Anisotropic mesh adaptation for numerical solution of boundary value problems. (English) Zbl 1060.65125

Dolejší, Vít; Felcman, Jiří

Numer. Methods Partial Differ. Equations 20, No. 4, 576-608 (2004).

The authors have a great experience of mesh adaptation techniques from the mathematical analysis point of view but also from numerical implementation practice. Here, they obtain a significant improvement of numerical results valid for a wide range of 2D problems of physics and engineering described by partial differential equations discretized on triangular grids. In this way, they combine the anisotropic mesh adaptation algorithm with a new smoothing procedure.
This new approach is really carefully analyzed and its efficiency is illustrated upon several problems: a singular diffusion problem, a Poisson problem with steep gradients and finally on the geometrically challenging industrial problem of an inviscid transonic flow around NACA0012 profile. Beyond these so attractive contents, the presentation is clear and very nice.

Reviewer: Michel Bernadou (Paris La Defense)

MSC:

65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76H05	Transonic flows
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
76M10	Finite element methods applied to problems in fluid mechanics

Keywords:

anisotropy mesh adaptation algorithm; boundary value problem; interpolation error function; edge optimisation; compressible flow; singular diffusion problem; Poisson problem; inviscid transonic flow

References:

[1]	and Anisotropic mesh adaptation: theory, validation and applications, and editors, Computational fluid dynamics ’96, Paris, 1996. Wiley, Chichester, 1996, pp 174-180.
[2]	and Anisotropic mesh optimization: Towards a solver-independent and mesh-independent CFD. Compressible fluid dynamics, VKI Lecture Series 1996-06. von Karman Institute for Fluid Dynamics, Concordia University, August 1996.
[3]	and Anisotropic adaptive mesch generation in two dimensions for CFD, and editors, Computational fluid dynamics ’96, Paris, Wiley, Chichester, 1996, pp 181-186.
[4]	Simpson, Appl Numer Math 14 pp 183– (1994)
[5]	Dolej??, Computing and Visualisation in Science 1 pp 165– (1998)
[6]	Numerical solver for compressible flows?Anisotropic mesh adaption, and editors, Numerical modelling in continuum mechanics, Matfyzpress, Praha, 1997, pp 246-253.
[7]	Adaptive methods for the numerical solution of the compressible Navier-Stokes equations, and editors, Computational Fluid Dynamics ’98, Vol. 1, ECCOMAS, John Wiley & Sons, New York, 1998, pp 393-397.
[8]	Anisotropic finite elements: local estimates and applications, Teubner, Stuttgart, 1999. · Zbl 0917.65090
[9]	and Finite element methods with anisotropic meshes near edges. and editors, Finite element methods for three-dimensional problems, Vol. 15 of GAKUTO Internat Series Math Sci Appl, Tokyo, 2001. University of Jyv?skyl?, Gakkotosho Co., Ltd., 2001, pp 1-8.
[10]	Mathematical methods in fluid dynamics, Longman Scientific & Technical, Harlow, 1993. · Zbl 0819.76001
[11]	Numerical schemes for conservation laws, Wiley Teubner, Stuttgart, 1997. · Zbl 0872.76001
[12]	Kr?ner, SIAM J Numer Anal 31 pp 324– (1994)
[13]	The finite elements method for elliptic problems, North-Holland, Amsterdam, 1979.
[14]	Ph., Appl Math 43 pp 263– (1998)
[15]	and Finite volume methods on unstructured meshes for compressible flows, and editors, Finite volumes for complex applications (problems and perspectives), Hermes, Rouen, 1996, pp. 667-674.
[16]	Navier-Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam, 1977. · Zbl 0383.35057
[17]	An introduction to the discontinuous Galerkin method for convection-dominated problems, et al., editors, Advanced numerical approximation of nonlinear hyperbolic equations, Lecture Notes in Mathematics, 1697, Springer, Berlin, 1998, pp 151-268.
[18]	D’Azevedo, SIAM J Sci Statist Comput 10 pp 1063– (1989)
[19]	D’Azevedo, Numer Math 59 pp 321– (1991)
[20]	Buscaglia, Int J Numer Methods Eng 40 pp 4119– (1977)
[21]	Felcman, Eng Mech 5 pp 327– (1998)
[22]	and Refinement algorithm and data structures for regular local mesh refinement, et al., editors, Scientific computing, Vol. 44, IMACS North-Holland, Amsterdam, 1983, pp 3-17.
[23]	and On an adaptive method for heat conduction problems with boundary layers. Preprint Hamburger Beitr?ge zur Angenwandten Mathematik, Comput Visual Sci 2003. Submitted.
[24]	Felcman, ZAMM 76 pp 301– (1996)
[25]	and Adaptive finite volume method for the numerical solution of the compressible Euler equations, and editors, Computational fluid dynamics ’94, John Wiley and Sons, Stuttgart, 1994, pp 894-901.
[26]	and Flow computations using combined finite volume-finite element method, Preprint, Charles University, Prague, 2001.