{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,13]],"date-time":"2023-09-13T15:26:25Z","timestamp":1694618785660},"reference-count":57,"publisher":"American Mathematical Society (AMS)","issue":"220","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"
A sensitive issue in numerical calculations for exterior flow problems, e.g. around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two\u2013dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl\u2013Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element\/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.<\/p>","DOI":"10.1090\/s0025-5718-97-00878-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:28Z","timestamp":1027721668000},"page":"1407-1440","source":"Crossref","is-referenced-by-count":6,"title":["Analysis of a FEM\/BEM coupling method for transonic flow computations"],"prefix":"10.1090","volume":"66","author":[{"given":"H.","family":"Berger","sequence":"first","affiliation":[]},{"given":"G.","family":"Warnecke","sequence":"additional","affiliation":[]},{"given":"W.","family":"Wendland","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1997]]},"reference":[{"key":"1","unstructured":"AGARD Advisory Report No. 211, Test Cases for Inviscid Flow Field Methods, Report of Fluid Dynamics Panel Working Group 07, 1985."},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/BF01389582","article-title":"The convergence of spline collocation for strongly elliptic equations on curves","volume":"47","author":"Arnold, Douglas N.","year":"1985","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"3","first-page":"1","article-title":"Survey lectures on the mathematical foundations of the finite element method","author":"Babu\u0161ka, Ivo","year":"1972"},{"key":"4","unstructured":"E. B. Becker, G. F. Carey and J. T. Oden, Finite Elements, Vol. VI: Fluid Mechanics, Prentice\u2013Hall Inc., Englewood Cliffs, New Jersey, 1984."},{"key":"5","unstructured":"H. Berger, \"Finite\u2013Element\u2013Approximationen f\u00fcr transsonische Str\u00f6mungen\", Doctoral Thesis, Universit\u00e4t Stuttgart, Germany, 1989."},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1007\/BF01396647","article-title":"A convergent finite element formulation for transonic flow","volume":"56","author":"Berger, Harald","year":"1989","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"204","key":"7","doi-asserted-by":"publisher","first-page":"493","DOI":"10.2307\/2153238","article-title":"Analysis of the finite element variational crimes in the numerical approximation of transonic flow","volume":"61","author":"Berger, Harald","year":"1993","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","unstructured":"H. Berger, G. Warnecke and W. Wendland, \"Finite Element\u2013Berechnungen f\u00fcr transsonische Str\u00f6mungen unter Ber\u00fccksichtigung verschiedener Fernfeldrandbedingungen.\" In: Str\u00f6mungen mit Abl\u00f6sungen, DGLR\u2013Bericht 88\u201305, Bonn (1988) 233-242."},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1002\/num.1690060103","article-title":"Finite elements for transonic potential flows","volume":"6","author":"Berger, H.","year":"1990","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"http:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"key":"10","first-page":"323","article-title":"Coupling of FEM and BEM for transonic flows","author":"Berger, H.","year":"1994"},{"key":"11","series-title":"Surveys in Applied Mathematics, Vol. 3","volume-title":"Mathematical aspects of subsonic and transonic gas dynamics","author":"Bers, Lipman","year":"1958"},{"key":"12","doi-asserted-by":"publisher","first-page":"611","DOI":"10.2307\/1968946","article-title":"On Frobeniusean algebras. I","volume":"40","author":"Nakayama, Tadasi","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"http:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"13","first-page":"9","article-title":"Subsonic flow of compressible fluid","author":"Bojarski, B.","year":"1967"},{"key":"14","first-page":"A129--A132","article-title":"Une nouvelle m\u00e9thode pour l\u2019\u00e9tude d\u2019\u00e9coulements stationnaires","volume":"276","author":"Br\u00e9zis, Ha\u00efm","year":"1973","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"http:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"key":"15","first-page":"203","article-title":"Application of optimal control and finite element methods to the calculation of transonic flows and incompressible viscous flows","author":"Bristeau, M. O.","year":"1980"},{"key":"16","unstructured":"M. O. Bristeau, R. Glowinski, J. Periaux, P. Perrier, O. Pironneau and G. Poirier, \"Transonic flow simulations by finite elements and least squares methods,\" in Finite Elements in Fluids IV, R. H. Gallagher, G. Carey, J. T. Oden, and O. C. Zienkiewicz (eds.), Wiley, Chichester, 453-482 (1982)."},{"issue":"1-3","key":"17","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/0045-7825(85)90039-8","article-title":"On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. II. Application to transonic flow simulations","volume":"51","author":"Bristeau, M. O.","year":"1985","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"http:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"18","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"http:\/\/id.crossref.org\/isbn\/0444850287"},{"issue":"2","key":"19","first-page":"257","article-title":"Une nouvelle approche dans le plan physique pour le calcul d\u2019\u00e9coulements subcritiques et stationnaires autour d\u2019un profil portant","volume":"16","author":"Ciavaldini, J.-F.","year":"1977","journal-title":"J. M\\'{e}canique","ISSN":"http:\/\/id.crossref.org\/issn\/0021-7832","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF00250842","article-title":"Existence and regularity of stream functions for subsonic flows past profiles with a sharp trailing edge","volume":"93","author":"Ciavaldini, J.-F.","year":"1986","journal-title":"Arch. Rational Mech. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"21","unstructured":"C. Coclici and W.L. Wendland, \"On the treatment of the Kutta\u2013Joukowski condition in transonic flow computations\", in preparation (Preprint 95\u201314 MIA Univ. Stuttgart)."},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1137\/0519043","article-title":"Boundary integral operators on Lipschitz domains: elementary results","volume":"19","author":"Costabel, Martin","year":"1988","journal-title":"SIAM J. Math. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"23","volume-title":"Methods of mathematical physics. Vol. II: Partial differential equations","author":"Courant, R.","year":"1962"},{"key":"24","doi-asserted-by":"publisher","first-page":"783","DOI":"10.2307\/2371336","article-title":"Ring homomorphisms which are also lattice homomorphisms","volume":"61","author":"Ward, Morgan","year":"1939","journal-title":"Amer. J. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"issue":"178","key":"25","doi-asserted-by":"publisher","first-page":"521","DOI":"10.2307\/2007825","article-title":"The stability in \ud835\udc3f_{\ud835\udc5d} and \ud835\udc4a\u00b9_{\ud835\udc5d} of the \ud835\udc3f\u2082-projection onto finite element function spaces","volume":"48","author":"Crouzeix, M.","year":"1987","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"26","doi-asserted-by":"crossref","unstructured":"M. Dauge and M. Pogu, \"Existence et r\u00e9gularit\u00e9 de la function potentiel pour des \u00e9coulements subcritiques s\u2019\u00e9tablissant autour d\u2019un corps \u00e0 singularit\u00e9 conique,\" Ann. Fac. des Sci. de Toulouse IX, 213-242 (1988).","DOI":"10.5802\/afst.658"},{"issue":"154","key":"27","doi-asserted-by":"publisher","first-page":"405","DOI":"10.2307\/2007650","article-title":"A method of calculation of lifting flows around 2-dimensional corner-shaped bodies","volume":"36","author":"Djaoua, M.","year":"1981","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"28","series-title":"Die Grundlehren der mathematischen Wissenschaften, Band 153","volume-title":"Geometric measure theory","author":"Federer, Herbert","year":"1969"},{"issue":"2","key":"29","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/0377-0427(92)90008-L","article-title":"Finite-element solution of flow problems with trailing conditions","volume":"44","author":"Feistauer, Miloslav","year":"1992","journal-title":"J. Comput. Appl. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"30","first-page":"47","article-title":"Analysis and numerical realization of coupled BEM and FEM for nonlinear exterior problems","author":"Feistauer, M.","year":"1995"},{"issue":"4","key":"31","doi-asserted-by":"publisher","first-page":"305","DOI":"10.4171\/ZAA\/155","article-title":"On the solvability of transonic potential flow problems","volume":"4","author":"Feistauer, M.","year":"1985","journal-title":"Z. Anal. Anwendungen","ISSN":"http:\/\/id.crossref.org\/issn\/0232-2064","issn-type":"print"},{"issue":"4","key":"32","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1007\/BF01396664","article-title":"Finite element solution of nonlinear elliptic problems","volume":"50","author":"Feistauer, Miloslav","year":"1987","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"33","series-title":"Prentice-Hall Series in Automatic Computation","volume-title":"An analysis of the finite element method","author":"Strang, Gilbert","year":"1973"},{"key":"34","unstructured":"F. I. Frankl and M. Keldysh, \"Die \u00e4u\u00dfere Neumannsche Aufgabe f\u00fcr nichtlineare elliptische Differentialgleichungen mit Anwendungen auf die Theorie der Fl\u00fcgel im kompressiblen Gas,\" Bull. Acad. Sci. URSS 12, 561-607 (1934)."},{"key":"35","doi-asserted-by":"crossref","unstructured":"G. N. Gatica and G. C. Hsiao, \"The coupling of boundary element and finite element methods for a nonlinear exterior boundary value problem,\" Numer. Math. 61, 171-214 (1992).","DOI":"10.1007\/BF01385504"},{"key":"36","unstructured":"R. Glowinski and O. Pironneau, \"On the computation of transonic flows,\" in Funct. Anal. and Num. Anal., H. Fujita (ed.), Jap. Soc. Prom. Sci., Tokyo\u2013Kyoto, 143-173 (1978)."},{"issue":"4","key":"37","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/BF01385706","article-title":"A shock indicator for adaptive transonic flow computations","volume":"66","author":"G\u00f6hner, U.","year":"1994","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"3","key":"38","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/0022-247X(77)90186-X","article-title":"A finite element method for some integral equations of the first kind","volume":"58","author":"Hsiao, George C.","year":"1977","journal-title":"J. Math. Anal. Appl.","ISSN":"http:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"152","key":"39","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.2307\/2006375","article-title":"On the coupling of boundary integral and finite element methods","volume":"35","author":"Johnson, Claes","year":"1980","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6-7","key":"40","doi-asserted-by":"publisher","first-page":"1197","DOI":"10.1080\/03605309108820795","article-title":"The existence of viscous profiles and admissibility for transonic shocks","volume":"16","author":"Keyfitz, Barbara Lee","year":"1991","journal-title":"Comm. Partial Differential Equations","ISSN":"http:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"key":"41","unstructured":"N. Kroll and R. K. Jain, Solution of Two\u2013Dimensional Euler Equation Experience with a Finite Volume Code, DFVLR\u2013Forschungsbericht 87\u201341, DFVLR Institut f\u00fcr Entwurfsaerodynamik, Braunschweig 1987."},{"key":"42","isbn-type":"print","volume-title":"Lehrbuch der theoretischen Physik (``Landau-Lifschitz''). Band IV","author":"Berestetzki, W. B.","year":"1991","ISBN":"http:\/\/id.crossref.org\/isbn\/3055000684","edition":"7"},{"key":"43","unstructured":"M. N. LeRoux, R\u00e9solution num\u00e9rique du probl\u00e8me du potential dans le plan par une m\u00e9thode variationelle d\u2019\u00e9l\u00e9ments finis, Doctoral Thesis, University Rennes, 1974."},{"issue":"5","key":"44","doi-asserted-by":"publisher","first-page":"985","DOI":"10.1137\/0724064","article-title":"Convergence of finite elements for transonic potential flows","volume":"24","author":"Mandel, Jan","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"45","doi-asserted-by":"publisher","first-page":"23","DOI":"10.2307\/1989990","article-title":"Steinitz field towers for modular fields","volume":"46","author":"MacLane, Saunders","year":"1939","journal-title":"Trans. Amer. Math. Soc.","ISSN":"http:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"46","doi-asserted-by":"publisher","first-page":"771","DOI":"10.2307\/2371335","article-title":"Infinite number fields with Noether ideal theories","volume":"61","author":"MacLane, Saunders","year":"1939","journal-title":"Amer. J. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"47","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1002\/cpa.3160170308","article-title":"Non-existence of transonic flow past a profile","volume":"17","author":"Morawetz, Cathleen S.","year":"1964","journal-title":"Comm. Pure Appl. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"6","key":"48","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1002\/cpa.3160380610","article-title":"On a weak solution for a transonic flow problem","volume":"38","author":"Morawetz, Cathleen S.","year":"1985","journal-title":"Comm. Pure Appl. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"3","key":"49","first-page":"309","article-title":"L\u2019injection du c\u00f4ne positif de \ud835\udc3b\u207b\u00b9 dans \ud835\udc4a^{-1,\ud835\udc5e} est compacte pour tout \ud835\udc5e<2","volume":"60","author":"Murat, Fran\u00e7ois","year":"1981","journal-title":"J. Math. Pures Appl. (9)","ISSN":"http:\/\/id.crossref.org\/issn\/0021-7824","issn-type":"print"},{"key":"50","unstructured":"J. Ne\u010das, Compacite par Entropie et Ecoulements de Fluides, Lecture Notes Universit\u00e9 de Charles et E.N-S., Masson, Paris (1989)."},{"issue":"4","key":"51","first-page":"317","article-title":"Sur une classe de probl\u00e8mes monotones pos\u00e9s dans des ouverts non born\u00e9s, m\u00e9thode d\u2019approche, algorithmes de r\u00e9solution et application","volume":"31","author":"Pogu, Marc","year":"1986","journal-title":"Rev. Roumaine Math. Pures Appl.","ISSN":"http:\/\/id.crossref.org\/issn\/0035-3965","issn-type":"print"},{"issue":"158","key":"52","doi-asserted-by":"publisher","first-page":"437","DOI":"10.2307\/2007280","article-title":"Some optimal error estimates for piecewise linear finite element approximations","volume":"38","author":"Rannacher, Rolf","year":"1982","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"53","series-title":"Notes on Numerical Fluid Mechanics","isbn-type":"print","volume-title":"Numerical methods for the computation of inviscid transonic flows with shock waves","volume":"3","year":"1981","ISBN":"http:\/\/id.crossref.org\/isbn\/3528080779"},{"key":"54","doi-asserted-by":"crossref","unstructured":"J. Smoller, Shock Waves and Reaction\u2013Diffusion Equations, Springer\u2013Verlag, Berlin (1983).","DOI":"10.1007\/978-1-4684-0152-3"},{"key":"55","unstructured":"I.L. Sofronov and S.V. Tscyncov, \"Application of the linear potential flow model to the artificial boundary conditions construction for the Euler equations.\" (Preprint No. 41 Nat. Center of Math. Simulation, Russian Acad. Sci. Moscow, 1991.) To appear."},{"key":"56","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1007\/978-1-4613-9049-7_19","article-title":"Admissibility of solutions to the Riemann problem for systems of mixed type\u2014transonic small disturbance theory","author":"Warnecke, Gerald","year":"1990"},{"key":"57","unstructured":"J. Zierep, Theoretische Gasdynamik, Braun, Karlsruhe (1976)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1997-66-220\/S0025-5718-97-00878-8\/S0025-5718-97-00878-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-220\/S0025-5718-97-00878-8\/S0025-5718-97-00878-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T23:32:42Z","timestamp":1635895962000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1997-66-220\/S0025-5718-97-00878-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997]]},"references-count":57,"journal-issue":{"issue":"220","published-print":{"date-parts":[[1997,10]]}},"alternative-id":["S0025-5718-97-00878-8"],"URL":"http:\/\/dx.doi.org\/10.1090\/s0025-5718-97-00878-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":["Applied Mathematics","Computational Mathematics","Algebra and Number Theory"],"published":{"date-parts":[[1997]]}}}