  In this paper, we present a parallel higher-order boundary integral method to
solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed
boundary integral formulation is used to ensure the fast convergence of Krylov
subspace linear solver such as GMRES. The molecular surfaces are first
discretized with flat triangles and then converted to curved triangles with the
assistance of normal information at vertices. To maintain the desired accuracy,
four-point Gauss-Radau quadratures are used on regular triangles and
sixteen-point Gauss-Legendre quadratures together with regularization
transformations are applied on singular triangles. To speed up our method, we
take advantage of the embarrassingly parallel feature of boundary integral
formulation, and parallelize the schemes with the message passing interface
(MPI) implementation. Numerical tests show significantly improved accuracy and
convergence of the proposed higher-order boundary integral Poisson-Boltzmann
(HOBI-PB) solver compared with boundary integral PB solver using often-seen
centroid collocation on flat triangles. The higher-order accuracy results
achieved by present method are important to sensitive solvation analysis of
biomolecules, particularly when accurate electrostatic surface potentials are
critical in the molecular simulation. In addition, the higher-order boundary
integral schemes presented here and their associated parallelization
potentially can be applied to solving boundary integral equations in a general
sense.
