FEM is a comutational method for approximate solution of engineering and physical problems
Simply because general and exact solutions only exist for simple geometrical shapes and material behaviours while real world applications have complex physical shapes and properties
Mechanics: Solids, Fluids and Thermo-mechanics
Civil, Structures and Geotechnics
Biomechanics
Electromagnetics
Finance
The geometric shapes of the objects (domain of the problem) is divided into a FINITE number of small and well-defined ELEMENTs.
Elements will have EDGES (sides) and NODES (vertices)
A relation to relate the quantities inside the element to those quantities at the nodes are defined (SHAPE FUNCTION)
This local (e.g. stiffness) matrix defines the physical behaviour of its ELEMENT
It relates the involving quantities (e.g. displacement and loads) to those values at the nodes of the elements
Element matrices will be assembled (if time is not involved) into a global matrix for all the elements
Transformation of coordinates from local to global is used for assembling
Boundary Conditions (BCs) are applied at nodes
Global stiffness matrix is modified accordingly
The global matrix equation is solved using comutational linear algebra (libraries) giving values at the nodes
Shape functions are used to determine values throughout elements
Convergence
Stability
Efficiency