A High Order Compact FD Framework for Elliptic BVPs Involving Singular Sources, Interfaces, and Irregular Domains

High order methods are preferred in many applications such as Helmholtz equations with large wave numbers to resolve the solution numerically. In this paper, a third order compact immersed interface method (IIM) based on the standard nine-point stencil is first proposed for solving Poisson/Helmholtz interface problems with discontinuous solutions and fluxes in two-space dimensions. Theoretically, new high order jump relations are derived, which are necessary for determining the correction terms of the finite difference scheme near or on an interface. Then, based on the developed third order compact IIM, an augmented third order compact finite difference method is further developed for elliptic interface problems with piecewise constant but discontinuous coefficients. In this approach, the jump in the normal derivative is set as an unknown so that the high order compact IIM can be applied. The co-dimension one augmented variable is solved by the Schur complement system via the GMRES iterative method. Various non-trivial examples are provided to show the performance of the new methods. One important feature of the new methods is that the computed normal derivative is also nearly third order accurate. Finally, the third order augmented method is applied to Poisson/Helmholtz equations on irregular domains with few changes along examples of Neumann, Robin, and Dirichlet boundary conditions.