Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

Figures & data

Figure 1. A solution domain D with boundary part ${\mathrm{\Gamma }}_1$ contained within the outer boundary surface ${\mathrm{\Gamma }}_2$.

Figure 2. The quadrature points ${\widehat{y}}_{s^{\prime }{p}^{\prime }}$ ($·$) with $n{}^{\prime }=8$.

Figure 3. The solution domain in Examples and .

Figure 4. L-curves ($n=12$) in Example .

Figure 5. The exact and numerical approximation ($n=12$) of function values on the boundary surface ${\mathrm{\Gamma }}_1$ with 3 % noise for Example .

Figure 6. The exact and numerical approximation ($n=12$) of the normal derivative on the boundary surface ${\mathrm{\Gamma }}_1$ with 3 % noise for Example .

Table 1. Errors in the case of exact data in Example .

$n=n{}^{\prime }$	$\mathit{\lambda }$	$\Vert u-u_n{\Vert }_{L^2({\mathrm{\Gamma }}_1)}$	${∥\frac{\mathit{\partial }u}{\mathit{\partial }\mathit{\nu }}-\frac{\mathit{\partial }{u}_n}{\mathit{\partial }\mathit{\nu }}∥}_{L^2({\mathrm{\Gamma }}_1)}$
4	${10}^{-5}$	1.72E$-02$	8.88E$-02$
6	${10}^{-7}$	8.86E$-04$	6.45E$-03$
8	${10}^{-7}$	5.26E$-05$	3.85E$-04$
10	${10}^{-9}$	2.12E$-06$	2.45E$-05$
12	${10}^{-10}$	1.21E$-07$	1.58E$-06$

Table 2. Errors in the case of data with 3 % noise in Example .

$n=n{}^{\prime }$	$\mathit{\lambda }$	$\Vert u-u_n{\Vert }_{L^2({\mathrm{\Gamma }}_1)}$	${∥\frac{\mathit{\partial }u}{\mathit{\partial }\mathit{\nu }}-\frac{\mathit{\partial }{u}_n}{\mathit{\partial }\mathit{\nu }}∥}_{L^2({\mathrm{\Gamma }}_1)}$
4	${10}^{-3}$	5.17E$-01$	9.38E$-01$
6	${10}^{-2}$	2.35E$-01$	6.51E$-01$
8	${10}^{-2}$	2.30E$-01$	4.41E$-01$
10	${10}^{-2}$	1.72E$-01$	6.76E$-01$
12	$8·{10}^{-3}$	1.61E$-001$	6.48E$-01$

Figure 7. L-curves ($n=12$) in Example .

Figure 8. The exact and numerical approximation ($n=12$) of function values on the boundary surface ${\mathrm{\Gamma }}_1$ with 3 % noise for Example .

Figure 9. The exact and numerical approximation ($n=12$) of the normal derivative on the boundary surface ${\mathrm{\Gamma }}_1$ with 3 % noise for Example .

Table 3. Errors in the case of exact data in Example .

$n=n{}^{\prime }$	$\mathit{\lambda }$	$\Vert u-u_n{\Vert }_{L^2({\mathrm{\Gamma }}_1)}$	${∥\frac{\mathit{\partial }u}{\mathit{\partial }\mathit{\nu }}-\frac{\mathit{\partial }{u}_n}{\mathit{\partial }\mathit{\nu }}∥}_{L^2({\mathrm{\Gamma }}_1)}$
4	${10}^{-3}$	9.29E$-02$	4.21E$-01$
6	${10}^{-3}$	2.12E$-02$	8.44E$-02$
8	${10}^{-4}$	5.39E$-03$	3.61E$-02$
10	${10}^{-4}$	3.47E$-03$	2.60E$-02$
12	${10}^{-5}$	1.31E$-03$	1.23E$-02$

Table 4. Errors in the case of data with 3 % noise in Example .

$n=n{}^{\prime }$	$\mathit{\lambda }$	$\Vert u-u_n{\Vert }_{L^2({\mathrm{\Gamma }}_1)}$	${∥\frac{\mathit{\partial }u}{\mathit{\partial }\mathit{\nu }}-\frac{\mathit{\partial }{u}_n}{\mathit{\partial }\mathit{\nu }}∥}_{L^2({\mathrm{\Gamma }}_1)}$
4	${10}^{-3}$	7.94E$-02$	3.86E$-01$
6	${10}^{-3}$	2.87E$-02$	1.28E$-01$
8	${10}^{-3}$	1.31E$-02$	9.79E$-02$
10	${10}^{-3}$	2.48E$-02$	7.56E$-02$
12	${10}^{-3}$	2.08E$-02$	9.19E$-02$