The Local Grid Refinement Method In Time And Space For Parabolic Problems

Some large-scale physical processes may possess highly localized properties both in space and in time. These local properties are due to stationary features such as wells, cracks, obstacles, domain boundaries, etc., which are fixed in space. In many other cases they are moving in time: moving point loads, sharp fronts, etc. These physical processes can be turned into the problems of solving large-scale partial differential equations. If we use uniform grids, the space-step and the time-step must be small enough to resolve these problems within a given accuracy. Then the computational cost can be increased a lot and the scope of the problem which can be solved can be limited. Prom 1982, the local refinement technique is put forward in foreign documents [1-5]. The local refinement technique is used on composite grids which are composed of fine grids in some subdomains where the problem possess highly localized properties and coarse grids in the others. Then we can resolve the localized characteristics within a given error tolerance using this technique. At present, the local grid refinement technique has been widely used in several engineering fields. In theory, many scholars have carried out a great deal of research and studied some kinds of approximations with local refinement.Composite grids with local refinement in time and space can be formed by means of regular local grid refinement [6,7]: first, one introduces a global time discretization and a global space discretization for the whole domain; next, in some subdomains, using some adaptive mesh refinement or some a prior information on potential rapid local change of the solution, one introduces local time steps and local space steps that are respectively fractions of the global time step and space step. In this way a composite time-space mesh is introduced. The first problem that arises in such situations is the construction of a stable and accurate approximation. Difficulties arise at the interface between the subregions with different steps. Second, one needs to construct and study efficient solution methods for the system resulting from the composite grid. Dawson and Du [8], on the basis of Galerkin discretization, investigate a domain decomposition procedure; Ewing, Lazarov, Pasciak and Vassilevski [9] use the discontinuous Galerkin method to construct a discretization scheme and investigate an iterative method for...