Error estimation of Newmark's method for conservative second order linear evolution equation (Q1318918)

Motivated by the study of the linear water wave equation, the following problem is considered on a Hilbert space \(X\): (1) \({d^ 2\over dt^ 2}\phi+ A\phi= \psi\), \(t>0\), \(\phi(0)= \phi^ 0\), \({d\phi\over dt}(0)= \phi^ 1\), where \(A\) is self-adjoint, positive definite. To develop error estimates an abstract method is employed for Newmark's approximations applied to (1). This method allows obtaining stability criteria and results (error convergence estimates) that could be gotten through the energy method or the semigroup theoretical method.