From formal numerical solutions of elliptic PDE's to the true ones
Mathematics of Computation
We propose a discretization scheme for a numerical solution of elliptic PDE's, based on local representation of functions, by their Taylor polynomials (jets). This scheme utilizes jet calculus to provide a very high order of accuracy for a relatively small number of unknowns involved. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 198 Z. WIENER AND Y. YOMDIN solution of the equation. Thus the solution process consists of finding the
... s of finding the values of the unknown parameters which minimize the discrepancy between the neighboring Taylor polynomials. This approach can be considered to be a discretized realization of Gromov's h-homotopy. In such a terminology the standard methods use a true function which approximately satisfies the differential relation. Our method uses objects which are not true functions but satisfy the differential relation exactly. 3. We implement the last stage of the solution as a certain relaxation procedure where the Taylor polynomial at each node is corrected according to the neighboring ones. The mere presence of several Taylor coefficients at each node (instead of the only one in standard schemes) allows one to find relaxation coefficients which "cancel" the discretization error of the solution up to an order m which is much higher than k. For example, for ∆u = 0 for the second degree Taylor polynomials, for an internal node we get the discretization error of order h 10 , where h is the step of the grid. 4. At the previous stage we got at each grid point a Taylor polynomial of degree k which agrees with the Taylor polynomial of the true solution at this point up to order m > k. It turns out that from this data one can usually reconstruct at each grid point the m-th degree Taylor polynomial of the true solution, with the same accuracy. For this reconstruction, the same neighboring nodes as those for relaxation, are used. Figure 1 gives a pictorial explanation of the method. Let us now discuss in more detail the reasons behind the choice of this scheme and some of its features. The question of an optimal ε-representation of smooth functions has been investigated by Kolmogorov in his work on ε-entropy of functional classes  . It was shown that asymptotically, the best way to memorize a C k -function up to accuracy ε > 0 is to store the coefficients of its k-th order Taylor polynomials at each point of some grid of size h = O(ε 1/(k+1) ). The following point implied by the Kolmogorov representation is very important in our approach: We assume that a required accuracy or tolerance ε > 0 is given from the very beginning. We require the discretized data to represent the "true" function up to this accuracy in a C k -norm. But we do not require the discretized data to be itself a C k -function. In particular, the Taylor polynomials at the neighboring grid points may disagree up to ε.