Using two different variational formulations some special finite elements as for example elements with circular and elliptic holes, with internal and external cracks have been developed and tested numerically. These elements are problem adapted as the trial functions satisfy not only the governing differential equations but also some boundary conditions on such influential boundary portions as crack or hole surfaces. So the local character of a solution is taken into account appropriately. Even problems with singularities as crack and sharp corner problems for example can be treated numerically in an effective manner. In both methods used one obtains symmetric element matrices which can be evaluated via simple numerical integrations along the element boundaries. The special problem-adapted finite elements can be coupled with conventional elements. The discussion of the special elements includes the variational formulations and element coupling procedures as well as the construction of special trial functions for 2- and 3-dimensional elasticity problems.