Patch Test

For the patch test, we select the following 6 problems shown in Figures 17.1(a) to 17.1(f).

Figure 17.1 Problems for Patch Test

All these problems have exact solutions. For problems (a) and (b), the exact solutions are given by :

,     for problem (a),
,    for problem (b).	(17.1)

The exact solution is a linear polynomial. Therefore, we choose p = 1 as the order of approximation. We obtain the FE solution using the FE code with p = 1 and number of elements as one. Then, we compare the nodal FE values with the exact values given by eq. (17.1).

       For problems (c) and (d), the exact solutions are given by :

,    for problem (c),
,  for problem (d)	(17.2)

The exact solution is a quadratic polynomial. Therefore, we obtain the FE solution using p = 2 (for one-element mesh) and compare the nodal FE values with the exact values given by eq. (17.2).

        For problems (e) and (f), the exact solutions are given by :

,   for problem 9(e),
,   for problem 9(f)	(17.3)

The exact solution is a cubic polynomial. Therefore, we obtain the FE solution using p = 3 (for one element mesh) and compare the nodal FE values with the exact values given by eq. (17.3)