After finite-element assembly and application of boundary conditions, a linear model has the algebraic form

$$\mathbf K\mathbf U=\mathbf F.$$

Solving it gives the nodal degrees of freedom. Because finite-element connectivity is local, $\mathbf K$ is usually sparse and should be handled with methods appropriate to sparse linear systems.

From nodal values to derived fields

Structural nodal displacements are interpolated inside each element. Differentiating the displacement approximation gives strain, and the material relation then gives stress.

Quantities such as principal stress or von Mises equivalent stress are therefore derived outputs, not independent unknowns of the basic displacement formulation.

Mesh convergence

A quantity of interest should be recomputed on successively refined meshes or with higher-order elements. The general theory of numerical convergence explains what it means for these approximations to approach a stable value; FEM adds the practical question of where and how to refine the mesh.

Local stress singularities or poorly chosen quantities of interest may fail to converge in a useful way even when global quantities behave well.

Numerical convergence only checks the discretized mathematical model. It does not validate the loads, geometry, boundary conditions or material assumptions of that model.