### Table 1: Finite element mesh description.

1998

"... In PAGE 19: ... To further ensure a fair comparison, the atlas was rst registered to the subject volumes using the same principal axes method applied in [1]. Using trilinear hexahedral elements, the volumes at the 3 lowest spatial scales were subdivided into 3 3 3 element domains and the volume at the nest scale into contiguous 5 5 5 regions (see Table1 ). In contrast, the same mesh size, h = 1 voxel, was used in the nite di erence implementation at each resolution level.... ..."

Cited by 6

### Table 1: Comparison of refinement techniques for unstructured meshes. An X indicates that the method is capable of refining the mesh entity.

2004

"... In PAGE 2: ... Parallel sheet refinement divides each edge into two, splitting a single element into eight hexes. As summarized in Table1 , both single sheet and parallel sheet techniques are limited as to the mesh entities each can refine. By combining the two techniques into a hybrid method and coupling it with the single sheet technique, all mesh entities within a mesh are then capable of refinement.... ..."

Cited by 1

### Table 3: Preconditioned convergence factors for the multi-level method (accelerated by CGS) applied to the `staircase apos; problem.

1997

"... In PAGE 20: ...and, as discussed in [27], re ects the error better than that of the residual itself since the preconditioned system is better-conditioned than the original one.) The results in Table3 are as expected from Theorem 1 in the sense that the preconditioned convergence factor increases slowly with L so long as L 3. The deterioration of the convergence rate when L = 4 is used is due to the large p3 obtained in this case, which implies a large upper bound in (18).... ..."

Cited by 2

### Table 3: Preconditioned convergence factors for the multi-level method (accelerated by CGS) applied to the `staircase apos; problem.

"... In PAGE 27: ... This is an advantage of left preconditioning over right and symmetric ones. The results in Table3 are as expected from Theorem 1 in the sense that the preconditioned convergence factor increases slowly with L so long as L 3. The deterioration of the convergence rate when L = 4 is used is due to the large p3 obtained in this case, which implies a large upper bound in (21).... ..."

### Table 1 Performance of CWS method for (4.18 on unstructured meshes CWS method quot; h #nodes

73

"... In PAGE 6: ... We call this kind of method to be the cross-wind strip (CWS) domain decomposition method. Table1 shows the performance of CWS method for problem (4.18) discretized by our monotone nite element scheme on unstructured grids on the unit square.... ..."

### Table 4. Finite element problem

"... In PAGE 14: ... The matrices were generated by the nite element code VECFEM, see [3]. Table4 shows the reduction of the calculation time by at least a factor of 5. The heavy oscillations of the timings are e ected by the random sample of the values for k.... ..."

### Table 1: The hook methods for the mesh element class in a four-point mesh.

2002

"... In PAGE 6: ... This is an example of error-reduction due to code generation that is discussed later in the paper. In general, there are nine different location-dependent mesh element methods that replace this single operation method, and they are shown in Table1 . However, for a given topology, the generated Mesh framework provides only a subset of these operation methods.... In PAGE 6: ... This Mesh template parameter determines the argument set for the operation methods that are generated. Table1 shows only four arguments for the four immediate neighbors. To support an eight-point mesh, the generated method code must have eight arguments instead.... In PAGE 7: ... The operate() method is also mapped to each mesh element, but the mapping results in one of nine appropriate operation methods being invoked on the element depending on the mesh topology and the location of the element. Table1 indicates all of these possible methods. The barrier() method does not map directly to each mesh element.... In PAGE 8: ... The fifth is to implement a method that applies a programmer-defined reducer object to a mesh element. The list of methods is shown in Table1 , where the methods in each row implements one of the tasks. For example, in the sample reaction-diffusion problem, the mesh element class is MorphogenPair.... ..."

Cited by 39

### Table 5: A Multilevel Scheme for Engineering Optimization Calculations

"... In PAGE 22: ... Once the location of an optimal value is determined, a high-fidelity model is used to determine its converged value. This methodology resembles the scheme proposed in Section 8 ( Table5 ) in that it provides a way to identify candidate optima ( regions of interest ) followed by local refinement of the candidate optima.... In PAGE 41: ... 8.3 A Multilevel Scheme for Engineering Optimization Calculations We therefore propose the following scheme for performing practical optimization calculations using a mesh-based engineering model (such as a finite-difference or finite- element code) ( Table5 ). The concept of the scheme is to use a coarser mesh to identify candidate optima, and then improve the objective function values at the candidate optima using a finer mesh, followed by a final ranking of the optima based on the converged value of the objective function.... In PAGE 49: ...) One must also be able to find a mesh which is fine enough to give reasonable approximations to the local optima, but coarse enough to run on an available platform in a practical amount of time. Having thus an improved CTH model for the BRL 81-mm shaped charge and an improved scheme for locating the jet tip, we employed the scheme presented in Table5 to find an optimum solution to the Sandia wave-shaper optimization problem.... ..."