New in Mathematica 10
Solving Partial Differential Equations with Finite Elements

Classical Partial Differential Equations

The Coefficient Form of Partial Differential Equations

What types of equations can be solved with the finite element method as implemented in NDSolve? Consider a single linear partial differential equation in :

The PDE is defined in . Here is the dependent variable for which a solution is sought. The coefficients , , , and are scalars; , , and are vectors; and is an × matrix.

Systems of Partial Differential Equations

The Coefficient Form of Systems of Partial Differential Equations

Consider a system of two equations with dependent variables and :

For a region in , each is an × matrix, and each of , , and are vectors.

Sources: http://reference.wolfram.com/language/FEMDocumentation/tutorial/SolvingPDEwithFEM.html#509267359 

http://reference.wolfram.com/language/FEMDocumentation/tutorial/FiniteElementOverview.html