Introduction to Finite Element Methods

Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function. Although unknowns are still associated to nodes, the function composed by piece-wise polynomials on each element and thus the gradient can be computed element-wise. Finite element spaces can thus be constructed on general triangulations and this method is able to handle complex geometries and boundaries. Boundary condition is naturally build into the weak formulation or the function space. The variational approach also give solid mathematical foundation and make the error analysis more systematic. Generally speaking, finite element methods is the method of choice in all types of analysis for elliptic equations in complex domains.