NAG Library Routine Document

d03eaf  (dim2_laplace)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{stage1}$ – LogicalInput

On entry: indicates whether the routine call is for stage $1$ of the computation as defined in Section 3.

$\mathbf{stage1}=\mathrm{.TRUE.}$

The call is for stage $1$.

$\mathbf{stage1}=\mathrm{.FALSE.}$

The call is for stage $2$.

2:     $\mathbf{ext}$ – LogicalInput

On entry: the form of the domain. If $\mathbf{ext}=\mathrm{.TRUE.}$, the domain is unbounded. Otherwise the domain is an interior one.

3:     $\mathbf{dorm}$ – LogicalInput

On entry: the form of the boundary conditions. If $\mathbf{dorm}=\mathrm{.TRUE.}$, the problem is a Dirichlet or mixed boundary value problem. Otherwise it is a Neumann problem.

4:     $\mathbf{n}$ – IntegerInput

On entry: the number of intervals into which the boundary is divided (see Sections 7 and 9).

5:     $\mathbf{p}$ – Real (Kind=nag_wp)Input

6:     $\mathbf{q}$ – Real (Kind=nag_wp)Input

On entry: to stage $2$, p and q must specify the $x$ and $y$ coordinates respectively of a point at which the solution is required.

When stage1 is .TRUE., p and q are ignored.

7:     $\mathbf{x}\left(\mathbf{n1p1}\right)$ – Real (Kind=nag_wp) arrayInput

8:     $\mathbf{y}\left(\mathbf{n1p1}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the $x$ and $y$ coordinates respectively of points on the one or more closed contours which define the domain of the problem.

Note: each contour is described in such a manner that the subscripts of the coordinates increase when the domain is kept on the left. The final point on each contour coincides with the first and, if a further contour is to be described, the coordinates of this point are repeated in the arrays. In this way each interval is defined by three points, the second of which (the nodal point) always has an even subscript. In the case of the interior Neumann problem, the outermost boundary contour must be given first, if there is more than one.

9:     $\mathbf{n1p1}$ – IntegerInput

On entry: the value $2\left(\mathbf{n}+M\right)-1$, where $M$ denotes the number of closed contours which make up the boundary.

10:   $\mathbf{phi}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: for stage $1$, phi must contain the nodal values of $\varphi$ or its normal derivative (into the domain) as prescribed in each interval. For stage $2$ it must retain its output values from stage $1$.

On exit: from stage $1$, it contains the constants which approximate $\varphi$ in each interval. It remains unchanged on exit from stage $2$.

11:   $\mathbf{phid}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: for stage $1$, $\mathbf{phid}\left(\mathit{i}\right)$ must hold the value $0.0$ or $1.0$ accordingly as $\mathbf{phi}\left(\mathit{i}\right)$ contains a value of $\varphi$ or its normal derivative, for $\mathit{i}=1,2,\dots ,\mathbf{n}$. For stage $2$ it must retain its output values from stage $1$.

On exit: from stage $1$, phid contains the constants which approximate the normal derivative of $\varphi$ in each interval. It remains unchanged on exit from stage $2$.

12:   $\mathbf{alpha}$ – Real (Kind=nag_wp)Input/Output

On entry: for stage $1$, the use of alpha depends on the nature of the problem:

• if $\mathbf{dorm}=\mathrm{.TRUE.}$, alpha need not be set;
• if $\mathbf{dorm}=\mathrm{.FALSE.}$ and $\mathbf{ext}=\mathrm{.TRUE.}$, alpha must contain the prescribed constant $c$ (see Section 3);
• if $\mathbf{dorm}=\mathrm{.FALSE.}$ and $\mathbf{ext}=\mathrm{.FALSE.}$, alpha must contain an appropriate value (often zero) for the integral of $\varphi$ around the outermost boundary.

For stage $2$, on every call alpha must contain the value returned at stage $1$.

On exit: from stage $1$:

• if $\mathbf{ext}=\mathrm{.FALSE.}$, alpha contains $0.0$;
• if $\mathbf{ext}=\mathrm{.TRUE.}$ and $\mathbf{dorm}=\mathrm{.FALSE.}$ alpha is unchanged;
• if $\mathbf{ext}=\mathrm{.TRUE.}$ and $\mathbf{dorm}=\mathrm{.TRUE.}$ alpha contains a computed estimate for $c$.

From stage $2$:

• alpha contains the computed value of $\varphi$ at the point (p,q).

13:   $\mathbf{c}\left(\mathbf{ldc},\mathbf{np4}\right)$ – Real (Kind=nag_wp) arrayWorkspace

14:   $\mathbf{ldc}$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which d03eaf is called.

Constraint: $\mathbf{ldc}\ge \mathbf{n}+1$.

15:   $\mathbf{np4}$ – IntegerInput

On entry: the value $\mathbf{n}+4$.

16:   $\mathbf{icint}\left(\mathbf{np1}\right)$ – Integer arrayWorkspace

17:   $\mathbf{np1}$ – IntegerInput

On entry: the value $\mathbf{n}+1$.

18:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

Invalid tolerance used in an internal call to an auxiliary routine:

$\mathbf{icint}\left(1\right)=0$

Indicates too large a tolerance.

$\mathbf{icint}\left(1\right)>0$

Indicates too small a tolerance.

Note:  this error is only possible in stage 1, and the circumstances under which it may occur cannot be foreseen. In the event of a failure, it is suggested that you change the scale of the domain of the problem and apply the routine again.

$\mathbf{ifail}=2$

Incorrect rank obtained by an auxiliary routine; $\mathbf{icint}\left(1\right)$ contains the computed rank.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

An interior Neumann problem to solve Laplace's equation in the domain bounded externally by the triangle with vertices $\left(3,0\right)$, $\left(-3,0\right)$ and $\left(0,4\right)$, and internally by the triangle with vertices $\left(2,1\right)$, ($-2,1$) and $\left(0,3\right)$, given that the normal derivative of the solution $\varphi$ is zero on each side of each triangle and, for uniqueness that the total integral of $\varphi$ around the outer triangle is $16$ (the length of the contour).

Figure 1

10.1

Program Text

Program Text (d03eafe.f90)

10.2

Program Data

Program Data (d03eafe.d)

10.3

Program Results

Program Results (d03eafe.r)