NAG FL Interface
d03eaf (dim2_​laplace)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{stage1}$ – Logical Input

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: $\mathbf{ext}$ – Logical Input

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

3: $\mathbf{dorm}$ – Logical Input

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: $\mathbf{n}$ – Integer Input

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) array Input

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

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: $\mathbf{n1p1}$ – Integer Input

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) array Input/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) array Input/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: $\mathbf{c}\left(\mathbf{ldc},\mathbf{n}+4\right)$ – Real (Kind=nag_wp) array Workspace

14: $\mathbf{ldc}$ – Integer Input

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: $\mathbf{np4}$ – Integer Input

On entry: this argument is no longer referenced, but is included for backwards compatability.

16: $\mathbf{icint}\left(\mathbf{n}+1\right)$ – Integer array Workspace

17: $\mathbf{np1}$ – Integer Input

On entry: this argument is no longer referenced, but is included for backwards compatability.

18: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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 $\mathbf{icint}\left(1\right)=〈\mathit{\text{value}}〉$:

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

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; $\mathbf{icint}\left(1\right)$ is the computed rank.

$\mathbf{ifail}=-99$

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

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

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

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface 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

10.2 Program Data

10.3 Program Results