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Time discretization of an integro-differential equation of parabolic type. (English) Zbl 0608.65096

The evolution equation ${u}_{t}+Au={\int }_{0}^{t}\kappa \left(t,s\right)Bu\left(s\right)ds+f\left(t\right)$, $t\ge 0$, $u\left(0\right)=v$ is considered for an unbounded positive-definite self-adjoint operator with dense domain D(A) in a Hilbert space and D(B)$\supset D\left(A\right)$. The kernel $\kappa$ is assumed to be a smooth real-valued function. Stability and convergence are investigated for the backward Euler scheme and the Crank-Nicolson scheme, both for the integral supplemented by quadrature rules consistent with the mentioned difference schemes. For quadrature the interval $\left[0,{t}_{n}\right]$ or $\left[0,{t}_{n-1/2}\right]$, respectively, is split into two subintervals: if k is the steplength of the difference scheme, the initial interval is discretized with a steplength ${k}_{1}=O\left({k}^{1/2}\right)$, and the final interval taken to have a length $<{k}_{1}$ is discretized with steplength k. Then in the initial interval the trapezoidal rule or Simpson’s rule, respectively, is taken, in the final interval, however, simply the rectangle rule.

If everything is smooth enough the consistency order of the difference scheme is preserved, the advantage being sparseness of the quadrature nodal points. Modifications are possible in cases of not so good regularity properties. In applications to parabolic problems in space and time (A then being an elliptic differential operator) space discretization should be done in such a way that consistency order is not worsened. It is described how this can be achieved in the context of finite elements.

Reviewer: R.Gorenflo

##### MSC:
 65R20 Integral equations (numerical methods) 45K05 Integro-partial differential equations