Elliptic equations method

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

In this chapter economical direct and iterative methods are designed for numerical solution of difference elliptic equations. 

Direct and iterative methods. Recall that the final results of the difference approximation of boundary-value problems associated with elliptic equations from Chapter 4 were various systems of linear algebraic equations (difference or grid equations). The sizes of the appropriate matrices are extra large and equal the total number N of the grid nodes. For... 

In the preceding sections this trend of research was due to serious developments of the Russian and western scientists. Specifically, the method for solving difference equations approximating an elliptic equation with variable coefficients in complex domains G of arbitrary shape and configuration is available in Section 8 with placing special emphasis on real advantages of MATM in the numerical solution of the difference Dirichlet problem for Poisson s equation in Section 9. 

Gibrid (combined) methods. In mastering the difficulties involved in solving difference elliptic equations, some consensus of opinion is to bring together direct and iterative methods in some or other aspects as well as to combine iterative methods of various types (two-step methods). All the tricks and turns will be clarified for the iteration scheme... 

Allen and Severn (A3, A4) demonstrate how relaxation methods, originally developed for elliptic partial differential equations, can be extended to the heat conduction equation. With elliptic equations, the value of the dependent variable at any mesh point is determined by all... 

This paper reports the mathematical modelling of electrochemical processes in the Soderberg aluminium electrolysis cell. We consider anode shape changes, variations of the potential distribution and formation of a gaseous layer under the anode surface. Evolution of the reactant concentrations is described by the system of diffusion-convection equations while the elliptic equation is solved for the Galvani potential. We compare its distribution with the C02 density and discuss the advantages of the finite volume method and the marker-and-cell approach for mathematical modelling of electrochemical reactions. 

Incompressible steady flows are commonly solved by pressure-based methods and methods based on the concept of artificial compressibility [183, 45]. The extension of pressure correction methods to steady flows, generally elliptic equations, has been performed by Patankar and Spalding [140] and Patankar [141]. The artificial compressibility method for calculating steady incompressible flows was proposed by Chorin [29]. In this method, an artificial compressibility term is introduced in the continuity equation, and the unsteady terms in the momentum equations are retained. Hence, the system of equations becomes hj perbolic and many of the methods developed for h3rperbolic systems can be applied. 

Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources./ Comput. Phys., 213(l) l-30,2006. 

Note the definition of the parameter p in Eq. 12.1786. Since our main concern is to solve elliptic equation, that is, the steady-state solution to Eq. 12.176, the solution can be obtained effectively by choosing a sequence of parameters p (Peaceman and Rachford 1955). The optimum sequence of this parameter is only found for simple problems. For other problems, it is suggested that this optimum sequence is obtained by numerical experiments. However, if the scalar parameter p is kept constant, the method has been proved to converge for all values of p. Interested readers should refer to Wachspress and Habetler (1960) for further exposition of this method. 

The previous section showed how straightforward the orthogonal collocation can be when solving partial differential equations, particularly parabolic and elliptic equations. We now present a variation of the orthogonal collocation method, which is useful in solving problems with a sharp variation in the profiles. 

Beginning with this section, we will now consider partial differential equations. Let us consider a general approach for constructing asymptotic expansions of the solutions of singularly perturbed linear partial differential equations, which was proposed in the well-known fundamental work of Vishik and Lyusternik [27]. We will illustrate the idea of this approach (known in the literature as the method of Vishik-Lyusternik) on a simple example of an elliptic equation in a bounded domain with smooth boundary. 

We illustrated the method of Vishik-Lyusternik with the example of a linear elliptic equation. Note that even for this simple problem it is impossible to find the solution explicitly for an arbitrary coefficient k(x, y)... 

The method of corner boundary functions is well developed also for equations of hyperbolic type [29], for systems of elliptic equations [30], for systems of parabolic equations [31], for partial differential equations in the multidimensional case [32], as well as for difference equations [33]. This method works successfully for a variety of applied problems. 

Gourlay AR, McKee S (1977) The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative. J Comput Appl Math 3 201-206... 

So far, we have not imposed any constraints on the functions x = x(, r ) and y = y( ii), or their inverses = (x,y) and p = ri(x,y). One well-known transformation is Thompson s mapping, discussed in Thompson (1978, 1984), Thompson, Warsi, and Mastin (1985), White (1982), and Tamamidis and Assanis (1991). It was originally developed to solve the Navier-Stokes equations for viscous flows past planar airfoils, and later extended to three dimensions to study wing-fuselage effects in aerospace applications (Thomas, 1982). This method was also used in Chin (1992a,b 2001A,B) to study non-Newtonian flows in eccentric annuli and noncircular pipes. In Thompson s approach, (x,y) and r (x,y) are defined as solutions to the elliptic equations... 

Pressure solution. Next, consider the corresponding pressure field. We recall from Equations 12-2 and 12-4a that g(x,y,z) = p(x,y,z) Vk(x,y,z) satisfies 9 g/9 + g/9y + g/9z = 0. If we assume that both the permeabilities and pressures are known at all well positions and boundaries, it follows that g = pVk can be prescribed as known Dirichlet boundary conditions. Then, the numerical methods devised in Chapter 7 for elliptic equations can be applied directly on the other hand, analytical separation of variables methods can be employed for problems with idealized pressure boundary conditions. The general approach in this example is desirable for two reasons. First, the analytical constructions devised for the permeability function (see Equations 12-5b, 12-10, and 12-11) allow us to retain full control over the details of small-scale heterogeneity. Second, the equation for the modified pressure g(x,y,z) (see Equation 12-4a) does not contain variable, heterogeneity-dependent coefficients. It is, in fact, smooth thus, it can be solved with a coarser mesh distribution than is otherwise possible. 

While we have demonstrated von Neumann s test for the transient heat equation, the stability test applies equally to iterative methods for elliptic equations describing steady flows. The (artificial) time levels t and t + At would refer to the approximate solutions obtained at consecutive iterations. The pressure solvers in Chapter 7 are examples of simple elliptic solvers that are stable in von Neumann s sense. Recall that the iterative method applied to single wells as it did to line fractures. Such a robust algorithm can be used to model general multilateral well drainhole trajectories where the overall topology can be arbitrarily defined by the driller or reservoir engineer. 

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