Condition numbers of finite element methods on a class of anisotropic meshes.

We study the behavior of the finite element condition numbers on a class of anisotropic meshes. These newly-developed mesh algorithms can produce numerical approximations with optimal convergence to isotropic and anisotropic singular solutions of elliptic boundary value problems in two- and three-dimensions. Despite the simplicity and fewer geometric constraints in implementation, these meshes can be highly anisotropic and do not maintain the maximum angle condition. We formulate a unified refinement principle and establish sharp estimates on the growth rate of the condition numbers of the stiffness matrix from these meshes. These results are important for effective applications of these meshes and for the design of fast numerical solvers. Numerical tests validate the theoretical analysis. [ABSTRACT FROM AUTHOR]