Comer boundary functions

To compensate for these discrepancies we introduce comer boundary functions. One corner boundary function must correspond to each of the vertices of the rectangle, and therefore P contains terms of four types ... 

Two types of comer boundary functions appear near each of the vertices (0, b) and (a, b). Let us consider them, for example, for the vertex (0, b). Comer boundary functions of the first type [we will denote... 

Note that the comer boundary functions Po(fj l) Pi( >v) have... 

As in the case of elliptic equations, comer boundary functions ) and t) are needed to compensate for the discrepancies introduced... 

By virtue of the corresponding exponential estimates, the boundary layer function Ilo and the comer boundary functions and P% become arbitrarily small for l 5 > 0, and an approximation of the solution with an accuracy of order s (such an approximation is often sufficient for practical purposes) is given by the sum... 

In the case when the boundary of the domain is no longer smooth, but contains corner points, the structure of the asymptotic solutions becomes more complicated in vicinities of these points. The boundary layer functions constructed in the previous section are not sufficient to describe the asymptotic behavior of the solution near the comers. We need to introduce a new type of boundary layer functions, corner boundary functions, in the vicinities of the comer points. We begin our discussion with a problem for the same elliptic equation as in Section VI, but now the domain ft is a rectangle, whose boundary has four comer points. 

Note that, due to the estimate (7.11), the functions P,(f> ) significant only in some small vicinity of the comer point (0,0) of the rectangle ft. These functions vanish exponentially as their arguments move away from the point (0,0). This explains the name of these functions—corner boundary functions. 

This problem is shown in Figure 4.5. The feasible region is defined by linear constraints with a finite number of comer points. The objective function, being nonlinear, has contours (the concentric circles, level sets) of constant value that are not parallel lines, as would occur if it were linear. The minimum value of/corresponds to the contour of lowest value having at least one point in common with the feasible region, that is, at xx = 2, x2 = 3. This is not an extreme point of the feasible set, although it is a boundary point. For linear programs the minimum is always at an extreme point, as shown in Chapter 7. 

Figure E.l represents a highly simplified view of an ideal structure for an application program. The boxes with the rounded borders represent those functions that are problem specific, while the square-comer boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically systems of nonlinear algebraic equations, ordinary differential equation initiator boundary-value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the analyst must write the subroutine that describes the particular system of equations. Moreover, for most numerical-solution algorithms, the system of equations must be written in a discrete form (e.g., a finite-volume representation). However, the equation-defining sub-...

The central approach to constmction is to iterate over each comer of the current polytope P introduce new hyperplanes that eliminate unattainable space. Sharp corners of the polytope are slowly smoothed out by the introduction of additional bounding planes, and the level of accuracy obtained is hence a strong function of the number of unique hyperplanes that are introduced. Curvature of a region, such as that generated by a PFR manifold on the AR boundary, may be approximated by the use of many bounding hyperplanes. 

The origin is in one comer of the cube and the coordinate axes along the edges. It is easily seen that sin(kx) is a solution of Equation 5.47. This function also satisfies the boundary condition at x = 0. To satisfy the boundary condition at x = L, we must have... 

Here we use the old notation y for the new variable y. We assume that all the given functions are sufficiently smooth, that a >0, and that the initial and boundary values (10.6) are matched at the comer points, that is,... 

This casts Equation 9-2a in a linear-like form for p , which facilitates numerical analysis, noting that full nonlinearity is assumed in this book. Now, let us consider the function p " (x.y.z,t) and its image P (, ri,z,t) in transformed curvilinear (, ri,z) coordinates. Such transformed coordinates are often called comer point geometries in the reservoir simulation literature for their ability to adapt to stratigraphic boundaries. In particular, we will examine the combination keeping the function p together as a... 

The function starts with the initialization section, which checks the inputs and sets the values required in the calculations. The solution of the equation follows next and consists of an outer loop on time interval. At each time interval, values of the dependent variable for inner grid points are being calculated based on Eq. (6.66), followed by calculation of the grid points on the boundaries according to the formula developed in the previous section. The values of the dependent variable at comer points are assumed to be the average of their adjacent points on the converging boundaries. 

---