Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains

Second and fourth order Cartesian grid based finite difference methods are proposed for elliptic and parabolic partial differential equations, and associated eigenvalue problems on irregular domains with general boundary conditions. Our methods are based on the continuation of a solution idea using multivariable Taylor's expansion of the solution about selected boundary points, and the core ideas of the immersed interface method. The methods offer systematic treatment of the general boundary conditions in two- and three-dimensional domains and are directly applied to semi-discretize heat equations on irregular domains. Convergence analysis and numerical examples are presented. The validity and effectiveness of the proposed methods are demonstrated through our numerical results including computations of the eigenvalues of the associated eigenvalue problem.