Error estimates in the Galerkin method for an abstract second-order evolution equation with nonsmooth right-hand side (Q2387972)

The (abstract) second order differential equation considered in this paper is of the following form \[ u{''}(t)+A(t)u{'}(t)+B(t)u(t)+C(t)u(t)=f(t),\;\;t\in[0,T], \] \[ u(0)=u{'}(0)=0. \] Here \(f:[0,T]\rightarrow H\), where \(H\) is a real separable Hilbert space, and \(A(t),\;B(t),\;C(t)\) the families of operators with common domain: \begin{itemize}\item{}\(A(t):H\rightarrow H\) are bounded, \item{}\(B(t):H\rightarrow H\) are unbounded selfadjoint, positive, and such that \(\| B_0\| _H\leq b_0\| B(t)\| _H\) with \(B_0\) selfadjoint, positive, with compact inverse, \item{}\(C(t):V\rightarrow H\), where \(V\subset H\) is the energy space defined by \(B_0\). The generalized solution of the problem is defined with help of a bilinear form, as an element of the space \(C^1([0,T],H)\cap C^0([0,T],V)\), satisfying the original initial conditions, the second argument of the bilinear form being taken from \(W^{1,1}(0,T,H)\cap L^1(0,T,V)\); moreover it is assumed that \(f\in L^1(0,T,H)\). The (generalized) solution \(u\) is approximated by a standard procedure of the Galerkin type: the sequence of approximate solutions \(u_n\) is defined using orthogonal projections \(Q_n\) of \(V\) onto elements of the sequence of subspaces \(V_n\subset V\) (assumed to be dense in \(V)\). Main result is an error estimate of the following form \[ \| u-u_n\| \leq C\omega_f(\rho^{1\over 2}). \] Here \[ \| v\| =\sup_{0\leq t\leq T}[\| v{'}(t)\| _H+\| v(t)\| _V],\;\;\rho_n=\| (I-Q_n)B_0^{-1}\| _{{\mathcal L}(H,V)}, \] \[ \omega_f(\epsilon)=\sup_{| \tau| \leq\epsilon}\int_0^T\| f(t+\tau)-f(t)\| _Hdt. \] \end{itemize}