A two-dimensional adaptive mesh generation method

Local block refinement with a multigrid flow solver

“…Other authors couple reÿned and non-reÿned regions at the same level, computing the whole grid simultaneously [23][24][25][26][27][28]. Special discretisation schemes are required for the coarse-ÿne grid transition.…”

An adaptive level set method

“…Such an effect is, no doubt, often the case, and that probably accounts for the success of similar methods [3] in reducing the L2 norm of the error for specific problems, but such limited successes in no way guarantee an universal applicability.…”

“…The fact that was noted about the peripheral points in [3] is that they are all surrounded by four uniformly spaced points. For "edge" children, these points lie along the coordinate axes; for "corner" children, they are rotated by %.…”

“…For the curvature evolution equation (Eq-1.5) the inclusion of the cross derivative presents a problem, since there does not exist a rotated frame for the peripheral cells which provides all of the points needed for the standard central difference approximations. The technique of [3] did not allow cross derivatives for this reason, but it can be extended in the following manner. Fig-6.3 shows two stencils (for the model problem) from which the cross derivative can be approximated by the centered but unproportionately distributed "corner" points.…”

“…Chapter 6 A Second Technique Altas and Stephenson mention a technique in [3] which can be adapted here. It is restricted in its application, however, because of the involvement of a trick that only applies to refinement ratios of two and only to a different style of refining the mesh than the one discussed in Chapter 3.…”