Least-squares finite element methods and algebraic multigrid solvers for linear hyperbolic PDEs

SIAM Journal on Scientific Computing

Volumn 26, Issue 1, 2005, Pages 31-54

  De Sterck, H ^a   Manteuffel, Thomas A ^a   Mccormicr, Stephen F ^a   Olson, Luke ^b  

^a UNIVERSITY OF COLORADO   (United States)

^b BROWN UNIVERSITY   (United States)

Author keywords

Algebraic multigrid; Finite element discretization; Hyperbolic problems; Least squares variational formulation

Indexed keywords

ALGORITHMS; BOUNDARY CONDITIONS; COMPUTATIONAL COMPLEXITY; CONVERGENCE OF NUMERICAL METHODS; LEAST SQUARES APPROXIMATIONS; LINEAR SYSTEMS; OSCILLATIONS; POLYNOMIALS;

ALGEBRAIC MULTIGRID; FINITE ELEMENT DISCRETIZATION; HYPERBOLIC PROBLEMS; LEAST-SQUARES VARIATIONAL FORMULATION;

FINITE ELEMENT METHOD;

EID: 14544278067     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S106482750240858X     Document Type: Article

References (30)

1. R. ABGRALL, Toward the ultimate conservative scheme: following the quest, J. Comput. Phys., 167 (2001), pp. 277-315.
2. T. J. BARTH, Numerical methods for gasdynamic systems on unstructured meshes, in An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (Freiburg/Littenweiler, 1997), Lect. Notes Comput. Sci. Eng. 5, Springer-Verlag, Berlin, 1999, pp. 195-285.
3. T. J. BARTH AND H. DECONINCK, EDS., High-order methods for computational physics, Lect. Notes Comput. Sci. Eng. 9, Springer-Verlag, Berlin, 1999.
4. M. BERNDT, T. A. MANTEUFFEL, AND S. F. MCCORMICK, Local error estimates and adaptive refinement for first-order system least squares (FOSLS), Electron. Trans. Numer. Anal., 6 (1997), pp. 35-43.
5. P. B. BOCHEV AND J. CHOI, A comparative study of least-squares, SUPG and Galerkin methods for convection problems, Int. J. Comput. Fluid Dyn., 15 (2001), pp. 127-146.
6. P. B. BOCHEV AND J. CHOI, Improved least-squares error estimates for scalar hyperbolic problems, Comput. Methods Appl. Math., 1 (2001), pp. 115-124.
7. P. B. BOCHEV AND M. D. GUNZBURGER, Finite element methods of least-squares type, SIAM Rev., 40 (1998), pp. 789-837.
8. D. BRAESS, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd ed., Cambridge University Press, Cambridge, UK, 2001.
9. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, Springer-Verlag, New York, 1994.
10. W. L. BRIGGS, V. E. HENSON, AND S. F. MCCORMICK, A Multigrid Tutorial, 2nd ed., SIAM, Philadelphia, 2000.
11. Z. CAI, R. LAZAROV, T. A. MANTEUFFEL, AND S. F. MCCORMICK, First-order system least squares for second-order partial differential equations: Part I, SIAM J. Numer. Anal., 31 (1994), pp. 1785-1799.
12. Z. CAI, T. A. MANTEUFFEL, AND S. F. MCCORMICK, First-order system least squares for second-order partial differential equations: Part II, SIAM J. Numer. Anal., 34 (1997), pp. 425-454.
13. G. F. CAREY AND B. N. JIANG, Least-squares finite elements for first-order hyperbolic systems, Internat. J. Numer. Methods Engrg., 26 (1988), pp. 81-93.
14. B. COCKBURN, Discontinuous Galerkin methods for convection-dominated problems, in High-Order Methods for Computational Physics, Lect. Notes Comput. Sci. Eng. 9, Springer-Verlag, Berlin, 1999, pp. 69-224.
15. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Computational Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
16. V. HENSON AND U. M. YANG, BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41 (2002), pp. 155-177.
17. P. HOUSTON, M. JENSEN, AND E. SÜLI, hp-discontinuous Galerkin finite element methods with least-squares stabilization, J. Sci. Comput., 17 (2002), pp. 3-25.
18. P. HOUSTON, J. A. MACKENZIE, E. SÜLI, AND G. WARNECKE, A posteriori error analysis for numerical approximations of Friedrichs systems, Numer. Math., 82 (1999), pp. 433-470.
19. B. N. JIANG, The Least-Squares Finite Element Method, Theory and Applications in Computational Fluid Dynamics and Electromagnetics, Scientific Computation, Springer-Verlag, Berlin, 1998.
20. C. JOHNSON, U. NÄVERT, AND J. PITKÄRANTA, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 45 (1984), pp. 285-312.
21. C. JOHNSON AND J. PITKÄRANTA, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46 (1986), pp. 1-26.
22. R. J. LEVEQUE, Numerical Methods for Conservation Laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992.
23. E. E. LEWIS AND J. W. F. MILLER, Computational Methods of Neutron Transport, American Nuclear Society, La Grange Park, IL, 1993.
24. T. A. MANTEUFFEL, K. J. RESSEL, AND G. STARKE, A boundary functional for the least-squares finite-element solution of neutron transport problems, SIAM J. Numer. Anal., 37 (2000), pp. 556-586.
25. S. F. MCCORMICK, Multilevel Adaptive Methods for Partial Differential Equations, Frontiers Appl. Math. 6, SIAM, Philadelphia, 1989.
26. J. RUGE, FOSPACK: A First-Order Systems Least-Squares (FOSLS) Code, manuscript.
27. J. RUGE AND K. STÜBEN, Efficient solution of finite difference and finite element equations, in Multigrid Methods for Integral and Differential Equations (Bristol, 1983), Oxford University Press, New York, 1985, pp. 169-212.
28. L. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
29. U. TROTTENBERG, C. W. OOSTERLEE, AND A. SCHÜLLER, Multigrid, Academic Press, San Diego, 2001.
30. I. YAVNEH, C. H. VENNER, AND A. BRANDT, Fast multigrid solution of the advection problem with closed characteristics, SIAM J. Sci. Comput., 19 (1998), pp. 111-125.