ABSTRACT
Up to this point, we have only considered equations whose independent variables are spatial variables in one or two (and three) dimensions and are independent of time. Therefore, their solutions require only discretization in space, which is accomplished using the techniques discussed so far. If the solutions change with time, the spatial discretization is no longer sufficient, and we must follow the evolution of the solutions in the time dimension. We will now address the basic techniques used to model time-dependent problems that are parabolic or first-order hyperbolic using finite elements, beginning with the time-dependent heat diffusion equation and working our way up to the nonlinear Navier-Stokes equations in subsequent chapters. We will also touch briefly on second-order hyperbolic equations. Implicit and explicit schemes will be discussed in terms of accuracy and stability properties. The concepts of mass matrix and lumped mass matrix are introduced. These appear when time derivatives are present in the governing equations.
