Error estimates for finite element methods for a class of nonlinear hyperbolic equations (Q2774094)

The author considers the \(H^1\)-finite element approximation for a second order hyperbolic equation in a bounded domain in \(I R^d\): NEWLINE\[NEWLINE\varphi (x,u)_{tt}-\partial_j (a_{ij}(x,u)\partial_i u) -b_i(x,u) \partial_i u=f(x,u) NEWLINE\]NEWLINE with initial data and Dirichlet boundary condition. When \(\varphi (x,u)=u\) and \(\varphi (x,u)=\varphi (x)u\), the finite element approximation has been analyzed by some people. In this paper, the semi-discrete and full-discrete \(H^1\)-finite element methods are studied. Assuming that the solutions of the above equation are sufficiently smooth, and that \(\varphi_{uu}\) is Lipschitz-continuous and \(\varphi_u\) positively bounded from below and above, and that certain conditions upon \(a_{ij}, b_i, f\) are satisfied, the author obtains the optimal error estimates in \(H^1\)- and \(L^2\)-norms. The main ingredients in the proof are the auxiliary elliptic projection to the finite element spaces and the discrete \(L^2\)-energy method.