Dirichlet problem

We now turn to the design of difference schemes for solving the Dirichlet problem in which it is required to find a continuous in G-f P function ii x)... 

Of our initial concern is the Dirichlet problem in the rectangle Go = Go + F for the Poisson equation... 

The Dirichlet difference problem in a domain of rather compHcated configuration. If a solution of the Dirichlet problem needs to be determined in a domain G with a nonlinear boundary, the grid ( f G) is, generally speaking, non-equidistant near the boundary. We describe below such a grid and give the possible classification of its nodes. 

Our purpose here is to construct a difference scheme for solving the Dirichlet problem in the domain G = G + F, the complete posing of which is to find an unknown solution to the equation... 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

In the case of the difference scheme for the Dirichlet problem (24)-(26) of Section 1 the definition (4) of connectedness coincides with another definition from Section 1. The very definition implies that the point P may be boundary and, hence, the connectedness is to be understood that every point of the boundary belongs to the neighborhood Patt [P) of at least one inner node. 

Equations with variable coefficients. The Dirichlet problem for the elliptic equation in the domain G + F = G comes next ... 

The next step is to put the operator Art = =1 w in correspondence with the operator Lu due to which the difference Dirichlet problem... 

In this section we consider higher-accuracy schemes for the Dirichlet problem (1) of Section 1 in a rectangle. 

Iterative methods of successive approximation are in common usage for rather complicated cases of arbitrary domains, variable coefficients, etc. Throughout the entire section, the Dirichlet problem for Poisson s equation is adopted as a model one in the rectangle G = 0 < x < l, a = 1,2 with the boundary P ... 

The statement of the difference Dirichlet problem associated with problem (1) is... 

Omitting more details on this point, we refer the readers to the well-developed algorithm of the fast Fourier transform, in the framework of which Q arithmetic operations, Q fa 2N log. N, N = 2 , are necessary in connection with computations of these sums (instead of 0 N ) in the case of the usual summation), thus causing 0(nilog,- 2) arithmetic operations performed in the numerical solution of the Dirichlet problem (2) in a rectangle. 

A model problem. Comparison of methods. Further comparison of various iterative methods will be conducted by having recourse to the Dirichlet problem associated with Poisson s equation in the square 0 < < 1,... 

ATM for the difference Diidchlet problem. The trace of those ideas can clearly be seen in tackling the difference Dirichlet problem associated with Poisson s equation in the rectangle G = 0 < < / , a = 1,2) ... 

A higher-accuracy scheme in a rectangle. In Chapter 4, Section 5, the Dirichlet problem... 

In the two-dimensional case the iterative alternating direction method or the direct decomposition method turns out to be more economical, but for the multidimensional Dirichlet problem ATM is the most economical one among other available methods. This advantage is stipulated by the special structure of the operator A (see Chapter 4, Section 5) ... 

Adopting those ideas, some progress has been achieved by means of MATAI in tackling the Dirichlet problem in an arbitrary complex domain G with the boundary T for an elliptic equation with variable coefficients ... 

Summarizing, the number of the iterations required during the course of MATM in an arbitrary complex domain is close to the number of the iterations performed for the same Dirichlet problem in a minimal rectangle containing the domain G and numerical realizations confirm this statement. 

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. 

It is worth noting here that the same estimate for no( ) was established before for ATM with optimal set of Chebyshev s parameters, but other formulas were used to specify r] in terms of and A - If R = —A, where A is the difference Laplace operator, and the Dirichlet problem is posed on a square grid in a unit square, then... 

The difiference Dirichlet problem for Poisson s equation in a p-dimen-sional unit cube such as... 

Vol. 1482 J. Chabrowski, The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations. VI, 173 pages. 1991. 

Since values of the fields z and z are fixed on the boundary 9S = C, we deal with the two well-defined two-dimensional (2D) Dirichlet problems. The solutions of the Dirichlet problems fix values of z and z inside S. Another, more singular, gauge-fixing term has been proposed [18]. 

The specific electro-diffusion phenomena, the field and force saturation and counterion condensation, as well as the corresponding features of the solutions to the Dirichlet problem for (2.1.2) to be addressed in this chapter, are closely related to those observed by Keller [7], [8] for the solutions of (2.1.3a) with f tp) positive definite, satisfying a certain growth condition. Keller considered f( 0, satisfying the condition... 

Here G(x, y) is the Green s function for the Dirichlet problem for the Laplace equation in ft and n is the outward unit normal. It is well known that... 

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