Finite element method for elliptic problems with edge singularities
The authors consider numerical approximations of tangentially regular solutions of the Dirichlet problem for homogeneous elliptic partial differential equations in three dimensions. The solution domain is a bounded polygonal domain in two dimensions extended infinitely in the third. Theoretical results employ Fourier transforms in the infinite direction with numerical results attained by truncated Fourier series. The bounded polygonal cross-section is treated using finite elements. For this configuration, the solution can exhibit singularities arising from corners in the cross-section and yet maintain regularity in the third direction. This situation can lead to slow convergence rates. Convergence rates can be improved by augmenting the finite element spaces with certain singular functions. These singular functions are easily constructed in two dimensions but their extension to three dimensions is more delicate. The authors construct the singular functions first in two space and one Fourier dimension and then use inverse transforms to return to three space dimensions. The transform variable adds an infinite dimension to the normally finite-dimensional finite element spaces, but such important features as completeness remain true. Error estimates using the finite element method augmented with singular functions are proved and demonstrate the effectiveness of the method.
scientific article; zbMATH DE number 1334653
scientific article