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6 Adaptive Methods     continued...

Thus we can derive the standard linear interpolation formulas on a triangle which represent the first two terms of the Taylor's series expansion. This means that the error due to discretization (from using linear elements) is proportional to the third term of the Taylor's expansion:

where is the exact solution. We can conjecture, then, that the error due to discretization for first order linear elements is proportional to the second derivative. If is a linear function over the element, then the first derivative is a constant and the second derivative is zero and there is no error due to discretization. This implies that the gradient must be constant over each element. If the function is not linear, or the gradient is not constant over an element, the second derivative will not be zero and is proportional to the error incurred due to ``improper'' discretization. Examining (80) we can easily see that one way to decrease the error is to decrease the size of h and k. As h and k go to zero, the error tends to zero as well. Thus, decreasing the mesh size in places of high errors due to high gradients decreases the error. As an aside, we note that if one divides equation (15) by hk, one can also express the error in terms of the elemental aspect ratio , which is a measure of the relative shape of the element. It is easy to see that one must be careful to maintain an aspect ratio as close to unity as possible.