A fast adaptive grid scheme for elliptic partial differential equations

Abstract

We describe the Recursive Subdivision (RS) method-an efficient and effective adaptive grid scheme for two-dimensional elliptic partial differential equations (PDEs). The RS method generates a new grid by recursively subdividing a rectangular domain. We use a heuristic approach which attempts to equidistribute a given density function over the domain. The resulting grid is used to generate an adaptive grid domain mapping (AGDM), which may be applied to transform the PDE problem to another coordinate system. The PDE is then solved in the transformed coordinate system using a uniform grid. We believe parallelism is most easily exploited when computations are carried out on uniform grids; the AGDM framework allows the power of adaptation to be applied while still preserving this uniformity. Our method generates good adaptive grid domain mappings at a small cost compared to the cost of the entire computation. We describe the RS algorithm in detail, briefly describe the AGDM framework, and illustrate the effectiveness of our scheme on several realistic test problems.