  We introduce a quadrature scheme--QBKIX--for the high-order accurate
evaluation of layer potentials associated with general elliptic PDEs near to
and on the domain boundary. Relying solely on point evaluations of the
underlying kernel, our scheme is essentially PDE-independent; in particular, no
analytic expansion nor addition theorem is required. Moreover, it applies to
boundary integrals with singular, weakly singular, and hypersingular kernels.
  Our work builds upon Quadrature by Expansion (QBX), which approximates the
potential by an analytic expansion in the neighborhood of each expansion
center. In contrast, we use a sum of fundamental solutions lying on a ring
enclosing the neighborhood, and solve a small dense linear system for their
coefficients to match the potential on a smaller concentric ring.
  We test the new method with Laplace, Helmholtz, Yukawa, Stokes, and Navier
(elastostatic) kernels in two dimensions (2D) using adaptive, panel-based
boundary quadratures on smooth and corner domains. Advantages of the algorithm
include its relative simplicity of implementation, immediate extension to new
kernels, dimension-independence (allowing simple generalization to 3D), and
compatibility with fast algorithms such as the kernel-independent FMM.
