Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes

SIAM Journal on Scientific Computing

Volumn 30, Issue 1, 2007, Pages 246-271

  Georgoulis, Emmanuil H ^a   Hall, Edward ^b   Houston, Paul ^b  

^a UNIVERSITY OF LEICESTER   (United Kingdom)

^b UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

Anisotropic mesh adaptation; Discontinuous Galerkin methods; Pdes with non negative characteristic form

Indexed keywords

ANISOTROPY; CONSTRAINED OPTIMIZATION; ERROR ANALYSIS; GALERKIN METHODS; MESH GENERATION;

ADVECTION-DIFFUSION-REACTION PROBLEMS; ANISOTROPIC MESH ADAPTATION; DISCONTINOUS GALERKIN METHODS; DISCONTINUOUS GALERKIN; INTERIOR PENALTY METHOD; NON NEGATIVES; PDES WITH NON-NEGATIVE CHARACTERISTIC FORM; POSTERIORI; PRIORI ERROR ANALYSIS; SECOND-ORDER PARTIAL DIFFERENTIAL EQUATION;

ADAPTIVE ALGORITHMS;

EID: 52749085293     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060672352     Document Type: Article

References (30)

1. T. APEL, Anisotropic Finite Elements: Local Estimates and Applications, Adv. Numer. Math., Teubner, Stuttgart, 1999.
2. D. N. ARNOLD, F. BREZZI, B. COCKBURN, AND L. D. MARINI, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749-1779.
3. G. A. BAKER, W. N. JUREIDINI, AND O. A. KARAKASHIAN, Piecewise solenoidal vector fields and the Stokes problem, SIAM J. Numer. Anal., 27 (1990), pp. 1466-1485.
4. C. BAUMANN, An hp-Adaptive Discontinuous Galerkin FEM for Computational Fluid Dynamics, Ph.D. thesis, Texas Institute for Computational and Applied Mathematics, UT Austin, Austin, TX, 1997.
5. R. BECKER AND R. RANNACHER, An optimal control approach to a-posteriori error estimation in finite element methods, in Acta Numerica, A. Iserles, ed., Cambridge University Press, Cambridge, UK, 2007, pp. 1-102.
6. W. CAO, On the error of linear interpolation and the orientation, aspect ratio, and internal angles of a triangle, SIAM J. Numer. Anal., 43 (2005), pp. 19-40.
7. M. J. CASTRO-DÍAZ, F. HECHT, B. MOHAMMADI, AND O. PIRONNEAU, Anisotropic unstructured mesh adaption for flow simulations, Internat. J. Numer. Methods Fluids, 25 (1997), pp. 475-491.
8. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
9. L. FORMAGGIA AND S. PEROTTO, New anisotropic a priori error estimates, Numer. Math., 89 (2001), pp. 641-667.
10. E. H. GEORGOULIS, Discontinuous Galerkin Methods on Shape-Regular and Anisotropic Meshes, D. Phil. Thesis, Computing Laboratory, University of Oxford, Oxford, UK, 2003.
11. E. H. GEORGOULIS, hp-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes, Int. J. Numer. Anal. Model., 3 (2006), pp. 52-79.
12. E. H. GEORGOULIS AND A. LASIS, A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems, IMA J. Numer. Anal., 26 (2006), pp. 381-390.
13. K. HARRIMAN. P. HOUSTON, B. SENIOR, AND E. SÜLI, hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form, in Recent Advances in Scientific Computing and Partial Differential Equations, C.-W. Shu, T. Tang, and S.-Y. Cheng, eds., Contemp. Math. 330, AMS, Providence, RI, 2003, pp. 89-119.
14. R. HARTMANN AND P. HOUSTON, Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws, SIAM J. Sci. Comput., 24 (2002), pp. 979-1004.
15. P. HOUSTON, E. H. GEORGOULIS, AND E. HALL, Adaptivity and a posteriori error estimation for DG methods on anisotropic meshes, in Proceedings of the International Conference on Boundary and Interior Layers (BAIL) - Computational and Asymptotic Methods, G. Lube and G. Rapin, eds., 2006.
16. P. HOUSTON, C. SCHWAB, AND E. SÜLI, Stabilized hp-finite element methods for first-order hyperbolic problems, SIAM J. Numer. Anal., 37 (2000), pp. 1618-1643.
17. P. HOUSTON, C. SCHWAB, AND E. SÜLI, Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39 (2002), pp. 2133-2163.
18. P. HOUSTON AND E. SÜLI, hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems, SIAM J. Sci. Comput., 23 (2001), pp. 1226-1252.
19. P. HOUSTON AND E. SÜLI, Stabilized hp-finite element approximation of partial differential equations with non-negative characteristic form, Comput., 66 (2001), pp. 99-119.
20. P. HOUSTON AND E. SÜLI, Adaptive finite element approximation of hyperbolic problems, in Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, T. Barth and H. Deconinck, eds., Lecture Notes in Comput. Sci. Engrg. 25, Springer, New York, 2002, pp. 269-344.
21. W. HUANG, Mathematical principles of anisotropic mesh adaptation, Comm. Comput. Phys., 1 (2006), pp. 276-310.
22. G. KUNERT, A Posteriori Error Estimation for Anisotropic Tetrahedral and Triangular Finite Element Meshes, Ph.D. thesis, TU Chemnitz, Chemnitz, Germany, 1999.
23. M. G. LARSON AND T. J. BARTH, A posteriori error estimation for discontinuous Galerkin approximations of hyperbolic systems, in Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes Comput. Sci. Engrg. 11, Springer, New York, 2000, pp. 363-368.
24. L. DE LATHAUWER, B. DE MOOR, AND J. VANDEWALLE, A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253-1278.
25. J. T. ODEN, I. BABUŠKA, AND C. E. BAUMANN, A discontinuous hp-finite element method for diffusion problems, J. Comput. Phys., 146 (1998), pp. 491-519.
26. O. A. OLEINIK AND E. V. RADKEVIČ, Second Order Equations with Nonnegative Characteristic Form, AMS, Providence, RI, 1973.
27. S. PRUDHOMME, F. PASCAL, J. T. ODEN, AND A. ROMKES, Review of A Priori Error Estimation for Discontinuous Galerkin Methods, Technical report, TICAM Report 00-27, Texas Institute for Computational and Applied Mathematics, Austin, TX, 2000.
28. W. RACHOWICZ, L. DEMKOWICZ, AND J. T. ODEN, Toward a universal h-p adaptive finite element strategy, Part 3. Design of h-p meshes, Comput. Methods Appl. Mech. Engrg., 77 (1989), pp. 181-212.
29. R. SCHNEIDER AND P. JIMACK, Toward Anisotropic Mesh Adaptation Based upon Sensitivity of A Posteriori Estimates, Technical Report 2005.03, School of Computing, University of Leeds, Leeds, UK, 2005.
30. E. SÜM, CH. SCHWAB, AND P. HOUSTON, hp-DGFEM for partial differential equations with nonnegative characteristic form, in Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes Comput. Sci. Engrg. 11, Springer, New York, 2000, pp. 221-230.