Pseudo-Harmonic Interpolation on Convex Domains

Practical interpolation schemes for functions of more than one variable need not be finite dimensional as in the case of univariate functions. The so-called “blending function methods” are one example of such transfinite schemes. Another is the method of “pseudo-harmonic interpolation” described in this paper. For any bounded, convex planar domain D, we show how to construct a bivariate function U which interpolates to arbitrary Dirichlet boundary conditions on the perimeter of D. The interpolant $U(Q)$ satisfies the familar “maximum principle” and, in the case in which the domain D is circular, is actually a harmonic function in D. Higher order schemes are described in which the function $U(Q)$ interpolates to specified normal derivatives on the boundary of D as well as to the Dirichlet boundary conditions. The interpolation schemes are also viewed as idempotent linear operators (i.e., projectors) on families of continuous functions on D̄, and the algebraic and interpolatory properties of such projectors are examined. Numerical; techniques for evaluating the interpolant U are described together with examples, illustrations and applications.