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6.0.2 Energy Norms     continued...

Two other methods, the p and the hp methods, have been found, in most cases, to converge faster. The p method of refinement requires that one increase the order of the basis function that was used to represent the interpolation (i.e. linear to quadratic to cubic, etc.). The hp method is a combination of the h and p methods and has recently been shown to converge the fastest of the three methods (but, as you might imagine, it is the hardest to implement). Take a minute to estimate the number of additional elements that will be added for each iteration and the total number of elements for say, five iterations of the adaptive algorithm. Remember that this is a two-dimensional problem and that the data you are working with is only one of 116 such MRI sections. You should be able to get an idea of how the size of the problem increases with accuracy. Obviously, due to limited CPU, memory, and time, one has to make some choices concerning accuracy versus computational costs. One possible practical solution to this problem is to calculate the number of nodes that your particular workstation can handle and then refine only up to that particular number.

(See exercise 8.)