A new approach to convergence analysis of linearized finite element method for nonlinear hyperbolic equation. (English) Zbl 1524.65605
Summary: We study a new way to convergence results for a nonlinear hyperbolic equation with bilinear element. Such equation is transformed into a parabolic system by setting the original solution \(u\) as \(u_t=q\). A linearized backward Euler finite element method (FEM) is introduced, and the splitting skill is exploited to get rid of the restriction on the ratio between \(h\) and \(\tau \). The boundedness of the solutions about the time-discrete system in \(H^2\)-norm is proved skillfully through temporal error. The spatial error is derived without the mesh-ratio, where some new techniques are utilized to deal with the problems caused by the new parabolic system. The final unconditional optimal error results of \(u\) and \(q\) are deduced at the same time. Finally, a numerical example is provided to support the theoretical analysis. Here \(h\) is the subdivision parameter, and \(\tau\) is the time step.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35L72 | Second-order quasilinear hyperbolic equations |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
Keywords:
nonlinear hyperbolic equation; parabolic system; bilinear element; linearized FEM; optimal error resultsReferences:
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