A Computational Framework for Constructing Adaptive Grid Domain Mappings

This paper describes a computational framework for efficiently constructing and applying adaptive grid domain mappinglJ for the numerical solution of partial differential equations (PDEs). These mappings are used to transform a PDE problem 80 that solving the new problem in the IItransrormed domain using a uniform grid is equivalent to solving the original problem using an adapted grid. Thus, it is possible to apply a moving or irregular grid, and yet compute on a. uniform grid. We believe that data structures and algorithms based on regular grids are the most promising for the development of good vector and parallel methods. We describe a very efficient method for constructing and evaluating a wide variety of adaptive grid domain mappings, given a redistribution of grid points in the original domain. The extra computation introduced by the mapping is small compared to the cost of solving a typical PDE problem. Using this framework, one can concentrate on the interesting and difficult problems of grid adaption. without having to write new algorithms to solve PDEs on these irregular grids. Good existing software may be used to solve the trallBformed problem on a uniform grid. Our method was developed as part; of an investigation of adaptive grid schemes for two-dimensiona~elliptic boundary value problems; it is applicable to grid based methods for other classes of PDEs as w,ll