| Summary: | Abstract 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 ut=q $u_{t}=q$. 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 τ. The boundedness of the solutions about the time-discrete system in H2 $H^{2}$-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 τ is the time step.
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