  We present two accurate and efficient algorithms for solving the
incompressible, irrotational Euler equations with a free surface in two
dimensions with background flow over a periodic, multiply-connected fluid
domain that includes stationary obstacles and variable bottom topography. One
approach is formulated in terms of the surface velocity potential while the
other evolves the vortex sheet strength. Both methods employ layer potentials
in the form of periodized Cauchy integrals to compute the normal velocity of
the free surface, are compatible with arbitrary parameterizations of the free
surface and boundaries, and allow for circulation around each obstacle, which
leads to multiple-valued velocity potentials but single-valued stream
functions. We prove that the resulting second-kind Fredholm integral equations
are invertible, possibly after a physically motivated finite-rank correction.
In an angle-arclength setting, we show how to avoid curve reconstruction errors
that are incompatible with spatial periodicity. We use the proposed methods to
study gravity-capillary waves generated by flow around several elliptical
obstacles above a flat or variable bottom boundary. In each case, the free
surface eventually self-intersects in a splash singularity or collides with a
boundary. We also show how to evaluate the velocity and pressure with spectral
accuracy throughout the fluid, including near the free surface and solid
boundaries. To assess the accuracy of the time evolution, we monitor energy
conservation and the decay of Fourier modes and compare the numerical results
of the two methods to each other. We implement several solvers for the
discretized linear systems and compare their performance. The fastest approach
employs a graphics processing unit (GPU) to construct the matrices and carry
out iterations of the generalized minimal residual method (GMRES).
