Uniform in $\epsilon$ discretization error estimates for convection dominated convection-diffusion problems

 Access to full text

 Full (PDF)

1. [1] O. AXELSSON, Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations, I.M.A. Journal Num. Anal., 1 (1981), 329-345 Zbl0508.76069MR641313
2. [2] O. AXELSSON, On the numerical solution of convection dominated convection-diffusion problems, in : Proc. Tagung Math. Physik, Karl-Marx-Stadt 1983,Teubner-Texte, Leipzig 1984. Zbl0563.76084MR781752
3. [3] C. BARDOS, J. RAUCH, Maximal positive boundary value problems as limits of singular perturbations problems, Transact. Amer. Math. Soc, 270 (1982) 2,377-400. Zbl0485.35010MR645322
4. [4] A. DEVINATZ, R. ELLIS, A. FRIEDMAN , The asymptotic behaviour of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives II. Indiana Univ. Math. J., 23 (1974), 991-1011. Zbl0263.35026MR344709
5. [5] A. FELGENHAUER, Application of a generalized maximum principle to estimate the corner layers in the n-dimensional case, in : Singularly perturbed differential equations and applications (J. Förste ed.), Akademie der Wissenschaften der DDR, Inst. f. Math., Report R-Mech 03/84, Berlin 1984, 1-8. 
6. [6] M. B. GILLES, M. E. ROSE, A numerical study of the steady scalar convective diffusion equation for small viscosity, J. Comput. Phys. 56 (1984), 513-529. Zbl0572.76087MR768674
7. [7] H. GOERING, A. FELGENHAUER, G. LUBE, H. G. ROOS, L. TOBISKA, Singularly perturbed differential equations, Math. Research, v. 13, Akademie-Verlag Berlin 1983. Zbl0522.35003MR718115
8. [8] T. J. R. HUGHES, A. BROOKS, multidimensional upwind scheme with no crosswind diffusion, in : AMD v. 34, Finite element methods for convection dominated flows (T. J. R. Hughes ed.), ASME, New York, 1979. Zbl0423.76067MR571679
9. [9] K. W. JEMELJANOV, On a difference scheme for the équation $\epsilon \Delta u+a{u}_{x_1}=f$, in : Difference methods for solving boundary value problems containing a small parameter and discontinuous boundary conditions, Isd. Uralskovo nacn. centra AN SSSR, Swerdlowsk 1976, 19-37 (russ.). 
10. [10] C. JOHNSON, U. NÄVERT, J. PITKARANTA, Finite element methods for linear hyperbolic problems, Comp. Meth. Appl. Mech. Engrg. 45 (1984), 285-312. Zbl0526.76087MR759811
11. [11] R. B. KELLOGG, Analysis of a difference approximation for a singular perturbation problem in two dimensions, in : Proc. Conf. Boundary and interior layers - computational and asymptotic methods (J. J. H. Miller ed.), Dublin 1980, Boole Press 1980, 113-117. Zbl0439.65081MR589355
12. [12] J. J. H. MILLER, On the convergence, uniformly in $\epsilon$, of difference schemes for a two point boundary value problem, in : Numerical analysis of singular perturbation problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, London, New York, San Francisco 1979, 467-474. Zbl0419.65051MR556537
13. [13] A. MIZUKAMI, T. J. R. HUGHES, A Petrov-Galerkin finite element method for solving convection dominated flows : an accurate upwinding technique for satisfying the maximum principle, Comp. Meth. Appl. Mech. Engrg. 50 (1985),181-193. Zbl0553.76075MR802339
14. [14] U. NÄVERT, A finite element method for convection-diffusion problems, Thesis, Chalmers Univ. of Technol., Gothenburg, Sweden 1982. 
15. [15] K. NIIJIMA, On a three-point difference scheme for a singular perturbation problem without a first derivative term, Mem. Num. Math. 7 (1980). Zbl0484.65054MR588462
16. [16] M. H. PROTTER, H. F. WEINBERGER, Maximum principles in differential equations, Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 1967. Zbl0153.13602MR219861
17. [17] U. RISCH, Ein hybrides upwind-FEM-Verfahren und dessen Anwendung auf schwach gekoppelte elliptische Differentialgleichungen mit dominanter Konvektion, Dissertation, Techn. Hochschule Magdeburg 1986. 
18. [18] A. M. SCHATZ, L. B. WAHLBIN, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimension, Math. Comp.40 (1983), 47-89. Zbl0518.65080MR679434
19. [19] F. SCHIEWECK, Eine asymptotisch angepafite Finite-Element-Methode fur singulär gestörte elliptische Randwertaufgaben, Dissertation, Techn. Hochschule Magdeburg 1986. 
20. [20] G. I. SHISHKIN, Solution of a boundary value problem for an elliptic equation with a small parameter affecting the highest derivatives, Shurnal Vytsch. Mat. Mat. Fis., 26 (1986), 1019-1031 (russ.). Zbl0622.65078MR851752
21. [21] G. I. SHISHKIN, V. A. TITOV, A différence scheme for a differential equation with two small parameters affecting the derivatives, Numer. Meth. Mechs. Cont. Media, 7 (1976), 145-155. (russ.) MR455427