The Efficient Generation of Simple Two-Dimensional Adaptive Grids

We present an efficient adaptive procedure for solving steady partial differential equations based on the simple two-dimensional quadrilateral grid generator of Huang and Sloan [SIAM J. Sci. Comput., 15 (1994), pp. 776--797]. The mesh equations are derived using an equidistribution principle and are discretized in the physical domain. This allows the application of the adaptive grid generator to finite element and finite volume methods. To efficiently solve the nonlinear algebraic equations describing the adapted grid, we propose an alternating line Gauss--Seidel relaxation procedure. The beneficial effect of adaptive grids is demonstrated by coupling the grid generator to finite volume and finite difference approximations of advection and advection-diffusion test problems. For some examples the proposed method is shown to be 50 times more efficient than previously used solution procedures.