TABLE OF CONTENTS The finite element method, developed for computations in structural mechanics in the 1950's, has been a subject of numerous monographs written with various levels of mathematical sophistication and application details. The classic book [Zienkiewicz and Taylor, 2000] provides a comprehensive overview of the method with emphasis on engineering applications and [Becker et al., 1981] and [Reddy, 1993] provide good introductions for those new to the field. The texts [Braess, 2001], [Brenner and Scott, 2002], and [Ciarlet, 2002] are more mathematically oriented.
The finite element method may be defined as a special type of Galerkin approximation ( c.f., Section 3.2.4.2), in which the finite-dimensional sub-space S h of the appropriate Sobolev space is subject to three additional requirements. First, the region ? in which the solution is sought, is partitioned into a set of polyhedral elements designated by { 
}. Strictly speaking, since the elements { 
} depend on the selected approximation, they should be denoted by { 
}. Typically elements are of triangular or quadrilateral shape in two-dimensional problems and triangular prisms, tetrahedra, or hexahedra for three-dimensional problems. Any two different elements within the domain either do not meet at all or meet only at vertices, edges, or faces. Second, the basis functions spanning the approximation space S h are piecewise polynomials in the sense that the restriction of each basis function to an element D e is a polynomial. Third, there should exist a basis in S h
