The finite element method is a numerical method for solving partial differential equations (PDEs), and has become particularly popular in engineering and physics. More recently, it has also gained interest in the quantitative finance community as a valuable tool to solve PDEs that arise in modeling financial instruments. The basic strategy of the finite element method is to divide the region of interest into smaller parts (finite elements), and to approximate the solution in each element by a simple function. The method can thus be seen as a piecewise approximation. Polynomial-type interpolation functions are most widely used for the element solutions (also called approximating functions or interpolation functions), due to
Although other types of interpolation functions, such as trigonometric functions, can be used, we restrict ourselves to polynomial functions in this book. The interested reader is referred to (Zienkiewicz and Taylor, 2000a) and (Segerlind, 1984) and the references therein.
If an approximate solution (trial function) is substituted into a differential equation a residual remains, since the approximate solution does not satisfy the equation. Consider, as an example, the one-dimensional second order partial differential equation