COMPUTATIONAL ASPECTS OF THE FINITE ELEMENT METHOD

Publisher Summary This chapter focuses on the computational aspects of the finite element method. The finite element procedure discussed in the chapter uses basis functions consisting of piecewise bicubic Hermite polynomials defined on a mesh which is refined in a well-defined manner in a neighborhood of each corner. The coefficients and right-hand side of the resulting linear algebraic system of equations involve integrals over two-dimensional rectangular elements that are approximated by the local nine-point product Gaussian quadrature scheme —the tensor product of the one-dimensional three-point Gaussian quadrature schemes. Finally, the approximate linear algebraic system of equations is symmetric and positive definite and is solved by either the band Cholesky or profile Cholesky decomposition procedure. The chapter presents theoretical justifications for the procedure used in the chapter. It is shown that asymptotically the procedure used in the chapter is far more efficient than the combination of the five-point central difference approximation and successive overrelaxation (SOR).