Chebyshev pseudospectral method in the reconstruction of orthotropic conductivity

Figures & data

Figure 1. Sparse structure of matrix ${\mathbf{M}}_{\mathbf{m}}$, for n + 1 = 16 grid points on both directions.

Figure 2. Numerical solutions for three time stages (top) and associated pointwise absolute error (bottom).

Figure 3. Example of grid, for n = 4, enumerated in accordance with Equation (42(42) $\ell =i+j(n+1)+1,i=0,1,\dots ,n,j=0,1,\dots ,n.$(42) ), i.e, lexicographic order.

Figure 4. Example of FEM mesh's structure with 185 nodes, 328 triangles and global mesh size h = 0.1 (left), respective solution with CN ($\mathrm{\Delta }t=0.01$) at time t = 1 (middle) and sparse structure of mass matrix $\stackrel{\mathbf{~}}{\mathbf{M}}$ (right), for data used in Figure 2.

Table 1. Errors associated with FEM-based solutions for two global mesh sizes and errors associated with CPM.

t	${\hat{E}}_h(t)$ $(h=0.1)$	${\hat{E}}_h(t)$ $(h=0.01)$	$E_q$ $(n=10)$
0.1	$9.3656\times {10}^{-3}$	$1.2089\times {10}^{-4}$	$5.9071\times {10}^{-6}$
0.5	$2.5849\times {10}^{-2}$	$4.2066\times {10}^{-4}$	$2.1342\times {10}^{-5}$
1.0	$3.6829\times {10}^{-2}$	$6.3101\times {10}^{-4}$	$3.0597\times {10}^{-5}$

Figure 5. Residual $\Vert \mathsf{\text{T}}({\mathbf{k}}^j)-\stackrel{\mathsf{~}}{\mathsf{T}}{\Vert }_2$ (left) and relative error between ${\mathbf{k}}^j$ and the exact conductivity $k(x,y)=(1+x+y)/12$ (right), $\mathrm{NL}=1\mathrm{\%}$.

Figure 6. Comparison between solutions for $k(x,y)=(1+x+y)/12$ calculated with different choices of $\mathbf{R}$ and α. For (d) and (e), $\alpha =0.1$.

Table 2. Errors and number of iterations until DP is satisfied, for different choices of $\mathbf{R}$ and α, considering as exact $k(x,y)=(1+x+y)/12$.

$\mathbf{R}$	I	I	${\mathcal{L}}_1$	${\mathcal{L}}_2$
α	0.1	0.35	0.1	0.1
RE	0.0946	0.0717	0.0128	0.0134
IE	0.0236	0.0250	0.0096	0.0084
Iterations	7	29	4	4

Figure 7. Contour plots for every exact $k_i(x,y)$ (top) and the respective approximation (bottom), $i=1,\dots ,4$.

Table 3. Relative errors RE, interior error IE and number of LMM iterations until DP is satisfied.

Function	$k_1(x,y)$	$k_2(x,y)$	$k_3(x,y)$	$k_4(x,y)$
RE	0.0151	0.0196	0.0300	0.0320
IE	0.0093	0.0117	0.0130	0.0310
Iterations	4	3	3	2

Figure 8. Behavior of reconstruction results solution for different noise levels. Top: exact conductivities and respective approximations along the line x = 0.5. Bottom: absolute value of reconstruction errors along the line x = 0.5 and relative error RE.

Table 4. Orthotropic test cases.

Case	$k_{11}(x,y)$	$k_{22}(x,y)$
1	$(1+x+y)/12$	$(1+0.5x+y)/12$
2	$(1+x^2)/12$	$(1+y^2)/12$
3	$(1+x^2+y^2)/12$	$(1+\sin (x)\sin (y))/12$

Table 5. Orthotropic results for test cases in Table 4.

	$k_{11}(x,y)$		$k_{22}(x,y)$
Case	RE	IE	RE	IE	Iterations
1	0.0223	0.0141	0.0100	0.0068	3
2	0.0277	0.0190	0.0257	0.0195	2
3	0.0221	0.0135	0.0179	0.0129	3

Figure 9. Contour plots for every test case: 1 (top row), 2 (middle row) and 3 (bottom row).

Figure 10. Computed temperature T (above) at time t = 1 using the reconstructions of $k_{11}$ and $k_{22}$ and the absolute value of the error between FEM's T and its approximation (below).