A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs (Communications in Computational Physics, 2020)
This work proposes a generalized boundary integral method for variable coefficientselliptic partial differential equations (PDEs), including both boundary valueand interface problems. The method is kernel-free in the sense that there is no needto know analytical expressions for kernels of the boundary and volume integrals inthe solution of boundary integral equations. Evaluation of a boundary or volume integralis replaced with interpolation of a Cartesian grid based solution, which satisfiesan equivalent discrete interface problem, while the interface problem is solved by afast solver in the Cartesian grid. The computational work involved with the generalizedboundary integral method is essentially linearly proportional to the numberof grid nodes in the domain. This paper gives implementation details for a second-orderversion of the kernel-free boundary integral method in two space dimensionsand presents numerical experiments to demonstrate the efficiency and accuracy ofthe method for both boundary value and interface problems. The interface problemsdemonstrated include those with piecewise constant and large-ratio coefficients andthe heterogeneous interface problem, where the elliptic PDEs on two sides of the interfaceare of different types.