The Approximation of Certain Parabolic Equations Backward in Time by Sobolev Equations

For any nonnegative, self-adjoint operator A, which does not depend on time, the backward solution to the parabolic equation, $u'(t) = - Au(t)$, $t \geqq 0$, in a cylinder can be approximated by the solution to the Sobolev equation, $u'(t) = - (I + \beta A)^{ - 1} Au(t)$. The solution to the backward Sobolev equation can be more readily computed than the solution to the adjoint of the parabolic equation. In a Hilbert space setting, if the norm of the solution is assumed to be bounded by a positive constant E at the base $t = 0$ of the cylinder and the data error at $t = T$ is less than a prescribed $\varepsilon > 0$, then the norm of the difference in the solutions is $O([ - \log ({\varepsilon / E})]^{ - 1} )$. This logarithmic continuity is essentially the best that can be obtained for this approximation.The above result can be generalized to operators A which are sectorial with semiangle ${\pi / 4}$ and such that $- A$ generates a contraction semigroup of operators. Simple numerical results for the heat...