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On uniform difference schemes for second-order singular perturbation problems in Banach spaces. (English) Zbl 0745.65037

The authors consider four singular perturbation problems for partial differential equations involving a small coefficient ${\epsilon }^{2}$ in higher derivatives, such as elliptic, parabolic, Schrödinger, and hyperbolic equations. In each of the four cases the equation is reduced by elementary transformations to (*) ${v}^{\text{'}\text{'}}\left(t\right)=Av\left(t\right)+g\left(t\right)$. The minimal assumption on the initial value problem for any of the equations is well posedness, this is satisfied if and only if $A$ is the infinitesimal generator of a strongly continuous cosine function.

The solution to (*) may be represented explicitly by solution operators, i.e. operator valued sine and cosine functions. Hence, the paper considers approximations by finite difference schemes of the original initial value problems, based on the reduction of each equation to (*) and on the simplest difference scheme for (*), a direct generalization of the Courant-Friedrichs-Lewy difference scheme for the one-dimensional wave equation. Everywhere approximation is shown of arbitrary order in the discretization step, uniform with respect to $\epsilon$.

##### MSC:
 65J10 Equations with linear operators (numerical methods) 65L12 Finite difference methods for ODE (numerical methods) 65N06 Finite difference methods (BVP of PDE) 34G10 Linear ODE in abstract spaces 65M06 Finite difference methods (IVP of PDE) 34E15 Asymptotic singular perturbations, general theory (ODE) 35B25 Singular perturbations (PDE)