Simplifying numerical solution of constrained PDE systems through involutive completion
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 5, p. 909-929

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.

DOI : https://doi.org/10.1051/m2an:2005040
Classification:  35G15,  35N10,  65M60,  65N30
Keywords: overdetermined PDEs, involution, discretization
@article{M2AN_2005__39_5_909_0,
     author = {Mohammadi, Bijan and Tuomela, Jukka},
     title = {Simplifying numerical solution of constrained PDE systems through involutive completion},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {5},
     year = {2005},
     pages = {909-929},
     doi = {10.1051/m2an:2005040},
     zbl = {1078.35010},
     mrnumber = {2178567},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_5_909_0}
}
Mohammadi, Bijan; Tuomela, Jukka. Simplifying numerical solution of constrained PDE systems through involutive completion. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 5, pp. 909-929. doi : 10.1051/m2an:2005040. http://www.numdam.org/item/M2AN_2005__39_5_909_0/

[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. | Zbl 0880.35001

[2] Å. Björck, Numerical methods for least squares problems, SIAM (1996). | MR 1386889 | Zbl 0847.65023

[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. | Zbl 0902.76057

[5] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955) 503-538. | Zbl 0066.08002

[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. | Zbl 0831.35113

[7] Femlab 3.0, http://www.comsol.com/products/femlab/

[8] FreeFem, http://www.freefem.org/

[9] P.L. George, Automatic mesh generation. Applications to finite element method, Wiley (1991). | MR 1165297 | Zbl 0808.65122

[10] R. Glowinski, Finite element methods for incompressible viscous flow. Handb. Numer. Anal. Vol. IX, North-Holland, Amsterdam (2003) 3-1176. | Zbl 1040.76001

[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. | Zbl 0848.65086

[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. | Zbl 0865.35092

[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. | Zbl 0973.76068

[17] O. Pironneau, Finite element methods for fluids, Wiley (1989). | MR 1030279 | Zbl 0712.76001

[18] J.F. Pommaret, Systems of partial differential equations and Lie pseudogroups. Math. Appl., Gordon and Breach Science Publishers 14 (1978). | MR 517402 | Zbl 0401.58006

[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). | MR 1299729 | Zbl 0803.65088

[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. | Zbl 0185.33801

[23] J. Tuomela and T. Arponen, On the numerical solution of involutive ordinary differential systems. IMA J. Numer. Anal. 20 (2000) 561-599. | Zbl 0982.65088

[24] J. Tuomela and T. Arponen, On the numerical solution of involutive ordinary differential systems: Higher order methods. BIT 41 (2001) 599-628. | Zbl 1001.65093

[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. | Zbl 1105.65083