A two-dimensional moving finite element method with local refinement based on a posteriori error estimates

Applied Numerical Mathematics

Volumn 46, Issue 1, 2003, Pages 75-94

  Lang, Jens ^a   Cao, Weiming ^b   Huang, Weizhang ^c   Russell, Robert D ^d  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF TEXAS AT SAN ANTONIO   (United States)

^c UNIVERSITY OF KANSAS   (United States)

^d SIMON FRASER UNIVERSITY   (Canada)

Author keywords

A posteriori error estimates; Local refinement; Moving mesh methods; Multilevel finite elements; Nonlinear time dependent PDEs; Rosenbrock methods

Indexed keywords

COMPUTATIONAL METHODS; DEGREES OF FREEDOM (MECHANICS); ERROR ANALYSIS;

MOVING MESH METHODS;

FINITE ELEMENT METHOD;

EID: 0038517141     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(03)00013-8     Document Type: Article

References (37)

1. Arney D.C., Flaherty J.E. An adaptive mesh-moving and local refinement method for time-dependent partial differential equations. ACM Trans. Math. Software. 16:1990;48-71.
2. Adjerid S., Flaherty J.E. A moving-mesh finite element method with local refinement for parabolic partial differential equations. Comp. Meth. Appl. Mech. Engrg. 55:1986;3-26.
3. Bank R.E. PLTMG: A Software Package for Solving Elliptic Partial Differential Equations - User's Guide 8.0. 1998;SIAM, Philadelphia, PA.
4. Bank R.E., Smith R.K. A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30:1993;921-935.
5. Babuška I., Suri M. The p and h-p version of the finite element method, an overview. Comp. Meth. Appl. Mech. Engrg. 80:1990;5-26.
6. Baines M.J. Moving Finite Elements. 1994;Oxford University Press, Oxford.
7. Bornemann F.A. An adaptive multilevel approach to parabolic equations. III. 2D error estimation and multilevel preconditioning. IMPACT Comput. Sci. Engrg. 4:1992;1-45.
8. Bornemann F.A., Erdmann B., Kornhuber R. A posteriori error estimates for elliptic problems in two and three space dimensions. SIAM J. Numer. Anal. 33:1996;1188-1204.
9. Cao W., Huang W., Russell R.D. An r-adaptive finite element method based upon moving mesh PDEs. J. Comput. Phys. 149:1999;221-244.
10. Cao W., Huang W., Russell R.D. A study of monitor functions for two-dimensional adaptive mesh generation. SIAM J. Sci. Comput. 20:1999;1978-1994.
11. Cao W., Huang W., Russell R.D. An error indicator monitor function for an r -adaptive finite element method. J. Comput. Phys. 170:2001;871-892.
12. P.J. Capon, P.K. Jimack, On the adaptive finite element solution of partial differential equations using h-r-refinement, Report 96.13, University of Leeds, School of Computer Studies, Leeds, 1996.
13. Carlson N., Miller K. Design and application of a gradient-weighted moving finite element method. SIAM J. Sci. Comput. 19:1998;798.
14. Dompierre J., Vallet M., Bourgault Y., Fortin M., Habashi W. Anisotropic mesh adaption: Towards user-independent, mesh-independent and solver-independent CFD. Part III: Unstructured grids. Internat. J. Numer. Methods Fluids. 39:2002;675-702.
15. Deuflhard P., Lang J., Nowak U. Adaptive algorithms in dynamical process simulation. Neunzert H. Progress in Industrial Mathematics at ECMI '94. 1996;122-137 Wiley-Teubner.
16. Deuflhard P., Leinen P., Yserentant H. Concepts of an adaptive hierarchical finite element code. IMPACT Comput. Sci. Engrg. 1:1989;3-35.
17. B. Erdmann, J. Lang, R. Roitzsch, KASKADE manual, Version 2.0, TR93-5, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, 1993.
18. Gropp W.D. Local uniform mesh refinement with moving grids. SIAM J. Sci. Statist. Comput. 8:1987;292-304.
19. Gustafsson K. Control-theoretic techniques for stepsize selection in implicit Runge-Kutta methods. ACM Trans. Math. Software. 20:1994;496-517.
20. Gustafsson K., Lundh M., Söderlind G. A PI stepsize control for the numerical solution of ordinary differential equations. BIT. 28:1988;270-287.
21. Habashi W., Dompierre J., Bourgault Y., Ait-Ali-Yahia D., Fortin M., Vallet M. Anisotropic mesh adaption: Towards user-independent, mesh-independent and solver-independent CFD. Part I: General principles. Internat. J. Numer. Methods Fluids. 32:2000;725-744.
22. Hairer E., Wanner G. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. 2nd revised edition:1996;Springer, Berlin.
23. Huang W. Practical aspects of formulation and solution of moving mesh partial differential equations. J. Comput. Phys. 171:2001;753-775.
24. Huang W. Variational mesh adaptation: Isotropy and equidistribution. J. Comput. Phys. 174:2001;903-924.
25. Huang W., Sun W. Variational mesh adaptation II: Error estimates and monitor functions. J. Comput. Phys. 184:2003;619-648.
26. Huang W., Russell R.D. A high dimensional moving mesh strategy. Appl. Numer. Math. 26:1997;63-76.
27. Huang W., Russell R.D. Moving mesh strategy based upon a gradient flow equation for two dimensional problems. SIAM J. Sci. Comput. 20:1999;998-1015.
28. Johnson A.A., Tezduyar T.E. Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces. Comp. Meth. Appl. Mech. Engrg. 119:1994;73-94.
29. A.P. Kuprat, Creation and Annihilation of Nodes for the Moving Finite Element Method, Ph.D. Thesis, University of California, Berkeley, CA, 1992.
30. Lang J. Adaptive FEM for reaction-diffusion equations. Appl. Numer. Math. 26:1998;105-116.
31. Lang J. Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications. Lecture Notes Comput. Sci. Engrg. 16:2000;Springer, Berlin.
32. Miller K., Miller R.N. Moving finite elements I. SIAM J. Numer. Anal. 18:1981;1019-1032.
33. B. Nkonga, H. Guillard, Godunov type method on non-structured meshes for three-dimensional moving boundary problems, INRIA Report 1883, 1993.
34. Rosenbrock H.H. Some general implicit processes for the numerical solution of differential equations. Comput. J. 5:1963;329-331.
35. Sandu A., Verwer J.G., Blom J.G., Spee E.J., Carmichael G.R., Potra F.A. Benchmarking stiff ODE solvers for atmospheric chemistry problems II: Rosenbrock solvers. Atmos. Environ. 31:1997;3459-3472.
36. van der Vorst H.A. BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. 13:1992;631-644.
37. Zienkiewicz O.C., Zhu J.G., Gong N.G. Effective and practical h-p adaptive analysis strategies for the finite element method. Internat. J. Numer. Meth. Engrg. 28:1989;879-891.