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The grid generation is very crucial for the accuracy of the numerical solution of PDEs, especially for problems with very rapid variations or sharp layers, such as shock waves, wing leading and trailing edges, regions of separation, and boundary layers. The adaptive grid generation is an iterative approach to accommodate these complex structures. In this paper, we introduce a deformation based adaptive grid generation method, in which a differentiable and invertible transformation from computational domain to physical domain is constructed such that the cell volume (Jacobian determinant) of the new grid is equal to a prescribed monitor function. A vector field is obtained by solving the div-curl system and can be used to move the grids to the desired locations. By computing the inverse of Jacobian, any deformed grids can also be transformed back to the uniform grid. Several numerical results in two dimensions are presented. Some applications in image registration are discussed. [ABSTRACT FROM AUTHOR]

We consider the one-parameter family of systems where F: ℝ × [0, 1] → ℝ is a continuous vector field. The solution x( t) = φ( t, y, λ) is uniquely determined by the initial condition x(0) = y = φ(0, y, λ) and can be continued to the whole axis (−∞, +∞) for all λ ∈ [0, 1]. We obtain conditions ensuring the preservation of the property of global asymptotic stability of the stationary solution of such a system as the parameter λ varies. [ABSTRACT FROM AUTHOR]