A note on space approximation of parabolic evolution equations
The problem is to find the numerical approximation of the abstract Cauchy problem \[ \dot u(t)= Au(t),\quad u(0)= x_0\in X, \] where \(A\) generates an analytic semigroup in a separable Banach space \(X\). Such equations include e.g. parabolic partial differential equations of arbitrary order defined on a bounded domain. The author investigates the accuracy of space approximation of solutions to parabolic evolution equations in arbitrary separable Banach space covering Besov spaces and \(L^p\) spaces, \(1< p< 2\). He formulates the problem in the abstract setting of semigroup theory and gives the rate of convergence depending on the smoothness of the initial condition in terms of real interpolation spaces. To handle regularity he introduces several scales of \(X\) which are available by interpolation methods.
scientific article; zbMATH DE number 2107896