\(W\)-methods for semilinear parabolic equations (Q1902086)

The author studies the temporal convergence behavior of \(W\)-methods applied to the initial value problem \[ u'(t)+ Au(t)= g(t, u(t)), \quad u(t_0) \text{ given}, \] in an arbitrary Banach space \(X\). The operator \(A\) is not necessarily bounded. The stability and convergence analysis uses the framework of analytic semigroups of linear operators and perturbation techniques of such operators. The convergence order is a small non-integer, depending on the operator and the boundary conditions. Under additional conditions on the \(W\)-method the convergence rate can be increased. Numerical results are given.