Finite element method for elliptic problems with edge singularities

摘要

We consider tangentially regular solution of the Dirichlet problem for an homogeneous strongly elliptic operator with constant coefficients, on an infinite vertical polyhedral cylinder based on a bounded polygonal domain in the horizontal xy-plane. The usual complex blocks of singularities in the non-tensor product singular decomposition of the solution are made more explicit by a suitable choice of the regularizing kernel. This permits to design a well-posed semi-discrete singular function method (SFM), which differs from the usual SFM in that the dimension of the space of trial and test functions is infinite. Partial Fourier transform in the z-direction (of edges) enables us to overcome the difficulty of an infinite dimension and to obtain optimal orders of convergence in various norms for the semi-discrete solution. Asymptotic error estimates are also proved for the coefficients of singularities. For practical computations, an optimally convergent full-disc! ! ! retization approach, which consists in coupling truncated Fourier series in the z-direction with the SFM in the xy-plane, is implemented. Other good (though not optimal) schemes, which are based on a tensor product form of singularities are investigated. As an illustration of the results, we consider the Laplace operator.