A Cartesian Grid-Based Boundary Integral Method for an Elliptic Interface Problem on Closely Packed Cells (Communications in Computational Physics, 2020)
In this work, we propose a second-order version and a fourth-order versionof a Cartesian grid-based boundary integral method for an interface problem of theLaplace equation on closely packed cells. When the cells are closely packed, the boundaryintegrals involved in the boundary integral formulation for the interface problembecome nearly singular. Direct evaluation of the boundary integrals has accuracy issues.The grid-based method evaluates a boundary integral by first solving an equivalent,simple interface problem on a Cartesian grid with a fast Fourier transform basedPoisson solver, then interpolating the grid solution to get values of the boundary integralat discretization points of the interface. The grid-based method presents itself asan alternative but accurate numerical method for evaluating nearly singular, singularand hyper-singular boundary integrals. This work can be regarded as a further developmentof the kernel-free boundary integral method [W.-J. Ying and C. S. Henriquez,A kernel-free boundary integral method for elliptic boundary value problems, Journalof Computational Physics, Vol. 227 (2007), pp. 1046-1074] for problems in unboundeddomains. Numerical examples with both second-order and fourth-order versions ofthe grid-based method are presented to demonstrate accuracy of the method.
Keywords: Laplace equation; cell suspension; inclusion of grains; boundary integral method; Cartesian grid method; finite difference method; fast Fourier transform.