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Least squares finite element simulation of transonic flows. (English) Zbl 0601.76071
Finite difference approximation of transonic flow problems is a well- developed and largely successful approach. Nevertheless, there is still a real need to develop finite element methods for applications arising from fluid-structure interactions and problems with complicated boundaries. In this paper we introduce a least squares based finite element scheme. It is shown that, if suitably formulated, such an approach can lead to physically meaningful results. Bottlenecks that arise from such schemes are also discussed.

##### MSC:
 76H05 Transonic flows 76M99 Basic methods in fluid mechanics
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##### References:
 [1] Fix, G.J.; Gunzburger, M.D.; Nicolaides, R.A., NASA-ICASE report no. 77-18, Comput. math. appl., 5, 87-98, (1979), revised version published in [2] Fix, G.J.; Gurtin, M., On patched variational methods, Numer. math., 28, 259-271, (1977) · Zbl 0348.65097 [3] Fix, G.J.; Gunzburger, M.D., On least squares approximation to indefinite problems of the mixed types, Internat. J. numer. methods engrg., 12, 453-470, (1978) · Zbl 0378.76046 [4] Cox, C.L.; Fix, G.J.; Gunzburger, M.D., A least squares finite element scheme for transonic flow around harmonically oscillating wings, J. comp. phys., 51, 3, 387-403, (1983) · Zbl 0521.76053 [5] Chen, T.-F., On finite element approximations to compressible flow problems, () [6] T.F. Chen, Least squares approximation to compressible flow problems, Comput. Math. Appl., submitted for publication. [7] Cox, C.L.; Fix, G.J., On the accuracy of least squares methods in the presence of corner singularities, Comput. math. appl., 10, 6, 463-476, (1984) · Zbl 0573.65081 [8] Fix, G.J.; Rose, M.E., A comparative study of finite element and finite difference methods for Cauchy-Riemann type equations, SIAM J. numer. anal., 22, 2, 250-260, (1985) · Zbl 0569.65077 [9] G.J. Fix, Least squares approximation of hyperbolic systems, SIAM J. Numer. Anal., submitted for publication. [10] Lax, P.D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, () · Zbl 0108.28203 [11] Osher, S.; Hafez, M.; Whitlow, W., Entropy conditions satisfying approximations for the full potential equation of transonic flow, Math. comp., 44, 1-29, (1969), (1985) · Zbl 0572.65076 [12] Eberle, A., Eine methode finiter elements berechnung der transsonischen potential—strömung un profile, MBB berech nr. UFE 1352(0), (1977) [13] Hafez, M.M.; Murman, E.M.; South, J.C., Artificial compressibility methods for numerical solution of transonic full potential equation, AIAA paper 78-1148, (1978), Seattle, Washington [14] Hafez, M.; Whitlow, W.; Osher, S., Improved finite difference schemes for transonic potential calculations, AIAA paper 84-0092, (1984), Reno, Nevada · Zbl 0635.76054 [15] Imai, I., On the flow of a compressible fluid past a circular cylinder, II, Proc. phys. math. soc. Japan, 23, 180-193, (1941) · JFM 67.0857.01 [16] Hasimoto, Z., On the subsonic flow of a compressible fluid past a circular cylinder between two parallel walls, Proc. phys. math. soc. Japan, 25, 563-574, (1943) · Zbl 0063.01958 [17] Jameson, A., Numerical solutions of nonlinear partial differential equations of mixed type, (), 275-320 [18] Bristeau, M.O.; Glowinski, R.; Périaux, J.; Perrier, P.; Pironneau, O.; Poirier, G., A finite element method for the numerical simulation of transonic potential flows, () · Zbl 0449.76002 [19] Pelz, R.; Jameson, A., Transonic flow calculations using triangular finite elements, Aiaa j., 23, 4, 569-576, (1985) · Zbl 0565.76059 [20] W.G. Habashi and M.M. Hafez, Finite element solution of transonic flow problems, AIAA Paper 81-1472. [21] Deconinck, H.; Hirsch, C., Finite element methods for transonic flow calculations, (), 66-77, Braunschweig, Friedr.
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