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Geometric Singularities of the Stokes Problem

DOI: 10.1155/2014/491326

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When the domain is a polygon of , the solution of a partial differential equation is written as a sum of a regular part and a linear combination of singular functions. The purpose of this paper is to present explicitly the singular functions of Stokes problem. We prove the Kondratiev method in the case of the crack. We finish by giving some regularity results. 1. Introduction The regularity of the solution of a partial differential equation depends on the geometry of the domain even when the data is smooth. Indeed, for each corner of the polygonal domain a countable family of singular functions can be defined, which depends only on the geometry of the domain. Then the solution of the equation can be written as the sum of a finite number of singular functions multiplied by appropriate coefficients and of a much more regular part. We refer to Kondratiev [1] and Grisvard [2] for their description. The purpose of our work is to study the singularities of the Stokes equation and the behavior of the solution in the neighborhood of a corner of a polygonal domain of . We are interested to nonconvex domains; we assume that there exists an angle equal either to or to (case of the crack). Handling the singular function is local process, so that there is no restriction to suppose that the nonconvex corner is unique; see Dauge [3]. We deduce the singular function of the velocity from those of the bilaplacian problem with a homogenous boundary conditions by applying the curl operator. We prove the Kondratiev method in the case of the crack. The singularities of the pressure are done by integration from the singular functions of the velocity near the corner. To approach these problems by a numerical method, we need to take into account the singular functions. Several numeric methods have been proposed in this context; see [4–9]. Since the singular functions are developed for the Stokes problem in this paper, we intend in future work to implement Strang and Fix algorithm, see [10], by the mortar spectral element method. It will be an extension of a work done on an elliptic operator [11, 12]. An outline of this paper is as follows. In Section 2, we present the geometry of the domain and the continuous problem. In Section 3, we give the singular functions and some regularity results. The Kondratiev method is described in Section 4. Section 5 is devoted to the conclusion. 2. The Continuous Problem We suppose that is a polygonal domain of simply connected and has a connected boundary . is the union of vertex for ; is positive integer. Let be the corner of between and ; is

References

[1]  V. A. Kondratiev, “Boundary value problems for elliptic equations in domains with conical or angular points,” Transactions of the Moscow Mathematical Society, vol. 16, pp. 227–313, 1967.
[2]  P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24, Pitman, Boston, Mass, USA, 1985.
[3]  M. Dauge, Elliptic Boundary Value Problems on Corner Domains, vol. 1341 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988.
[4]  M. Amara, C. Bernardi, and M. A. Moussaoui, “Handling corner singularities by the mortar spectral element method,” Applicable Analysis, vol. 46, no. 1-2, pp. 25–44, 1992.
[5]  S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions,” Communications on Pure and Applied Mathematics, vol. 12, pp. 623–727, 1959.
[6]  M. Amara and M. Moussaoui, “Approximation de coefficient de singularités,” Comptes Rendus de l'Académie des Sciences I, vol. 313, no. 5, pp. 335–338, 1991.
[7]  I. Babu?ka, “Finite element method for domains with corners,” Computing, vol. 6, no. 3-4, pp. 264–273, 1970.
[8]  J. Lelièvre, “Sur les éléments finis singuliers,” Comptes Rendus de l'Académie des Sciences A, vol. 283, no. 15, pp. 1029–1032, 1967.
[9]  M. Suri, “The -version of the finite element method for elliptic equations of order ,” ESAIM, vol. 24, no. 2, pp. 265–304, 1990.
[10]  G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973.
[11]  N. Chorfi, “Handling geometric singularities by the mortar spectral element method: case of the Laplace equation,” SIAM Journal of Scientific Computing, vol. 18, no. 1, pp. 25–48, 2003.
[12]  F. Ben Belgacem, C. Bernardi, N. Chorfi, and Y. Maday, “Inf-sup conditions for the mortar spectral element discretization of the Stokes problem,” Numerische Mathematik, vol. 85, no. 2, pp. 257–281, 2000.
[13]  I. Babu?ka, “The finite element method with Lagrangian multipliers,” Numerische Mathematik, vol. 20, no. 3, pp. 179–192, 1973.
[14]  O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, NY, USA, 1962.
[15]  R. Temam, Theory and Numerical Analysis of the Navier-Stokes Equations, North-Holland, Amsterdam, The Netherlands, 1977.
[16]  O. Pironneau, Méthode des éléments finis pour les fluides, Masson, Paris, France, 1988.
[17]  V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 5, Springer, New York, NY, USA, 1986.
[18]  L. Cattabriga, “Su un problema al contorno relativo al sistema di equazioni di Stokes,” Rendiconti del Seminario Matematico della Università di Padova, vol. 31, pp. 1–33, 1961.
[19]  P. Grisvard, Singularities in Boundary Value Problems, vol. 22, Springer, Amesterdam, The Netherlands, 1992.
[20]  C. Bernardi and G. Raugel, “Méthodes d'éléments finis mixtes pour les équations de Stokes et de Navier-Stokes dans un polygone non convexe,” Calcolo, vol. 18, no. 3, pp. 255–291, 1981.

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