Patch-adaptive multilevel iteration (Q1371678)

The present paper is devoted to an adaptive mesh construction for elliptic differential equations. The authors introduce virtual global grids as a systematic way to view the adaptive mesh refinement. These ideas are modified and extended to obtain a practical highly efficient implementation, to construct an adaptive system with as few restrictions as possible, and to stay as close as possible to the performance which can be obtained on a uniform mesh data structure. The basic idea of the proposed technique is based on using patches, and the constructed patch-adaptive multigrid method is intended to be a compromise between rigid uniform mesh multigrid and adaptive multigrid based on unstructured grids. The idea of patch-adaptive multigrid is the partitioning of the grid layers in a number of non-overlapping subgrids, called patches. Each patch has its own uniform mesh of grid points. Within the patch this permits a dense storage format for all grid points in arrays such that numerical algorithms can run efficiently. Some computational experiments are demonstrated on a model problem to illustrate the features of the proposed method. Experiments show that the patch-adaptive approach requires less points when the patch-size become smaller (or, by extrapolation, when an unstructured grid is used), but this advantage is only significant for a small problem size, when accuracy is low.