Local and parallel finite element algorithms for eigenvalue problems

Acta Mathematicae Applicatae Sinica

Volumn 18, Issue 2, 2002, Pages 185-200

  Xu, Jinchao ^a   Zhou, Aihui ^b  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

Author keywords

Eigenvalue; Finite element; Local algorithm; Parallel algorithm

Indexed keywords

EID: 16244369178     PISSN: 01689673     EISSN: 16183932     Source Type: Journal    
DOI: 10.1007/s102550200018     Document Type: Article

References (38)

1. Adams, R.A.: Sobolev spaces. Academic Press, New York, 1975
2. Axelsson, O., Layton, W.: A two-level discretization of nonlinear boundary value problems. SIAM J. Numer. Anal., 33: 2359-2374 (1996)
3. Babuska, I., Osborn, J.E.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comp., 1989, 52: 275-297
4. Babuska, I., Osborn, J.E.: Eigenvalue problems, handbook of numerical analysis, vol.2, finite element methods (part 1). Ciarlet, P.G., Lions, J.L., eds., Elsevier, 1991, 641-792
5. Baker, N., Holst, M., Wang, F.: Adaptive multilevel finite element solution of the Poisson-Boltzmann equation IT: refinement at solvent accessible surfaces in biomolecular systems. J. Comput. Chem., 21: 1343-1352 (2000)
6. Bank, R.E.: Hierarchical bases and the finite element method. Acta Numerica, 5: 1-43 (1996)
7. Bank, R.E., Holst, M.: A new paradigm for parallel adaptive meshing algorithms. SIAM J. Sci. Comput., 22: 1411-1443 (2001)
8. Bedivan, D.M.: A two-grid method for solving elliptic problems with inhomogeneous boundary conditions. Comput. Math. Appl., 29: 59-66 (1995)
9. Bramble, J.H.: Multigrid methods, pitman research notes in mathematics, Vol.294. London Co-published in the USA with Wiley, New York, 1993
10. Chan, T., Mathew, T.: Domain decomposition algorithms. Acta Numerica, 3: 61-143 (1994)
11. Chatelin, F. Spectral approximations of linear operators. Academic Press, New York, 1983
12. Ciarlet, P.G., Lions J.L. Handbook of numerical analysis, Vol.2, finite element methods (Part I). North-Holland, 1991
13. Dawson, C.N., Wheeler, M.F.: Two-grid methods for mixed finite element approximations of nonlinear parabolic equations. Contemp. Math., 180: 191-203 (1994)
14. Dawson, C.N., Wheeler, M.F., Woodward, C.S. A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal., 35: 435-452 (1988)
15. Grisvard, P.: Elliptic problems in nonsmooth domains. Pitman, Boston, MA, 1985
16. Hackbusch, W.: Multigrid methods and applications. Springer-Verlag, New York, 1985
17. Holst, M., Baker, N., Wang, F.: Adaptive multilevel finite element solution of the Poisson-Poltzmann equation I: algorithms and examples. J. Comput. Chem., 21: 1319-1342 (2000)
18. Layton, W., Lenferink, W.: Two-level picard and modified Picard methods for the Navier-Stokes equations. Appl. Math. Comp., 69: 263-274 (1995)
19. Marion, M., Xu, J.: Error estimates on a new nonlinear Galerkin method based on two-grid finite elements. SIAM J. Numer. Anal., 32: 1170-1184 (1995)
20. Nitsche, J., Schatz, A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comp., 28: 937-955 (1974)
21. Rannacher, R., Scott, R.: Some optimal error estimate for piecewise linear finite element approximations. Math. Comp., 38: 437-445 (1982)
22. Schatz, A.H., Wahlbin, L.B.: Interior maximum-norm estimates for finite element methods. Math. Comp., 31: 414-442 (1977)
23. Schatz, A.H., Wahlbin, L.B.: Interior maximum-norm estimates for finite element methods, Part II. Math. Comp., 1995, 64: 907-928
24. Schatz, A.H., Wang, J.: Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions. Math. Comp., 65: 19-27 (1996)
25. Utnes, T.: Two-grid finite element formulations of the incompressible Navier-Stokes equations. Comm. Numer. Methods Engrg., 34: 675-684 (1997)
26. Wahlbin, L.B. Local behavior in finite element methods, handbook of numerical analysis, Vol. II, finite element methods (Part 1). Ciarlet, P.G., Lions, J.L., eds. Elsevier, 1991, 355-522
27. Wahlbin, L.B. Superconvergence in Galerkin finite element methods, Vol.1605. Lecture Notes in Math., Springer-Verlag, Berlin-Heidelberg, 1995
28. Xu, J.: A new class of iterative methods for nonselfadjoint or indefinite problems. SIAM J. Numer. Anal., 29: 303-319 (1992)
29. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Review, 34(4): 581-613 (1992)
30. Xu, J. A novel two-grid method for semilinear equations. SIAM J. Sci. Comput., 15: 231-237 (1994)
31. Xu, J. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal., 33: 1759-1777 (1999)
32. Xu, J., Zhou, A.: Some local and parallel properties of finite element discretizations. In: Proc. 11 International Conference on DDM, Eds., C.H. Lai, P.E. Bjøsted, M. Cross & O.B. Widlund, DDM.org, 1999, 140-147
33. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comp., 69: 881-909 (2000)
34. Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comp., 70: 17-25 (2001)
35. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comp. Math., 14: 293-327 (2001)
36. Xu, J., Zou, J.: Some non-overlapping domain decomposition methods. SIAM Review, 40: 857-914 (1988)
37. Yserentant, H.: Old and new proofs for multigrid algorithms. Acta Numerica, 2: 285-326 (1993)
38. Zhou, A., Liem, C.L., Shih, T.M., Lu, T.: Error analysis on bi-parameter finite elements. Comput. Methods Appl. Mech. Engrg., 158: 329-339 (1998)