Simplifying numerical solution of constrained PDE systems through involutive completion

Bijan Mohammadi; Jukka Tuomela

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 5, page 909-929
  • ISSN: 0764-583X

Abstract

top
When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.

How to cite

top

Mohammadi, Bijan, and Tuomela, Jukka. "Simplifying numerical solution of constrained PDE systems through involutive completion." ESAIM: Mathematical Modelling and Numerical Analysis 39.5 (2010): 909-929. <http://eudml.org/doc/194293>.

@article{Mohammadi2010,
abstract = { When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes. },
author = {Mohammadi, Bijan, Tuomela, Jukka},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Overdetermined PDEs; involution; discretization.; elliptic systems; involutive form; overdetermined systems; discretization; auxiliary variables},
language = {eng},
month = {3},
number = {5},
pages = {909-929},
publisher = {EDP Sciences},
title = {Simplifying numerical solution of constrained PDE systems through involutive completion},
url = {http://eudml.org/doc/194293},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Mohammadi, Bijan
AU - Tuomela, Jukka
TI - Simplifying numerical solution of constrained PDE systems through involutive completion
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 5
SP - 909
EP - 929
AB - When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.
LA - eng
KW - Overdetermined PDEs; involution; discretization.; elliptic systems; involutive form; overdetermined systems; discretization; auxiliary variables
UR - http://eudml.org/doc/194293
ER -

References

top
  1. M.S. Agranovich, Elliptic boundary problems, Partial differential equations IX. M.S. Agranovich, Yu.V. Egorov and M.A. Shubin, Eds., Springer. Encyclopaedia Math. Sci.79 (1997) 1–144.  
  2. Å. Björck, Numerical methods for least squares problems, SIAM (1996).  
  3. H. Borouchaki, P.L. George and B. Mohammadi, Delaunay mesh generation governed by metric specifications. Parts i & ii. Finite Elem. Anal. Des., Special Issue on Mesh Adaptation (1996) 345–420.  
  4. M. Castro-Diaz, F. Hecht and B. Mohammadi, Anisotropic grid adaptation for inviscid and viscous flows simulations. Int. J. Numer. Meth. Fl.25 (1995) 475–491.  
  5. A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math.8 (1955) 503–538.  
  6. P.I. Dudnikov and S.N. Samborski, Linear overdetermined systems of partial differential equations. Initial and initial-boundary value problems, Partial Differential Equations VIII, M.A. Shubin, Ed., Springer-Verlag, Berlin/Heidelberg. Encyclopaedia Math. Sci.65 (1996) 1–86.  
  7. Femlab 3.0,  URIhttp://www.comsol.com/products/femlab/
  8. FreeFem,  URIhttp://www.freefem.org/
  9. P.L. George, Automatic mesh generation. Applications to finite element method, Wiley (1991).  
  10. R. Glowinski, Finite element methods for incompressible viscous flow. Handb. Numer. Anal. Vol. IX, North-Holland, Amsterdam (2003) 3–1176.  
  11. F. Hecht and B. Mohammadi, Mesh adaptation by metric control for multi-scale phenomena and turbulence. American Institute of Aeronautics and Astronautics 97-0859 (1997).  
  12. B. Jiang, J. Wu and L. Povinelli, The origin of spurious solutions in computational electromagnetics. J. Comput. Phys.7 (1996) 104–123.  
  13. K. Krupchyk, W. Seiler and J. Tuomela, Overdetermined elliptic PDEs. J. Found. Comp. Math., submitted.  
  14. E.L. Mansfield, A simple criterion for involutivity. J. London Math. Soc. (2)54 (1996) 323–345.  
  15. B. Mohammadi and J. Tuomela, Involutivity and numerical solution of PDE systems, in Proc. of ECCOMAS 2004, Vol. 1, Jyväskylä, Finland. P. Neittaanmäki, T. Rossi, K. Majava and O. Pironneau, Eds., University of Jyväskylä (2004) 1–10.  
  16. F. Nicoud, Conservative high-order finite-difference schemes for low-Mach number flows. J. Comput. Phys.158 (2000) 71–97.  
  17. O. Pironneau, Finite element methods for fluids, Wiley (1989).  
  18. J.F. Pommaret, Systems of partial differential equations and Lie pseudogroups. Math. Appl., Gordon and Breach Science Publishers 14 (1978).  
  19. R.F. Probstein, Physicochemical hydrodynamics, Wiley (1995).  
  20. A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Ser. Comput. Math.23 (1994).  
  21. W.M. Seiler, Involution — the formal theory of differential equations and its applications in computer algebra and numerical analysis, Habilitation thesis, Dept. of Mathematics, Universität Mannheim (2001) (manuscript accepted for publication by Springer-Verlag).  
  22. D. Spencer, Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc.75 (1969) 179–239.  
  23. J. Tuomela and T. Arponen, On the numerical solution of involutive ordinary differential systems. IMA J. Numer. Anal.20 (2000) 561–599.  
  24. J. Tuomela and T. Arponen, On the numerical solution of involutive ordinary differential systems: Higher order methods. BIT41 (2001) 599–628.  
  25. J. Tuomela, T. Arponen and V. Normi, On the numerical solution of involutive ordinary differential systems: Enhanced linear algebra. IMA J. Numer. Anal., submitted.  

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.