On finite element approximations of problems having inhomogeneous essential boundary conditions. (English) Zbl 0531.65056
Summary: The analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered. The results are given for linear second order elliptic partial differential equations and for the nonlinear stationary Navier-Stokes equations. For certain easily implemented boundary treatments, optimal error estimates and numerical examples are provided for problems posed on polyhedral domains.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
inhomogeneous essential boundary conditions; nonlinear stationary Navier- Stokes equations; optimal error estimates; numerical examples; polyhedral domainsReferences:
| [1] | Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101 |
| [2] | Babǔska, I.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A. K., The Mathematical Foundations of the Finite Element Method with Application to P.D.E. (1972), Academic Press: Academic Press New York), 1-359 · Zbl 0268.65052 |
| [3] | Blair, J. J., Higher order approximations to the boundary conditions for the finite element method, Math. Comp., 30, 250-262 (1976) · Zbl 0342.65068 |
| [4] | J. M. Boland and R. A. Nicolaides, Stability of incompressibility conditions with low order finite elements. to appear.; J. M. Boland and R. A. Nicolaides, Stability of incompressibility conditions with low order finite elements. to appear. · Zbl 0578.65123 |
| [5] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [6] | Ciarlet, P. G.; Raviart, P.-A., Interpolation theory over curved elements, with applications to finite element methods, Comput. Methods Applic. Mech. Engng, 1, 217-249 (1972) · Zbl 0261.65079 |
| [7] | Ciarlet, P. G.; Raviart, P.-A., The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, (Aziz, A. K., The Mathematical Foundation of the Finite Element Method with Application to P.D.E. (1972), Academic Press: Academic Press New York), 409-474 · Zbl 0262.65070 |
| [8] | G. J. Fix and M. Suri, The construction of stable conforming finite elements for the Navier-Stokes equations by the discrete Leray process. to appear.; G. J. Fix and M. Suri, The construction of stable conforming finite elements for the Navier-Stokes equations by the discrete Leray process. to appear. |
| [9] | M. D. Gunzburger and J. S. Peterson, On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations. to appear.; M. D. Gunzburger and J. S. Peterson, On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations. to appear. · Zbl 0559.76026 |
| [10] | Jamet, P., Estimation de l’erreur d’interpolation dans un domain variable et application aux éléments finis quadrilatéraux dégénérés, (Méthodes Numériques en Mathématiques Appliquées (1976), l’Université de Montréal: l’Université de Montréal Montreal), 55-100 · Zbl 0374.65007 |
| [11] | Jamet, P.; Raviart, P.-A., Numerical solution of the stationary Navier-Stokes equations by finite element methods, (Glowinski, R.; Lions, J.-L., Computing Methods in Applied Sciences and Engineering, Part 1, Lecture Notes in Computer Science (1979), Springer-Verlag: Springer-Verlag Berlin), 10 · Zbl 0285.76007 |
| [12] | O. A. Karakashian, On a Galerkin-Lagrange multiplier method for the stationary Navier-Stokes equations. SIAM J. Numer. Anal.; O. A. Karakashian, On a Galerkin-Lagrange multiplier method for the stationary Navier-Stokes equations. SIAM J. Numer. Anal. · Zbl 0496.76032 |
| [13] | Lions, J. L.; Magenes, E., Non-homonegeous Boundary Value Problems and Applications, 1 (1972), Springer-Verlag: Springer-Verlag Berlin · Zbl 0223.35039 |
| [14] | Schlichting, H., Boundary Layer Theory (1960), McGraw-Hill: McGraw-Hill New York · Zbl 0096.20105 |
| [15] | Scott, R., Interpolated boundary conditions in the finite element method, SIAM J. Numer. Anal., 12, 404-427 (1975) · Zbl 0357.65082 |
| [16] | Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0278.65116 |
| [17] | Teman, R., Navier-Stokes Equations (1979), North-Holland: North-Holland Amsterdam · Zbl 0406.35053 |
| [18] | Zlamal, M., Curved elements in the finite element method—I, SIAM J. Numer. Anal., 10, 229-240 (1973) · Zbl 0285.65067 |
| [19] | Zlamal, M., Curved elements in the finite element method—II, SIAM J. Numer. Anal., 11, 347-363 (1974) · Zbl 0277.65064 |
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