On Linear Differential Equations

Abstract

Application of the Finite Element Method to the solution of linear differential equations leads to a system of linear algebraic equations of the form b Ax  ; with non-linear differential equations one arrives at a system of non-linear equations, which cannot be solved by elementary elimination methods. Thus, much of the focus here is on methods of solving the resulting systems of FE non-linear equations. 5.1 Methods for the Solution of Non-Linear Equations There are a number of basic techniques for solving non-linear equations. For example, there are the 1. Substitution method 2. Newton-Raphson method 3. Incremental (step by step) method-Initial Stress Method-Modified Newton-Raphson method The Substitution Method is not commonly used, but is given in the Appendix to this Chapter. The Newton-Raphson method is the primary solution scheme for the non-linear equations which arise in the FEM and will be discussed in detail. 5.1.1 The Newton-Raphson Method Consider first the one-dimensional case: the non-linear equation () 0 R u  , (5.1) whose exact solution is () e u. Suppose one has an initial estimate of the solution, (0) u. Using a Taylor expansion and dropping higher order terms, (0) () (0) () () e u R R u R u u u     , (5.2)