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Simplifying numerical solution of constrained PDE systems through involutive completion

Published online by Cambridge University Press:  15 September 2005

Bijan Mohammadi  and
Jukka Tuomela
Bijan Mohammadi
Affiliation:
Mathematics and Modeling Institute, Montpellier University, France and Department of Mathematics, University of Joensuu, PO Box 111, 80101 Joensuu, Finland. [email protected]; [email protected]
Jukka Tuomela
Affiliation:
Mathematics and Modeling Institute, Montpellier University, France and Department of Mathematics, University of Joensuu, PO Box 111, 80101 Joensuu, Finland. [email protected]; [email protected]
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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 formof the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution ofseveral flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes.

Keywords

Overdetermined PDEsinvolutiondiscretization.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 5 , September 2005 , pp. 909 - 929
Copyright
© EDP Sciences, SMAI, 2005

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References

Agranovich, M.S., 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. CrossRef
Å. Björck, Numerical methods for least squares problems, SIAM (1996).
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.
Castro-Diaz, M., Hecht, F. and Mohammadi, B., Anisotropic grid adaptation for inviscid and viscous flows simulations. Int. J. Numer. Meth. Fl. 25 (1995) 475–491. 3.0.CO;2-6>CrossRef
Douglis, A. and Nirenberg, L., Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955) 503–538.
Dudnikov, P.I. and Samborski, S.N., 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. CrossRef
Femlab 3.0 , http://www.comsol.com/products/femlab/
FreeFem , http://www.freefem.org/
P.L. George, Automatic mesh generation. Applications to finite element method, Wiley (1991).
R. Glowinski, Finite element methods for incompressible viscous flow. Handb. Numer. Anal. Vol. IX, North-Holland, Amsterdam (2003) 3–1176.
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).
Jiang, B., Wu, J. and Povinelli, L., The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 7 (1996) 104–123. CrossRef
K. Krupchyk, W. Seiler and J. Tuomela, Overdetermined elliptic PDEs. J. Found. Comp. Math., submitted.
Mansfield, E.L., A simple criterion for involutivity. J. London Math. Soc. (2) 54 (1996) 323–345. CrossRef
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.
Nicoud, F., Conservative high-order finite-difference schemes for low-Mach number flows. J. Comput. Phys. 158 (2000) 71–97. CrossRef
O. Pironneau, Finite element methods for fluids, Wiley (1989).
J.F. Pommaret, Systems of partial differential equations and Lie pseudogroups. Math. Appl., Gordon and Breach Science Publishers 14 (1978).
R.F. Probstein, Physicochemical hydrodynamics, Wiley (1995).
A. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer Ser. Comput. Math. 23 (1994).
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).
Spencer, D., Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc. 75 (1969) 179–239. CrossRef
Tuomela, J. and Arponen, T., On the numerical solution of involutive ordinary differential systems. IMA J. Numer. Anal. 20 (2000) 561–599. CrossRef
Tuomela, J. and Arponen, T., On the numerical solution of involutive ordinary differential systems: Higher order methods. BIT 41 (2001) 599–628. CrossRef
J. Tuomela, T. Arponen and V. Normi, On the numerical solution of involutive ordinary differential systems: Enhanced linear algebra. IMA J. Numer. Anal., submitted.