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In multi-phase fluid flow, fluid-structure interaction, and otherapplications, partial differential equations (PDEs) often arise withdiscontinuous coefficients and singular sources (e.g., Dirac delta functions).These complexities arise due to changes in material properties at an immersedinterface or embedded boundary, which may have an irregular shape.Consequently, the solution and its gradient can be discontinuous, and numericalmethods can be difficult to design. Here a new method is presented andanalyzed, using a simple formulation of one-dimensional finite differences on aCartesian grid, allowing for a relatively easy setup for one-, two-, orthree-dimensional problems. The method preserves a sharp interface withdiscontinuous solutions, obtained from a small number of iterations(approximately five) of solving a symmetric linear system with updates to theright- hand side. Second-order accuracy is rigorously proven in one spatialdimension and demonstrated through numerical examples in two and three spatialdimensions. The method is tested here on the variable-coefficient Poissonequation, and it could be extended for use on time-dependent problems of heattransfer, fluid dynamics, or other applications.

Simple Second-Order Finite Differences for Elliptic PDEs with Discontinuous Coefficients and Interfaces