Spectral approximation of the free-space heat kernel

Many problems in applied mathematics, physics, and engineering require the solution of the heat equation in unbounded domains. Integral equation metho ds are particularly appropriate in this setting for several reasons: they a re unconditionally stable, they are insensitive to the complexity of the ge ometry, and they do not require the artificial truncation of the computatio nal domain as do finite difference and finite element techniques. Methods o f this type, however, have not become widespread due to the high cost of ev aluating heat potentials. When m points are used in the discretization of t he initial data, M points are used in the discretization of the boundary, a nd N time steps are computed, an amount of work of the order O((NN2)-N-2 NMm) has traditionally been required. In this paper, we present an algorith m which requires an amount of work of the order O(N M log M + m log m) and which is based on the evolution of the continuous spectrum of the solution. The method generalizes an earlier technique developed by Greengard and Str ain (1990, Comm. Pure Appl. Math. 43, 949) for evaluating layer potentials in bounded domains. (C) 2000 Academic Press.