Higher-Order Elements

Abstract

In the earlier seven chapters, we solved by the Green element method a variety of engineering problems in 1-D. spatial dimensions ranging from steady to transient, linear to nonlinear, and from those which apply in homogeneous to heterogeneous media. For all these problems, line segments were used to discretize the physical domain, and functional quantities were approximated by linear interpolation polynomials within each line segment (element). These interpolating polynomials have zero-order continuity in the sense that only the functional quantity is continuous, while its first derivative is discontinuous across elements. Where the variation of the functional quantities with respect to the spatial dimension is marginal, the use of linear interpolation can be adequate in approximating these functional quantities. However, there are certain problems, earlier encountered in chapters 6, 7, and 8, whose solutions have significant spatial gradients, and for which the use of linear interpolation is inadequate. In chapter 6, such a problem is the advection-dominant transport one that exhibits the unique feature of retaining the initial concentration profile as time progresses, so that steep gradients in the initial concentration profile persist throughout the solution history. The same can be said of the momentum transport problem of chapter 7 when viscous effects are negligible. For the unsaturated flow problem which we encountered in chapter 8, we are faced with steep gradients of the soil constitutive relations and that of the solution variable.