  We present a Trefftz-type finite element method on meshes consisting of
curvilinear polygons. Local basis functions are computed using integral
equation techniques that allow for the efficient and accurate evaluation of
quantities needed in the formation of local stiffness matrices. To define our
local finite element spaces in the presence of curved edges, we must also
properly define what it means for a function defined on a curved edge to be
"polynomial" of a given degree on that edge. We consider two natural choices,
before settling on the one that yields the inclusion of complete polynomial
spaces in our local finite element spaces, and discuss how to work with these
edge polynomial spaces in practice. An interpolation operator is introduced for
the resulting finite elements, and we prove that it provides optimal order
convergence for interpolation error under reasonable assumptions. We provide a
description of the integral equation approach used for the examples in this
paper, which was recently developed precisely with these applications in mind.
A few numerical examples illustrate this optimal order convergence of the
finite element solution on some families of meshes in which every element has
at least one curved edge. We also demonstrate that it is possible to exploit
the approximation power of locally singular functions that may exist in our
finite element spaces in order to achieve optimal order convergence without the
typical adaptive refinement toward singular points.
