A numerical method for solving the Oseen-type problem in an $L$-shaped domain
Keywords:Oseen problem with singularity, weighted finite element method
In this paper we introduce the notion of an Rν-generalized solution to the Oseen-type problem in rotation form in a polygonal domain with a reentrant obtuse angle on its boundary. We propose a weighted finite element method to solve this problem approximately. A number of numerical experiments performed on a model problem in an L-shaped domain show the advantages of the proposed method.
- V. A. Rukavishnikov, “On Differentiability Properties of an Rν-Generalized Solution of the Dirichlet Problem,” Dokl. Akad. Nauk SSSR 309 (6), 1318-1320 (1989) [Sov. Math. Dokl. 40 (3), 653-655 (1990)].
- H. K. Moffatt, “Viscous and Resistive Eddies near a Sharp Corner,” J. Fluid Mech. 18 (1), 1-18 (1964).
- M. Dauge, “Stationary Stokes and Navier-Stokes Systems on Two- or Three-Dimensional Domains with Corners. Part I. Linearized Equations,” SIAM J. Math. Anal. 20 (1), 74-97 (1989).
- H. Blum, “The Influence of Reentrant Corners in the Numerical Approximation of Viscous Flow Problems,” in Notes on Numerical Fluid Mechanics (Springer, Wiesbaden, 1990), Vol. 30, pp. 37-46.
- M. Orlt and A.-M. S854ndig, “Regularity of Viscous Navier-Stokes Flows in Nonsmooth Domains,” in Lecture Notes in Pure and Applied Mathematics (Marcel Dekker, New York, 1995), Vol. 167, pp. 185-201.
- V. A. Kondrat’ev, “Boundary Value Problems for Elliptic Equations in Domains with Conical or Angular Points,” Tr. Mosk. Mat. Obshch. 16, 209-292 (1967) [Trans. Moscow Math. Soc. 16, 227-313 (1967)].
- V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities (AMS Press, Providence, 1997).
- V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations (AMS Press, Providence, 2001).
- V. A. Rukavishnikov and E. I. Rukavishnikova, “The Finite Element Method for the First Boundary Value Problem with Compatible Degeneracy of the Initial Data,” Dokl. Akad. Nauk 338 (6), 731-733 (1994) [Dokl. Math. 50 (2), 335-339 (1995)].
- V. A. Rukavishnikov and E. V. Kuznetsova, “A Finite Element Method Scheme for Boundary Value Problems with Noncoordinated Degeneration of Input Data,” Sib. Zh. Vych. Mat. 12 (3), 313-324 (2009) [Numer. Anal. Appl. 2 (3), 250-259 (2009)].
- V. A. Rukavishnikov and H. I. Rukavishnikova, “The Finite Element Method for a Boundary Value Problem with Strong Singularity,” J. Comput. Appl. Math. 234 (9), 2870-2882 (2010).
- V. A. Rukavishnikov and A. O. Mosolapov, “New Numerical Method for Solving Time-Harmonic Maxwell Equations with Strong Singularity,” J. Comput. Phys. 231 (6), 2438-2448 (2012).
- V. A. Rukavishnikov and H. I. Rukavishnikova, “On the Error Estimation of the Finite Element Method for the Boundary Value Problems with Singularity in the Lebesgue Weighted Space,” Numer. Funct. Anal. Optim. 34 (12), 1328-1347 (2013).
- V. A. Rukavishnikov and S. G. Nikolaev, “Weighted Finite Element Method for an Elasticity Problem with Singularity,” Dokl. Akad. Nauk 453 (4), 378-382 (2013) [Dokl. Math. 88 (3), 705-709 (2013)].
- V. A. Rukavishnikov, “On the Uniqueness of an Rν-Generalized Solution to Boundary Value Problems with Inconsistently Degenerate Initial Data,” Dokl. Akad. Nauk 376 (4), 451-453 (2001) [Dokl. Math. 63 (1), 68-70 (2001)].
- V. A. Rukavishnikov and E. V. Kuznetsova, “The Rν-generalized solution of a boundary value problem with a singularity belongs to the space W2+κ+1/κ+22,ν+β (Ω, δ),” Differ. Uravn. 45 (6), 894-898 (2009) [Differ. Equ. 45 (6), 913-917 (2009)].
- V. A. Rukavishnikov, “On the Existence and Uniqueness of an Rν-Generalized Solution of a Boundary Value Problem with Uncoordinated Degeneration of the Input Data,” Dokl. Akad. Nauk 458 (3), 261-263 (2014) [Dokl. Math. 90 (2), 562-564 (2014)].
- V. A. Rukavishnikov and S. G. Nikolaev, “On the Rν-Generalized Solution of the Lamé System with Corner Singularity,” Dokl. Akad. Nauk 463 (2), 137-139 (2015) [Dokl. Math. 92 (1), 421-423 (2015)].
- L. R. Scott and M. Vogelius, “Norm Estimates for a Maximal Right Inverse of the Divergence Operator in Spaces of Piecewise Polynomials,” Math. Model. Numer. Anal. 19 (1), 111-143 (1985).
- J. Qin, On the Convergence of Some Low Order Mixed Finite Elements for Incompressible Fluids , PhD Thesis (Pennsylvania State University, State College, 1994).
- L. I. Sedov, Mechanics of Continuous Media (Nauka, Moscow, 1970; World Scientific, River Edge, 1997).
- M. Benzi, G. H. Golub, and J. Liesen, “Numerical Solution of Saddle Point Problems,” Acta Numerica 14, 1-137 (2005).
- Ph. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978; Mir, Moscow, 1980).
- J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, “Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems,” SIAM J. Numer. Anal. 1997. 34 (3), 1072-1092 (1997).
- Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, Philadelphia, 2003; Mosk. Gos. Univ., Moscow, 2013).
- M. A. Olshanskii and A. Reusken, “Analysis of a Stokes Interface Problem,” Numer. Math. 103 (1), 129-149 (2006).
- R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley and Teubner, Chichester-Stuttgart, 1996).