A third-order boundary condition for the exterior Stokes problem in three dimensions

Author:
Georges H. Guirguis

Journal:
Math. Comp. **49** (1987), 379-389

MSC:
Primary 65N30; Secondary 76-08, 76D07

MathSciNet review:
906177

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We approximate the Stokes operator on an exterior domain in three dimensions by a truncated problem on a finite subdomain. A third-order artificial boundary condition is introduced. We discuss the approximating behavior of the truncated problem and its discretization in a finite element space. Combined errors arising from truncation and discretization are considered.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**J. Boland,*Finite Element and the Divergence Constraint for Viscous Flow*, Ph.D. thesis, Carnegie-Mellon University, 1983.**[3]**F. Brezzi,*On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**8**(1974), no. R-2, 129–151 (English, with loose French summary). MR**0365287****[4]**Murray Cantor,*Numerical treatment of potential type equations on 𝑅ⁿ: theoretical considerations*, SIAM J. Numer. Anal.**20**(1983), no. 1, 72–85. MR**687368**, 10.1137/0720005**[5]**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. MR**0520174****[6]**M. Crouzeix and P.-A. Raviart,*Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge**7**(1973), no. R-3, 33–75. MR**0343661****[7]**V. Girault and P.-A. Raviart,*Finite element approximation of the Navier-Stokes equations*, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR**548867****[8]**C. I. Goldstein,*The finite element method with nonuniform mesh sizes for unbounded domains*, Math. Comp.**36**(1981), no. 154, 387–404. MR**606503**, 10.1090/S0025-5718-1981-0606503-5**[9]**Georges H. Guirguis,*On the existence, uniqueness and regularity of the exterior Stokes problem in 𝑅³*, Comm. Partial Differential Equations**11**(1986), no. 6, 567–594. MR**837276**, 10.1080/03605308608820437**[10]**Georges H. Guirguis and Max D. Gunzburger,*On the approximation of the exterior Stokes problem in three dimensions*, RAIRO Modél. Math. Anal. Numér.**21**(1987), no. 3, 445–464 (English, with French summary). MR**908240****[11]**G. H. Guirguis,*On the Existence, Uniqueness, Regularity and Approximation of the Exterior Stokes Problem in*, Ph.D. Thesis, University of Tennessee, Knoxville, 1983.**[12]**Alvin Bayliss, Max Gunzburger, and Eli Turkel,*Boundary conditions for the numerical solution of elliptic equations in exterior regions*, SIAM J. Appl. Math.**42**(1982), no. 2, 430–451. MR**650234**, 10.1137/0142032**[13]**B. Hanouzet,*Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace*, Rend. Sem. Mat. Univ. Padova**46**(1971), 227–272 (French). MR**0310417****[14]**O. A. Ladyzhenskaya,*The mathematical theory of viscous incompressible flow*, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. MR**0254401****[15]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Value Problems and Applications*, Springer-Verlag, New York, 1972.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
76-08,
76D07

Retrieve articles in all journals with MSC: 65N30, 76-08, 76D07

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1987-0906177-5

Article copyright:
© Copyright 1987
American Mathematical Society