The Approximation of Harmonic and Parabolic Functions on Half-Spaces from Interior Data

Abstract

The object of this paper is to discuss the numerical approximation of functions either harmonic or parabolic in a half-space given the approximate values of the functions on some subset of the open half-space. Two things are immediately obvious. One is that, without some knowledge of the global behavior of the functions, nothing can be said about the functions in the whole half-space. The second is that, if there exists one function extending the data to the half-space, in general there are infinitely many. Throughout the paper the notation is based on approximating any one of the solutions; it is clear that the error estimates apply equally well to every solution.

The global constraint imposed on harmonic functions is a convenient generalization of boundedness (with a known bound) on the half-space; in the parabolic case the solutions are assumed non-negative. These different assumptions lead to minor differences in the arguments and results; however, each assumption could be employed in either case. Two cases are treated for the measurement of the data. First, it is assumed that the function is measured with a known accuracy on an entire hyperplane parallel to the boundary of the half-space. Later the data are measured with a prescribed accuracy on a rectangle on the same hyperplane.