An Algorithm for Non-Matching Grid Projections with Linear Complexity

Non-matching grids are becoming more and more common in scientific computing. Examples are the Chimera methods proposed by [20] and analyzed in [2], the mortar methods in domain decomposition by [1], and the patch method for local refinement by [6], and [17], which is also known under the name ’numerical zoom’, see [9]. In the patch method, one has a large scale solver for a particular partial differential equation, and wants to add more precision in certain areas, without having to change the large scale code. One thus introduces refined, possibly non-matching patches in these regions, and uses a residual correction iteration between solutions on the patches and solutions on the entire domain, in order to obtain a more refined solution in the patch regions. The mortar method is a domain decomposition method that permits an entirely parallel grid generation, and local adaptivity independently of neighboring subdomains, because grids do not need to match at interfaces. The Chimera method is also a domain decomposition method, specialized for problems with moving parts, which inevitably leads to non-matching grids, if one wants to avoid regridding at each step. Contact problems in general lead naturally to nonmatching grids. In all these cases, one needs to transfer approximate solutions from one grid to a non-matching second grid by projection. This operation is known in the literature under the name mesh intersection problem in [12], intergrid communication problem in [16], grid transfer problem in [18], and similar algorithms are also needed when one has to interpolate discrete approximations, see [13, Chap. 13].