The elements must have enough approximation power to capture the analytical solution in the limit of a mesh refinement process. The element shape functions must represent exactly all polynomial terms of order ≤ m in the Cartesian coordinates. A set of shape functions that satisfies this condition is called m-complete. Note that this requirement applies at the element level and involves all shape functions of the element. The completeness is satisfied if the sum of the shape functions is unity and the element is compatible.
Compatibility.
The shape functions should provide displacement continuity between elements. Physically these insure that no material gaps appear as the elements deform. As the mesh is refined, such gaps would multiply and may absorb or release spurious energy.
Stability.
The system of finite element equations must satisfy certain well posedness conditions that preclude non-physical zero-energy modes in elements, as well as the absence of excessive element distortion. Stability may be informally characterised as ensuring that the finite element model enjoys the same solution uniqueness properties of the analytical solution of the mathematical model. For example, if the only motions that produce zero internal energy in the mathematical model are rigid body motions, the finite element model must inherit that property. Since FEM can handle arbitrary assemblies of elements, including individual elements, this property is required to hold at the element level. Completeness and compatibility are two aspects of the so-called consistency condition between the discrete and mathematical models. A finite element model that passes both completeness and continuity requirements is called consistent. This is the FEM analog of the famous Lax-Wendroff theorem, which says that consistency and stability imply convergence
