  This paper presents a "two-dimensional Fourier Continuation" method (2D-FC)
for construction of bi-periodic extensions of smooth non-periodic functions
defined over general two-dimensional smooth domains. The approach can be
directly generalized to domains of any given dimensionality, and even to
non-smooth domains, but such generalizations are not considered here. The 2D-FC
extensions are produced in a two-step procedure. In the first step the
one-dimensional Fourier Continuation method is applied along a discrete set of
outward boundary-normal directions to produce, along such directions,
continuations that vanish outside a narrow interval beyond the boundary. Thus,
the first step of the algorithm produces "blending-to-zero along normals" for
the given function values. In the second step, the extended function values are
evaluated on an underlying Cartesian grid by means of an efficient, high-order
boundary-normal interpolation scheme. A Fourier Continuation expansion of the
given function can then be obtained by a direct application of the
two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy
can be obtained by this method. The usefulness and performance of the proposed
two-dimensional Fourier Continuation method are illustrated with applications
to the Poisson equation and the time-domain wave equation within a bounded
domain. As part of these examples the novel "Fourier Forwarding" solver is
introduced which, propagating plane waves as they would in free space and
relying on certain boundary corrections, can solve the time-domain wave
equation and other hyperbolic partial differential equations within general
domains at computing costs that grow sublinearly with the size of the spatial
discretization.
