CMES-2018 A highly accurate corrected scheme in solving the Laplace ' s equation on a rectangle

A pointwise error estimation of the form OO(ρρh8), h is the mesh size, for the approximate solution of the Dirichlet problem for Laplace's equation on a rectangular domain is obtained as a result of three stage (9-point, 5-point and 5-point) finite difference method; here ρρ = ρρ(xx, yy) is the distance from the current grid point (xx, yy) ∈ ΠΠh to the boundary of the rectangle ΠΠ.


Introduction
A highly accurate corrected scheme is one of the compact approximations for decreasing the number of unknowns, that gives the effective numerical solution of differential equations.
In [1] a two-stage difference method was given on the way of solving the Dirichlet problem for Laplace's equation on a rectangular parallelepiped.It was assumed that the sixth order derivative of the boundary functions which are given on the faces of a parallelepiped satisfy the Hölder condition, and on the edges, besides the continuity they satisfy the compatibility condition for second derivatives, which results from the Laplace equation.It was proved that by using a simple 7-point scheme in two stages the order of uniform error can be improved up to (ℎ⁴ ℎ⁻¹).
In this paper, a new three-stage difference method to solve the Dirichlet problem for Laplace's equation on the rectangular domain is given.By using at the first stage the 9-point, and at the second and the third stages the 5point schemes for error of the approximate solution a pointwise estimation of order (ℎ⁸) is obtained, where  = (, ) is the distance from the current grid point (, ) ∈  ℎ to the boundary of the rectangle .Numerical experiment is given to support the theoretical analysis made.
The weighted estimates for the approximate solutions by the finite difference method of the Dirichlet problem for the Laplace and Poisson equations, at the rate as ( + ℎ)  ℎ², 0 <  ≤ 1, were obtained in [2].A highly accurate finite difference approximations by correcting the right hand term using the high order differences of the numerical solution of differential equation without the weighted estimates was constructed in [3], [4].
It is easy to check that for each of  = 4,8 the function ( 4) is the solution of following boundary value problem where From ( 2), ( 3) and ( 8) it follows that the following (12 − ) − ℎ order derivatives are bounded on : Let ℎ > 0, /ℎ ≥ 8  /ℎ ≥ 8 where /ℎ and /ℎ are integers.We define  ℎ a square grid net on  with step size ℎ, obtained by the lines ,  = 0, ℎ, 2ℎ, . ... Let   ℎ be a set of nodes on the interior of   , and let Let the operator   ,  ∈ {0,1} be defined as follows: (, ) = ( We consider for the approximation of problems ( 5)-( 7) by the following system of difference equations: By the maximum principle (see [5], Chap.4), problems (11) and ( 12) have the unique solution.
Consider the following systems: where   ,  ∈ {0,1} is the operator defined by ( 10

Proof.Lemma 1 follows by the Comparison theorem (see Chapter 4 in [5]).∎
In what follows and for simplicity , ₀, ₁, ... defined the positive constants which are independent of ℎ and the nearest factor, the similar notation is eonvenient for varied constants.

Lemma 2. For the solution of problem
the following inequality holds: where  is a positive constant defined by follows, ℓ > 2 is an integer,  = {, },   = (, ) is the distance between the boundary of the rectangle  and the current point (, ) ∈  ̅ ℎ .
3 Approximation of the solution using three stage method Lemma 3. The estimation holds where  ℎ  is the solution of system (11) when  = 4, and of system (12) when  = 8,   is the trace of exact solution of problem ( 5)-(7) on  ̅ ℎ .

Proof.
Let It is obvious that Theorem 1.On  ℎ , the pointwise estimation is true where  is the trace of the problem defined by (1) on  ℎ ,  ℎ is a solution of system (24).
We put where  is the solution of Dirichlet problem (1) and  ℎ is the solution to the system (24).
In the view of (24), ( 26) and (29), we get  Remark 1.The proposed method can be generalised for the mixed boundary value problem for Laplace's equation.Furthermore, the idea of the proposed method can be applied to increase an accuracy of the finite difference approximation of KdV equation used in [6].