×

zbMATH — the first resource for mathematics

Fourier series and integral equation method for the exterior Stokes problem. (English) Zbl 1161.65087
One of the objectives of this paper is to get more insights on the theoretical and constructive treatments of edge singularities of the three-dimensional Stokes system. The author investigates the following Stokes system of partial differential equations \[ \begin{aligned} -\nu\Delta U + \nabla p & = 0 \quad \text{ in } Q',\\ \nabla \cdot U & = 0 \quad \text{ in } Q', \end{aligned} \] with the boundary condition \[ U = F\text{ on } \Gamma\times[0,2\pi], \] and the periodic condition \[ U(\cdot,0) = U(\cdot,2\pi)\quad \text{ on }\mathbb{R}^{2}. \] Here the unbounded domain \(Q' := \Omega'\times(0, 2\pi)\) is the exterior to the prism \(Q:=\Omega\times(0, 2\pi)\), where \(\Omega := \mathbb{R}^{2}\setminus\overline{\Omega}'\) is a bounded simply connected domain with polygonal boundary \(\Gamma\equiv\partial\,\Omega\).
First, the author proves the well-posedness of the coupled exterior-interior problem for the Stokes operator in suitably defined weighted Sobolev spaces. Second, by means of Fourier series in the \(z\)-variable, the problem is reduced to finding Fourier coefficients via boundary integral equations of hydrodynamic potential theory. The global regularity of the solutions of the integral equations is investigated in appropriate weighted Sobolev spaces of traces. Finally, the last part of the paper is devoted to an optimal convergent boundary element method for the integral equations. This provides optimal convergent semi- and fully-discrete spectral methods of Fourier-Galerkin type.
MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
76M15 Boundary element methods applied to problems in fluid mechanics
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] and , editors, Handbook of mathematical functions with formulas, graphs and mathematical tables, 9th ed., Dover, New York, 1972. · Zbl 0543.33001
[2] Babuska, Computing 6 pp 264– (1970)
[3] Bernardi, Calcolo 18 pp 255– (1981)
[4] and , Spectral methods, Handbook of numerical analysis, Vol. V, and , editors, Techniques of scientifc computing, Part 2, North-Holland, Amsterdam, 1997, pp. 209–485.
[5] Bernardi, Comput Methods Appl Mech Eng 175 pp 267– (1999)
[6] Solution of some problems of vector analysis associated with the operators div and grad (in Russian), In: Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, pp. 5–40, 149, Trudy Sem SL Soboleva, 1, 1980. Akad. Nauk SSSR Sibirsk Otdel. Inst. Mat. Novosibirsk MR82m:26014.
[7] Bourlard, SIAM J Numer Anal 28 pp 728– (1991)
[8] , , and , Spectral methods in fluid dynamics, Springer-Verlag, Berlin, 1988. · Zbl 0658.76001
[9] Costabel, SIAM J Math Anal 19 pp 613– (1988)
[10] and , Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximations, Mathematical models and methods in mechanics 1981, and , editors, Banach Center Publications no. 15, PWN Polish Publishers, Warsaw, 1985, pp. 175–251.
[11] Costabel, Ann Scuola Norm Sup Pisa Cl Sci 10 pp 197– (1983)
[12] Costabel, J Reine Angewandte Math 372 pp 39– (1986)
[13] Elliptic boundary value problems on corner domains, Lecture Notes in Math 1341, Springer-Verlag, Berlin, 1988. · Zbl 0668.35001
[14] and , Mathematical analysis and numerical methods for science and technology: functional and variational methods, Vol. 2, Springer-Verlag, Berlin, 1988.
[15] and , Mathematical analysis and numerical methods for science and technology: physical origins and classical methods, Vol. 1, Springer-Verlag, Berlin, 1990.
[16] and , Mathematical analysis and numerical methods for science and technology: integral equations and numerical methods, Vol. 4, Springer-Verlag, Berlin, 1990.
[17] Deuring, J Math Fluid Mech 2 pp 353– (2000)
[18] Feistauer, Math Models Methods Appl Sci 4 pp 657– (1998)
[19] Fredholm, Jrnl öfver Kongl Vet-Akad Förh 57 pp 39– (1900)
[20] An introduction to the mathematical theory of the Navier–Stokes equations, Vol. 1: linearised steady problems, Springer-Verlag, Berlin, 1994.
[21] and , Finite element methods for Navier–Stokes equations: theory and algorithms, Springer-Verlag, Berlin, 1986. · Zbl 0585.65077
[22] Elliptic problems in nonsmooth domains, Pitman, London, 1985. · Zbl 0695.35060
[23] Singularities in boundary value problems, Masson, Paris and Springer-Verlag, Berlin, 1992.
[24] Guirguis, Comm Partial Differential Eq 11 pp 567– (1986)
[25] La théorie du potentiel et ses applications aux problèmes fondamentaux de la physique mathématique, Gauthier-Villars, Paris, 1934.
[26] Heinrich, SIAM J Numer Anal 33 pp 1885– (1996)
[27] Heinrich, Math Nachr 186 pp 147– (1997)
[28] Heinrich, Adv Math Sci Appl 10 pp 571– (2000)
[29] Heywood, Acta Math 136 pp 61– (1976)
[30] Hildebrandt, Comm Pure Appl Math 17 pp 369– (1964)
[31] Hsiao, Math Methods Appl Sci 6 pp 280– (1984)
[32] Hsiao, SIAM Rev 15 pp 687– (1973)
[33] Hsiao, J Math Anal Appl 58 pp 449– (1977)
[34] and , Introduction to fluid mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1980.
[35] Kondratiev, Trans Moscow Math Soc 16 pp 227– (1967)
[36] The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1969.
[37] le Roux, C R Acad Sci Paris Sér A 278 pp 541– (1974)
[38] Lubuma, J Comput Appl Math 61 pp 13– (1995)
[39] Lubuma, Proc R Soc Edinb 130A pp 107– (2000)
[40] Lubuma, Numer Funct Anal Optimiz 21 pp 743– (2000)
[41] Maz’ya, Trans Moscow Math Soc 37 pp 49– (1980)
[42] Maz’ya, Math Nachr 138 pp 27– (1988)
[43] Mercier, RAIRO Anal Numér 16 pp 405– (1982)
[44] Wavelet and fast numerical algorithms, Handbook of numerical analysis, Vol. V, and , editors, Techniques of scientifc computing. Part 2, North-Holland, Amsterdam, 1997, pp. 639–713.
[45] Mathematical physics: an advanced course, North-Holland, Amsterdam, 1970.
[46] Theoretical hydrodynamics, Macmillan & Co LTD, London, 1968. · Zbl 0164.55802
[47] Approximation des équations intégrales en mécanique et en physique, Lecture Notes, Ecole Polytechnique, Palaiseau, 1977.
[48] Polygonal interface problems, Peter Lang, Berlin, 1993.
[49] Théorie des distributions, Hermann, Paris, 1966.
[50] Sequeira, Math Methods Appl Sci 5 pp 356– (1983)
[51] editor, Navier–Stokes equations and related nonlinear problems, Proceedings of the third international conference held in Funchal, 21–27 May 1994, Plenum Press, New York, 1995.
[52] Solonnikov, Pacific J Math 93 pp 443– (1981) · Zbl 0413.35062
[53] Navier–Stokes equations: theory and numerical analysis (revised ed.), North-Holland, Amsterdam, 1979. · JFM 07.0137.01
[54] On Galerkin collocation methods for integral equations of elliptic boundary value problems, Numerical treatment of integral equations, ISNM 53, , and , editors, Birkhauser Verlag, Basel, 1980, pp. 244–275.
[55] Fluid mechanics, 2nd ed., McGraw-Hill, NewYork, 1986.
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.