Convergence of nonconforming finite element approximations to first-order linear hyperbolic equations
Author:
Noel J. Walkington
Journal:
Math. Comp. 58 (1992), 671-691
MSC:
Primary 65N30; Secondary 65M60, 76M10
DOI:
https://doi.org/10.1090/S0025-5718-1992-1122082-8
MathSciNet review:
1122082
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Abstract: Finite element approximations of the first-order hyperbolic equation ${\mathbf {U}} \bullet \nabla u + \alpha u = f$ are considered on curved domains $\Omega \subset {\mathbb {R}^2}$. When part of the boundary of $\Omega$ is characteristic, the boundary of numerical domain, ${\Omega _h}$, may become either an inflow or outflow boundary, so it is necessary to select an algorithm that will accommodate this ambiguity. This problem was motivated by a problem in acoustics, where an equation similar to the one above is coupled to three elliptic equations. In the last section, the acoustics problem is briefly recalled and our results for the first-order equation are used to demonstrate convergence of finite element approximations of the acoustics problem.
- J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI https://doi.org/10.1137/0707006
- 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
- Jim Douglas Jr. and Thomas F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982), no. 5, 871–885. MR 672564, DOI https://doi.org/10.1137/0719063
- Richard S. Falk and Gerard R. Richter, Analysis of a continuous finite element method for hyperbolic equations, SIAM J. Numer. Anal. 24 (1987), no. 2, 257–278. MR 881364, DOI https://doi.org/10.1137/0724021
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038 C. Johnson, Streamline diffusion methods for problems in fluid mechanics, Finite Elements in Fluids, Vol. 6 (R. H. Gallagher, G. F. Carey, J. T. Oden, and O. C Zienkiewicz, eds.), Wiley, Chichester, 1985.
- C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1–26. MR 815828, DOI https://doi.org/10.1090/S0025-5718-1986-0815828-4
- P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. Publication No. 33. MR 0658142
- K. Oswatitsch, Physikalische Grundlagen der Strömungslehre, Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, pp. 1–124 (German). MR 0108115
- Lars B. Wahlbin, A dissipative Galerkin method for the numerical solution of first order hyperbolic equations, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 147–169. Publication No. 33. MR 0658322
- Noel J. Walkington, Acoustic wave propagation through flows with vorticity, SIAM J. Numer. Anal. 25 (1988), no. 3, 533–549. MR 942206, DOI https://doi.org/10.1137/0725034
- Ragnar Winther, A stable finite element method for initial-boundary value problems for first-order hyperbolic systems, Math. Comp. 36 (1981), no. 153, 65–86. MR 595042, DOI https://doi.org/10.1090/S0025-5718-1981-0595042-6
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Additional Information
Keywords:
Hyperbolic equations,
nonconforming approximation
Article copyright:
© Copyright 1992
American Mathematical Society