An Iterative Grid Redistribution Method for Singular Problems in Multiple Dimensions

We introduce an iterative grid redistribution method based on the variational approach. The iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations. The method is particularly effective for solving partial differential equations with singular solutions (e.g., blowup solutions). Our method requires little prior information of the singular solutions and can handle multiple singularities. The method is successfully applied to the nonlinear Schrödinger equation and the Keller–Segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time.