A Comparison of Adaptive Refinement Techniques for Elliptic Problems

ACM Transactions on Mathematical Software (TOMS)

Volumn 15, Issue 4, 1989, Pages 326-347

  Mitchell, William F ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Adaptive refinement; error estimate; triangle division

Indexed keywords

MATHEMATICAL TECHNIQUES--DIFFERENTIAL EQUATIONS;

ADAPTIVE REFINEMENT; ELLIPTIC EQUATIONS; FINITE ELEMENT METHODS; MATHEMATICAL SOFTWARE; NUMERICAL ALGORITHMS; PARTIAL DIFFERENTIAL EQUATIONS;

COMPUTER PROGRAMMING;

EID: 0024883616     PISSN: 00983500     EISSN: 15577295     Source Type: Journal    
DOI: 10.1145/76909.76912     Document Type: Article

References (20)

1. Babuška, I., and Aziz, A. K. On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976), 214-226.
2. Babuska, I., and Rheinboldt, W. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978), 736-754.
3. Bank, R. E. PLTMG User’s Guide, Edition 4.0. Tech. Rep., Dept. of Mathematics, Univ. of California at San Diego, La Jolla, Calif., 1985.
4. Bank, R. E., and Sherman, A. H. The use of adaptive grid refinement for badly behaved elliptic partial differential equations. In Advances in Computer Methods for Partial Differential Equations III, R. Vichnevetsky and R. S. Stepleman, Eds. IMACS, New Brunswick, 1979, pp. 33-39.
5. Bank, R. E., and Sherman, A. H. An adaptive multilevel method for elliptic boundary value problems. Computing 26 (1981), 91-105.
6. Bank, R. E., and Weiser, A. Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985), 283-301.
7. De, J. P., Gago, S. R., Kelly, D. W., Zienkiewicz, O. C., and Babuska, I. A posteriori error analysis and adaptive processes in the finite element method: Part II—Adaptive mesh refinement. Int. J. Numer. Meth. Eng. 19 (1983), 1621-1656.
8. Fried, I. Condition of finite element matrices generated from nonuniform meshes. AIAA J. 10 (1972), 219-221.
9. Kelly, D. W., De, J. P., Gago, S. R., Zienkiewicz, O. C., and Babuska, I. A posteriori error analysis and adaptive processes in the finite element method: Part I—Error analysis. Int. J. Numer. Meth. Eng. 19 (1983), 1593-1619.
10. Mitchell, W. F. A comparison of adaptive refinement techniques for elliptic problems. Tech. Rep. UIUCDCS-R-87-1375, Dept. of Computer Science, Univ. of Illinois, Urbana, 1987.
11. Rice, J. R., Houstis, E. N., and Dyksen, W. R. A population of linear, second order elliptic partial differential equations on rectangular domains. Math. Comput. 36 (1981), 475-484.
12. Rivara, M. C. Mesh refinement processes based on the generalized bisection of simplices. SIAM J. Numer. Anal. 21 (1984), 604-613.
13. Rivara, M. C. Algorithms for refining triangular grids suitable for adaptive and multigrid techniques. Int. J. Numer. Meth. Eng. 20 (1984), 745-756.
14. Rivara, M. C. Design and data structure of fully adaptive multigrid, finite-element software. ACM Trans. Math. Softw. 10, 3 (Sept. 1984), 242-264.
15. Rivara, M. C. Adaptive finite element refinement and fully irregular and conforming triangulations. In Accuracy Estimates and Adaptive Refinements in Finite Element Computations, I. Babuska, O. C Zienkiewicz, J. Gago, and E. R. de A. Oliveira, Eds. Wiley, 1986, pp. 359-369.
16. Rosenberg, I. G., and Stenger, F. A lower bound on the angles of triangles constructed by bisecting the longest side. Math. Comput. 29 (1975), 390-395.
17. Sewell, E. G. Automatic generation of triangulations for piecewise polynomial approximation. Ph.D. dissertation, Purdue Univ., West Lafayette, Ind., 1972.
18. Sewell, E. G. A finite element program with automatic user-controlled mesh grading. In Advances in Computer Methods for Partial Differential Equations III, R. Vichnevetsky, and R. S. Stepleman, Eds. IMACS, New Brunswick, 1979, pp. 8-10.
19. Sewell, E. G. Analysis of a Finite Element Method: PDE/PROTRAN. Springer-Verlag, New York, 1985.
20. Zienkiewicz, O. C., Kelly, D. W., Gago, J., and Babuska, I. Hierarchical finite element approaches, error estimates and adaptive refinement. In The Mathematics of Finite Elements and Applications IV, J. R. Whiteman, Ed. Academic Press, Orlando, Fla., 1982, pp. 313-346.