Nonlinearity

In the previous chapters we have addressed the solution of linear differential equations, and established a finite element methodology that can be applied to a wide variety of linear problems. However, in many situations the mathematical models lead to nonlinear equations or systems of equations representing the physical system that cannot be reduced to linear approximations without oversimplification. We must therefore extend our method to encompass the solution of nonlinear problems. This represents a formidable challenge. Unlike the linear case where general theories can be developed that apply to all problems governed by the same type of equations, individual nonlinearities require different algorithms; the nonlinearity itself can be of many different types. It is difficult to pretend that we can address all kinds of nonlinearities here. In fact, we will restrict ourselves to a few important techniques that are applicable to a large number of situations. These are nonlinear material properties (coefficients); product nonlinearities like the convective nonlinearity in the Navier-Stokes equations; transcendental functions of the dependent variable such as the exponential dependence in chemical reactions; and nonlinearities occurring on the boundaries, as in the case of radiative heat transfer. We will not address other important cases such as the determination of the position of a free surface or nonlinear constitutive relations, even though these are also important subjects. Such topics are complex and would 210require lengthy treatment. Our intent is that readers interested in nonlinear topics not covered in this book will still be able to obtain sufficient information and references to be able to undertake them on their own.