# zbMATH — the first resource for mathematics

Analysis of the finite element variational crimes in the numerical approximation of transonic flow. (English) Zbl 0786.76051
A convergence analysis of a piecewise linear finite element method for transonic potential flow is given. The full potential equation is considered in a bounded domain with mixed Dirichlet-Neumann boundary conditions. A new version of entropy compactification of transonic flow and the theory of variational crimes enable the authors to develop a detailed theory of a conforming piecewise linear FEM in which the occuring integrals are treated by means of quadrature rules. In the convergence proof of the authors assume the existence of a solution of the physical problem.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76H05 Transonic flows 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] B. Arlinger, Axisymmetric transonic flow computations using a multigrid method, Lecture Notes in Phys., vol. 141, Springer-Verlag, Berlin and New York, 1981, pp. 55-60. [2] H. Berger, Finite-Element-Approximationen für transonische Strömungen, Dr. rer. nat. Dissertation, Universität Stuttgart, 1989. · Zbl 0734.76033 [3] Harald Berger, A convergent finite element formulation for transonic flow, Numer. Math. 56 (1989), no. 5, 425 – 447. · Zbl 0681.76066 [4] H. Berger, G. Warnecke, and W. Wendland, Finite elements for transonic potential flows, Numer. Methods Partial Differential Equations 6 (1990), no. 1, 17 – 42. · Zbl 0692.76061 [5] M. O. Bristeau, O. Pironneau, R. Glowinski, J. Periaux, P. Perrier, and G. Poirier, Application of optimal control and finite element methods to the calculation of transonic flows and incompressible viscous flows, Numerical methods in applied fluid dynamics (Reading, 1978) Academic Press, London-New York, 1980, pp. 203 – 312. · Zbl 0449.76002 [6] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058 [7] P. G. Ciarlet and P.-A. Raviart, General Lagrange and Hermite interpolation in \?$$^{n}$$ with applications to finite element methods, Arch. Rational Mech. Anal. 46 (1972), 177 – 199. · Zbl 0243.41004 [8] J. D. Cole and E. M. Murman, Calculation of plane steady transonic flows, AIAA J. 9 (1971), 199-206. · Zbl 0249.76033 [9] H. Deconinck, The numerical computation of transonic potential flows, Ph.D. thesis, Vriji Universiteit, Brussel, 1983. [10] H. Deconinck and C. Hirsch, Transonic flow calculations with higher finite elements, Lecture Notes in Phys., vol. 141, Springer-Verlag, Berlin and New York, 1981, pp. 138-143. [11] Ronald J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), no. 1, 1 – 30. · Zbl 0533.76071 [12] P. Doktor, On the density of smooth functions in certain subspaces of Sobolev space, Comment. Math. Univ. Carolinae 14 (1973), 609 – 622. · Zbl 0268.46036 [13] Miloslav Feistauer, On irrotational flows through cascades of profiles in a layer of variable thickness, Apl. Mat. 29 (1984), no. 6, 423 – 458 (English, with Russian and Czech summaries). · Zbl 0598.76061 [14] Miloslav Feistauer, On the finite element approximation of a cascade flow problem, Numer. Math. 50 (1987), no. 6, 655 – 684. · Zbl 0646.76085 [15] Miloslav Feistauer, Jan Mandel, and Jindřich Nečas, Entropy regularization of the transonic potential flow problem, Comment. Math. Univ. Carolin. 25 (1984), no. 3, 431 – 443. · Zbl 0563.35006 [16] M. Feistauer and J. Nečas, On the solvability of transonic potential flow problems, Z. Anal. Anwendungen 4 (1985), no. 4, 305 – 329 (English, with German and Russian summaries). · Zbl 0621.76069 [17] Miloslav Feistauer and Jindřich Nečas, Viscosity method in a transonic flow, Comm. Partial Differential Equations 13 (1988), no. 7, 775 – 812. · Zbl 0657.35090 [18] Miloslav Feistauer and Jindřich Nečas, Remarks on the solvability of transonic flow problems, Manuscripta Math. 61 (1988), no. 4, 417 – 428. · Zbl 0661.76054 [19] Miloslav Feistauer and Veronika Sobotíková, Finite element approximation of nonlinear elliptic problems with discontinuous coefficients, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 4, 457 – 500 (English, with French summary). · Zbl 0712.65097 [20] Miloslav Feistauer, On the finite element approximation of a cascade flow problem, Numer. Math. 50 (1987), no. 6, 655 – 684. · Zbl 0646.76085 [21] Miloslav Feistauer and Alexander Ženíšek, Compactness method in the finite element theory of nonlinear elliptic problems, Numer. Math. 52 (1988), no. 2, 147 – 163. · Zbl 0642.65075 [22] R. Glowinski, Lectures on numerical methods for nonlinear variational problems, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 65, Tata Institute of Fundamental Research, Bombay; sh Springer-Verlag, Berlin-New York, 1980. Notes by M. G. Vijayasundaram and M. Adimurthi. · Zbl 0456.65035 [23] -, Numerical methods for nonlinear variational problems, Springer-Verlag, Berlin and New York, 1984. [24] R. Glowinski and O. Pironneau, On the computation of transonic flows, Functional Analysis and Numerical Analysis , Japan Soc. for the Promotion of Science, 1978, pp. 143-173. [25] A. Jameson, Acceleration of transonic potential flow calculations on arbitrary meshes by the multiple grid method, AIAA Paper 79-1458 (1979). [26] A. Kufner, O. John, and S. Fučík, Function spaces, Academia, Praha, 1977. [27] Jan Mandel and Jindřich Nečas, Convergence of finite elements for transonic potential flows, SIAM J. Numer. Anal. 24 (1987), no. 5, 985 – 996. · Zbl 0635.76052 [28] R. Mani and A. J. Acosta, Quasi two-dimensional flows through a cascade, Trans. ASME Ser. A, J. Engrg. for Power 90 (1968), No. 2. [29] Cathleen S. Morawetz, On a weak solution for a transonic flow problem, Comm. Pure Appl. Math. 38 (1985), no. 6, 797 – 817. · Zbl 0615.76070 [30] François Murat, L’injection du cône positif de \?$$^{-}$$\textonesuperior dans \?^{-1,\?} est compacte pour tout \?<2, J. Math. Pures Appl. (9) 60 (1981), no. 3, 309 – 322 (French, with English summary). · Zbl 0471.46020 [31] J. Nečas, Les méthodes directes en théorie des équations Elliptiques, Masson, Paris, 1967. · Zbl 1225.35003 [32] -, Ecoulements de fluide: compacité par entropie, Masson, Paris, 1988. [33] Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437 – 445. · Zbl 0483.65007 [34] Gilbert Strang, Variational crimes in the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 689 – 710. · Zbl 0264.65068 [35] Alexander Ženíšek, Discrete forms of Friedrichs’ inequalities in the finite element method, RAIRO Anal. Numér. 15 (1981), no. 3, 265 – 286 (English, with French summary). · Zbl 0475.65072 [36] Miloš Zlámal, Curved elements in the finite element method. I, SIAM J. Numer. Anal. 10 (1973), 229 – 240. · Zbl 0285.65067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.