;
T. V. Kalpush — post-graduate student of Institute of Computational Modelling of Russian Academy of Sciences.

During this period new results were obtained in the following directions:

increasing accuracy of finite-element schemes for convection-diffusion equations;
adaptive triangulations in finite-elements and finite-difference methods;
increasing accuracy and multigrid (cascadic) algorithms for second-order elliptic equations;
numerical algoritms for time-dependent Navier-Stokes equations.

These results were reported at 7 international congresses and conferences:

Numerical Methods for Singular Perturbations. Oberwolfach, April, 1998;
International Congress of Mathematicians. Berlin, August, 1998;
International Workshop on the Analytical and Computational Methods for Convection-Dominated and Singular Perturbed Problems. Lozenets, Bulgaria, August, 1998;
International GAMM-Workshop on Multigrid Methods. Bonn, October, 1998;
International Conference on Numerical Methods for Transport-Dominated and Related Problems. Schloss Wendgrcbben, Germany, September, 1999;
Sixth European Multigrid Conference. Gent, Belgium, October, 1999;
Numerical Methods for Singular Perturbation Problems. Oberwolfach, April, 2001.

Several talks and communications were made at 3 russian congresses and conference with foreing participants:

Third Siberian Congress on Applied and Industrial Mathematics. Novosibirsk, Russia, June, 1998;
Mathematical Models and the Methods of Their Investigation. Krasnoyarsk, Russia, August, 1999;
Fourth Siberian Congress on Applied and Industrial Mathematics. Novosibirsk, Russia, June, 2000.

3 young specialists (E. G. Bykova, E. D. Karepova, T. V. Kalpush) made several communications at 5 regional conferences for young scientists. During this period 9 visits of the russian participants to Germany have been conducted including joint scientific work at

Erlangen-Nurnberg Friedrich-Alexander University,
Heidelberg Ruprecht-Karls University,
Augsburg University,
Magdeburg Otto-von-Guericke University,
Dresden Technological University,
Leipzig Max-Planck Institute for Mathematics in the Sciences,
Oberwolfach Mathematical Institute.

Two business trips of two german participants to Russia have been conducted including participation in congress at Novosibirsk and joint scientific work in Institute of Computational Modelling of Russian Academy of Sciences in Krasnoyarsk. Owing to financial support for russian participants, icluding participation in international conferences, and owing to computer up-grade, all russian members had successful progress in scientific level:

I. V. Kireev defended kandidat thesis [30] in 1997;
E. G. Bykova defended kandidat thesis [3] in 1998;
L. V. Gilyova defended kandidat thesis [15] in 2000;
E. D. Karepova defended kandidat thesis [29] in 2000;
S. F. Pyataev defended kandidat thesis in 2001;
T. V. Kalpush finished post-graduate course in 2001, will represent thesis, and will defend it in 2001 or 2002.

The most part of the results of this Project was published (see for [1]-[36]). Concerning well-known journals ([34]-[36]), we do not repeat papers from them in present report. Other journals and books, especially in Russian, are not so widely known, therefore we translate the paper from them to English, if necessary, and brought in this report. Some results presented here are only submitted in journals and are publushed here for the first time.

The first volume of this book is devoted to the results concerning the method of approximation of the convection-diffusion equations with convection dominated and the method of increasing accuracy for the second-order self-adjoint elliptic equations. The second volume deals with the multigrid iterative methods for solving the finite-element analogues of the second-order self-adjoint equations and the finite element method for solving the Navier-Stokes time-dependent equations.

In the first part of the present volume new results are presented which are related to the method of fitting and adaptation of grids for approximation of the convection-diffusion equations. The method of fitting for the coefficients of the finite-element grid problem is similar to the difference method of fitting for approximation of the solutions of the boundary layer type. Three different techniques of the adaptation of grids are realized on the basis of a priori or a posteriori estimates of solution derivatives.

In the second part of this volume new results concerning the nonhomogeneous difference schemes of increased accuracy are presented for the second-order elliptic equations. Besides, using the solution of the Poisson equation as an example, the well-known difference and finite element schemes of the fourth order of accuracy are compared in efficiency.

Russian participations of Project are very grateful to Volkswagen Foundation for the financial support. We tried to use it for most scientific benefit. Many scientists helped us in Russia and Germany, but we would like to thank Prof. L. Tobiska for active participation in joint work, Prof. R. Ran-nacher for initialization of this work and discussions, and Prof. H.-G. Roos for fruitful discussions. Coordinator of Project, Prof. U. Rude and his team made many things for effective scientific work. We are very thankful them, but the special thanks to Prof. U. Rude for his great organizing and scientific work.