Approximation of Quasilinear Hyperbolic Problems with Discontinuous Coefficients: An Optimal Error Estimate

We develop a theoretical framework for the approximation of a class of second-order quasilinear hyperbolic interface problems on quadratic finite element. Time discretization based on linearized implicit difference scheme with degree two polynomials for interface approximation is proposed. Sufficient conditions, on the input data, that guarantee the existence of a unique solution are given. Under these assumptions, the stability of the scheme is established and convergence rate of optimal order is proved. It is assumed that the interface is arbitrary but smooth.