Many elliptic PDEs can be recast as integral equations involving surface integrals $S(x)=\int_\Gamma G(x,y) q(y) \, dS(y)$ of a Green's function $G(x,y)$ multiplying a layer density $q(y)$. Such an integral, called a layer potential, must often be evaluated on the surface itself, $x \in \Gamma$, in which case the Green's function is singular (but $S(x)$ itself is finite). This calls for specialized methods to be used on-surface. We discuss and compare three such specialized methods.

The first method is based on local corrections to a standard quadrature method (the trapezoidal rule), in a few grid points around the singularity. For instance, correcting at nine points around the singularity leads to a 5th order scheme.

The second method is based on regularizing the Green's function so that a standard quadrature method can be used. For a suitable choice of the regularization, the method can be observed to have 3rd or even 5th order convergence on-surface.

The third method is based on evaluating the layer potential along a line off-surface, and extrapolating onto the surface. In particular, we consider extrapolation using rational functions, and compare with polynomials.

All methods are discussed in the context of simulating deformable capsules in Stokes flow. The fact that the surface deforms rules out any precomputation that depends on its particular shape. A partition of unity discretization is used to avoid the second-order boundary error of the trapezoidal rule.