Alternating and Linearized Alternating Schwarz Methods for Equidistributing Grids

The solution of partial differential equations (PDEs) with disparate space and time scales often benefit from the use of nonuniform meshes and adaptivity to successfully track local solution features. In this paper we consider the problem of grid generation using the so– called equidistribution principle (EP) [3] and domain decomposition (DD) strategies. In the time dependent case, the EP is used to evolve an initial (often uniform) grid by relocating a fixed number of mesh nodes. This leads to a class of adaptive methods known as r–refinement or moving mesh methods. A thorough recent review of moving mesh methods for PDEs can be found in the book [11]. In general, the appropriate grid for a particular problem depends on features of the (typically unknown) solution of the PDE. Here we will focus on the grid generation problem for the time independent, given function u(x) of a single spatial variable x ∈ [0, 1]. Given some positive measure M(x) of the error or difficulty in the solution u(x), the EP requires that the mesh points are chosen so that the error contribution on each interval [xi−1, xi] is the same. The function M is known as the monitor or mesh density function. Mathematically, we may write this as ∫ xi