HP-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form

Andrea Toselli

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 1, page 91-115
  • ISSN: 0764-583X

Abstract

top
We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements.

How to cite

top

Toselli, Andrea. "HP-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 91-115. <http://eudml.org/doc/194158>.

@article{Toselli2010,
abstract = { We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements. },
author = {Toselli, Andrea},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Advection–diffusion; hyperbolic problems; stabilization; domain decomposition; non-matching grids; discontinuous Galerkin; hp-finite elements.; Advection-diffusion equation; domain decomposition; discontinuous Galerkin method; hp-finite elements; error bounds; numerical results},
language = {eng},
month = {3},
number = {1},
pages = {91-115},
publisher = {EDP Sciences},
title = {HP-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form},
url = {http://eudml.org/doc/194158},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Toselli, Andrea
TI - HP-finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 91
EP - 115
AB - We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements.
LA - eng
KW - Advection–diffusion; hyperbolic problems; stabilization; domain decomposition; non-matching grids; discontinuous Galerkin; hp-finite elements.; Advection-diffusion equation; domain decomposition; discontinuous Galerkin method; hp-finite elements; error bounds; numerical results
UR - http://eudml.org/doc/194158
ER -

References

top
  1. Y. Achdou, The mortar element method for convection diffusion problems. C.R. Acad. Sci. Paris Sér. I Math.321 (1995) 117-123.  Zbl0835.65127
  2. Y. Achdou, C. Japhet, Y. Maday and F. Nataf, A new cement to glue non-conforming grids with Robin interface conditions: The finite volume case. Numer. Math.92 (2002) 593-620.  Zbl1019.65086
  3. T. Arbogast, L.C. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids. SIAM J. Numer. Anal.37 (2000) 1295-1315.  Zbl1001.65126
  4. T. Arbogast and I. Yotov, A non-mortar mixed finite element method for elliptic problems on non-matching multiblock grids. Comput. Methods Appl. Mech. Engrg.149 (1997) 255-265.  Zbl0923.76100
  5. I. Babuska and M. Suri, The hp version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér.21 (1987) 199-238.  Zbl0623.65113
  6. R. Becker and P. Hansbon, A finite element method for domain decomposition with non-matching grids. Technical Report N° 3613, INRIA, January 1999.  
  7. F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional finite elements. RAIRO Modél. Math. Anal. Numér.31 (1997) 289-302.  Zbl0868.65082
  8. F. Ben Belgacem and Y. Maday, Coupling spectral and finite element for second order elliptic three dimensional equations. SIAM J. Numer. Anal.31 (1999) 1234-1263.  Zbl0942.65132
  9. A. Ben Abdallah, F. Ben Belgacem, Y. Maday and F. Rapetti, Mortaring the two-dimensional Nédélec finite element for the discretization of the Maxwell equations. M2AS (submitted).  Zbl1100.78017
  10. F. Ben Belgacem, The mortar element method with Lagrange multipliers. Numer. Math.84 (1999) 173-197.  Zbl0944.65114
  11. C. Bernardi, Y. Maday and A.T. Patera, A new non conforming approach to domain decomposition: The mortar element method, in Collège de France Seminar, H. Brezis and J.-L. Lions Eds., Pitman (1994).  Zbl0797.65094
  12. K.S. Bey, A. Patra and J.T. Oden, hp-version discontinuous Galerkin methods for hyperbolic conservation laws: A parallel adaptive strategy. Internat. J. Numer. Methods Engrg.38 (1995) 3889-3908.  Zbl0855.65106
  13. F. Brezzi and D. Marini, A three-field domain decomposition method, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, A. Quarteroni, Y.A. Kuznetsov, J. Périaux and O.B. Widlund Eds., AMS. Contemp. Math.157 (1994) 27-34. Held in Como, Italy, June 15-19, 1992.  Zbl0801.65116
  14. F. Brezzi and D. Marini, Error estimates for the three-field formulation with bubble functions. Math. Comp.70 (2001) 911-934.  Zbl0970.65118
  15. B. Cockburn, G.E. Karniadakis and Chi-Wang Shu (Eds.), Discontinuous Galerkin Methods. Springer-Verlag, Lect. Notes Comput. Sci. Eng. 11 (2000).  Zbl0989.76045
  16. P. Houston, C. Schwab and E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal.39 (2002) 2133-2163.  Zbl1015.65067
  17. P. Houston and E. Süli, Stabilised hp-finite element approximation of partial differential equations with nonnegative characteristic form. Computing66 (2001) 99-119. Archives for scientific computing. Numerical methods for transport-dominated and related problems, Magdeburg (1999).  Zbl0985.65136
  18. T.J.R. Hughes, L.P. Franca and G.M. Hulbert, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg.73 (1989) 173-189.  Zbl0697.76100
  19. C. Johnson, Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987).  
  20. C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg.45 (1984) 285-312.  Zbl0526.76087
  21. C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp.46 (1986) 1-26.  Zbl0618.65105
  22. P. Le Tallec and T. Sassi, Domain decomposition with nonmatching grids: Augmented Lagrangian approach. Math. Comp.64 (1995) 1367-1396.  Zbl0849.65087
  23. A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer-Verlag, Berlin (1994).  Zbl0803.65088
  24. C. Schwab, p- and hp-finite element methods. Oxford Science Publications (1998).  Zbl0910.73003
  25. R. Stenberg, Mortaring by a method of J.A. Nitsche, in Computational Mechanics: New trends and applications, S. Idelshon, E. Onate and E. Dvorkin Eds., Barcelona (1998). @CIMNE.  
  26. M.F. Wheeler and I. Yotov, Physical and computational domain decompositions for modeling subsurface flows, in Tenth International Conference on Domain Decomposition Methods, J. Mandel, C. Farhat and X.-C. Cai Eds., AMS. Contemp. Math.218 (1998) 217-228.  Zbl0957.76037
  27. B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal.38 (2000) 989-1012.  Zbl0974.65105
  28. I. Yotov, Mixed Finite Element Methods for Flow in Porous Media. Ph.D. thesis, TICAM, University of Texas at Austin (1996).  
  29. I. Yotov, A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow. East-West J. Numer. Math.5 (1997) 211-230.  Zbl0897.76057

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.