Analysis of the Implicit Euler Local Uniform Grid Refinement Method

Attention is focused on parabolic problems having solutions with sharp moving transitions in space and time. An adaptive grid method is analysed that refines the space grid locally around sharp spatial transitions, so as to avoid discretization on a very fine grid over the entire physical domain. This method is based on static-regridding and local uniform grid refinement. Static-regridding means that for evolving time the space grid is adapted at discrete times. Local uniform grid refinement means that the actual adaptation of the space grid takes place using nested locally and uniformly refined grids. The present paper concentrates on stability and error analysis while using the implicit Euler method for time integration. Maximum norm stability and convergence results are proved for a certain class of linear and nonlinear partial differential equations. The central issue is a refinement condition with a strategy that distributes spatial interpolation and discretization errors in such a way that the spatial accuracy obtained is comparable to the spatial accuracy on the finest grid if this grid would be used without any adaptation. The analysis is confirmed with a numerical illustration.

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