ABSTRACT
In this chapter we present a technique that uses mimetic operators to build mimetic schemes on nonuniform structured meshes, thereby achieving the same accuracy for the solution in the entire domain (as is the case for the Castillo{Grone operators). By utilizing the idea proposed by D. Batista and J. Castillo [161], we are
able to locally transform the various cells, as opposed to transforming the entire mesh all at once. The combined use of the 1-D div and grad high-order operators, and local transformations allows us to avoid the normal diculty of keeping the order of accuracy p, for p 3, which occurs when using a global reference mesh. As in the case of the 1-D uniform staggered grid addressed earlier, we now
consider a geometric mesh U , with the cells of this mesh being the intervals [xi1; xi], with i = 1; 2; :::; n. The edges of the cells in U are xi, with i = 0; 1; :::; n, and are called G-points, because they are the points at which we dene the gradient operator G. We also dene the divergence operator D at the centers that can be described as: xi+1=2 of these cells, calling them D-points, and xi+1=2 = (xi + xi+1)=2. As with the uniform grid, the complete set of all G-points and D-points
is called a staggered grid. The cell [xi1; xi] is referred to as having the cell number (i 1=2). For the second-order, one-dimensional operators D and G, we must dene the elements CD and CG (which are explained in the following section).
