Direct Computation of Elliptic Singularities Across Anisotropic, Multi-Material Edges

We characterize the singularity of two-dimensional elliptic div-grad operators at a vertex where several materials meet on a Dirichlet or Neumann part of the boundary. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several three- and four-material constellations. These bounds then are used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains by identifying the edge singularities of the three-dimensional problem as the singularity of an associated two-dimensional one of the afore mentioned type. We exemplify this for the benchmark L-shape problem. Bibliography: 67 titles. Illustration: 10 figures. However, one can find these eigenvalues very seldom [. . . ]even in the case of the Laplace operator [1, p. 36]. © 2010 Springer Science+Business Media, Inc.