Example 5 In a $2013 \times 2013$ grid, some cells are colored black such that any $19 \times 19$ subgrid contains at least 21 black cells. Find the minimum number of black cells.
【Analysis】Notice that, $2013=19 \times 105+18$.
If we color the cells based on a certain condition of coordinates modulo 19, then the number of black cells in each $19 \times 19$ square is the same. By classifying the cells in the $2013 \times 2013$ grid according to the coordinates modulo 19, the category containing $(19,19)$ has $105 \times 105$ cells, the categories containing $(19, y)$ or $(x, 19) (1 \leqslant x, y < 19)$ have $105 \times 106$ cells, and the other categories have $106 \times 106$ cells. Thus, by selecting the 21 categories with the fewest cells, we can ensure the condition of the problem, and the number of black cells is $105 \times 106 \times 20 + 105 \times 105$, denoted as $k$.

Below is the proof that any number of black cells satisfying the problem's condition is at least $k$.

From $19 \times 2013$, we know that the $2013 \times 2013$ grid cannot be completely divided into non-overlapping $19 \times 19$ grids. Each $19 \times 19$ grid (drawer) can ensure at least 21 black cells. Estimating the size of $k$ requires $105 \times 106$ drawers, while allowing a maximum of 105 overlapping cells. These $19 \times 19$ grids can be arranged as follows: below the diagonal from the top left to the bottom right, place the drawers from left to right and from bottom to top until they do not exceed the diagonal; above the diagonal, place the drawers in the positions reflected about the diagonal from the already placed drawers. It is easy to verify that this arrangement places $105 \times 106$ $19 \times 19$ grids with 105 overlapping cells on the diagonal. The black cells in the overlapping cells are counted twice.

Therefore, the number of black cells in the overlapping cells plus the total number of black cells is at least $105 \times 106 \times 21$, so the total number of black cells is at least $105 \times 106 \times 21 - 105 = k$.