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Stabilized $$hp$$-finite element methods for first-order hyperbolic problems. (English) Zbl 0957.65103
The authors consider the approximate solution of the first-order hyperbolic differential equation $(\underline a.\underline\nabla)u= bu=f$ over a domain with a boundary condition $u= g.$ They extend the work of K. S. Bey and J. T. Oden [Comput. Methods Appl. Mech. Eng. 144, No. 3-4, 259-286 (1996; Zbl 0894.76036)] on the discontinuous Galerkin and the streamline-diffusion methods, and obtain error estimates in terms of $$h$$ the mesh interval size and $$p$$ the polynomial degree. They obtain bounds for $$u$$ in terms of $$f$$ and $$g$$ and provide error estimates, and treat the problem of hanging nodes. Some numerical results obtained show that the methods discussed are satisfactory.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35L50 Initial-boundary value problems for first-order hyperbolic systems
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