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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Finite element approximations of the first-order hyperbolic equation are considered on curved domains . When part of the boundary of is characteristic, the boundary of numerical domain, , 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.

**[1]**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**(1979), 112-124. MR**0263214 (41:7819)****[2]**P. G. Ciarlet,*The finite element method for elliptic problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[3]**J. Douglas, Jr. and T. F. Russel,*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), 871-885. MR**672564 (84b:65093)****[4]**R. S. Falk and G. R. Richter,*Analysis of a continuous finite element method for hyperbolic equations*, SIAM J. Numer. Anal.**24**(1987), 257-278. MR**881364 (88d:65133)****[5]**P. Hartman,*Ordinary differential equations*, Wiley, New York, 1964. MR**0171038 (30:1270)****[6]**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.**[7]**C. Johnson and J. Pitkäranta,*An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation*, Math. Comp.**46**(1986), 1-26. MR**815828 (88b:65109)****[8]**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 (C. de Boor, ed.), Academic Press, 1974, pp. 89-123. MR**0658142 (58:31918)****[9]**J. Serrin,*Mathematical principles of classical fluid mechanics*, Handbuch der Physik, Volume VIII/I, Springer-Verlag, Berlin, 1959. MR**0108116 (21:6836b)****[10]**L. B. Wahlbin,*A dissipative Galerkin method for the numerical solution of first order hyperbolic equations*, Mathematical Aspects of the Finite Element Method in Partial Differential Equations (C. de Boor, ed.), Academic Press, 1974, pp. 147-170. MR**0658322 (58:31929)****[11]**N. J. Walkington,*Acoustic wave propagation through flows with vorticity*, SIAM J. Numer. Anal.**25**(1988), 533-539. MR**942206 (89h:76034)****[12]**R. Winther,*A stable finite element method for initial-boundary value problems for first-order hyperbolic systems*, Math. Comp.**36**(1981), 65-68. MR**595042 (81m:65181)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
65M60,
76M10

Retrieve articles in all journals with MSC: 65N30, 65M60, 76M10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1122082-8

Keywords:
Hyperbolic equations,
nonconforming approximation

Article copyright:
© Copyright 1992
American Mathematical Society