A high-order immersed interface method for elliptic PDEs with variable or discontinuous coefficients

We present a high-order immersed interface method for discretizing elliptic PDEs on complex domains with variable or discontinuous coefficients. High-order accuracy is achieved by combing fourth and sixth-order dimension-split finite difference schemes with a high-order weighted least-squares reconstruction of the solution near domain boundaries and material interfaces. The approach does not require the derivation of jump conditions at material interfaces, which greatly facilitates the construction of discretization up to sixth order. We evaluate the spectra and conditioning of the resulting non-symmetric linear systems, and discuss appropriate matrix-based and matrix-free preconditioners that can be used when solving these systems iteratively. We also evaluate the accuracy of surface quantities on domain boundaries and material interfaces, particularly normal gradients that represent surface tractions or viscous fluxes. We conclude by applying the discretization to to scalar and vector elliptic PDEs that appear in simulations of incompressible flows with fluid-structure interaction.