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Convergence4

Convergence for Eigenvalue Problems with Mesh Adaptivity

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Numerics:

In this example we investigate how our method copes with discontinuous coefficients. We inserted a square subdomain of side 0.5 in the center of the unit square domain. We also choose the function $ \mathcal{A}$ to be piecewise constant and to assume the value 100 inside the subdomain and the value 1 outside it.

The jump in the value of $ \mathcal{A}$ could produce a jump in the gradient of the eigenfunctions all along the boundary of the subdomain. So the regularity of the eigenfunctions in each subdomain, in the sense of Assumption 1.1, is now between $ 3/2\leq s+1<2$. Using uniform refinement, the rate of convergence for eigenvalues should be at least $ \mathcal{O}(H^\mathrm{max}_n)^{2s}$ or equivalently $ \mathcal{O}(N^{-s})$, where $ N$ is the number of DOFs. Instead, using our method we obtain greater orders of convergence for big enough value of $ \theta$ and $ \tilde{\theta}$ , as can be seen from Table 3. We measure the rate of convergence computing the value of $ \beta$ as before. In fact the rate of convergence for $ \theta=\tilde{\theta}=0.5$ or $ 0.8$ is close to the rate of convergence for smooth problems. In this case the exact eigenvalue $ \lambda$ is unknown, but we approximate it by computing the eigenvalue on a very fine mesh involving about half a million of DOFs.

In Figure 3 we depict the mesh coming from the fourth iteration of Algorithm 1 with $ \theta=\tilde{\theta}=0.8$. This mesh is the result of multiple refinements using both marking strategies 1 and 2 each time. As can be seen the corners of the subdomain are much more refined than the rest of the mesh. This is clearly the effect of the first marking strategy, since the edge residuals have detected the discontinuity in the gradient of the eigenfunction along the interface.

Finally in Figure 4 we depict the eigenfunction corresponding to the smallest eigenvalue of the problem with discontinuous coefficients. This eigenfunction is the one used to refine the mesh in Figure 3.

$ \theta=\tilde{\theta}=0.2$$ \theta=\tilde{\theta}=0.5$$ \theta=\tilde{\theta}=0.8$
Iteration$ \vert\lambda-\lambda_n\vert$DOFs$ \beta$$ \vert\lambda-\lambda_n\vert$DOFs$ \beta$ $ \vert\lambda-\lambda_n\vert$DOFs$ \beta$
11.1071811.1071811.107181
21.02001030.34100.87381990.26320.48343560.5597
31.01051290.04160.58483140.88050.22447990.9494
41.00391470.04980.39834910.85910.099022350.7957
50.89681670.88430.27666731.15640.040147641.1932
60.80761940.69960.19339750.96650.0180123750.8372
70.80082170.07470.134614760.87220.0065123751.1888
80.75022370.74010.094820801.02370.0020653871.4482
Table 3: Comparison of the reduction of the error and DOFs of the adaptive method for the smallest eigenvalue for the problem with discontinuous coefficients.

Figure 3: A refined mesh from the adaptive method corresponding to the first eigenvalue of the problem with discontinuous coefficients.
Figure 4: The eigenfunction corresponding to the first eigenvalue of the problem with discontinuous coefficients.