Title: A BDDC algorithm for problems with mortar discretization


(NYU-CS-TR873)

Authors:  Hyea Hyun Kim, Maksymilian Dryja, and Olof B. Widlund

Abstract:

A BDDC (balancing domain decomposition by constraints) algorithm is developed for elliptic problems with mortar discretizations for geometrically non-conforming partitions in both two and three spatial dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face which is an intersection of the boundaries of a pair of subdomains. A condition number bound of the form $C \max_i \left\{ (1+\text{log} (H_i/h_i) )3 \right\}$ is established. In geometrically conforming cases, the bound can be improved to $C \max_i \left\{ (1+\text{log} (H_i/h_i) )2 \right\}$. This estimate is also valid in the geometrically nonconforming case under an additional assumption on the ratio of mesh sizes and jumps of the coefficients. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet preconditioner for the FETI--DP algorithms of \cite{K-04-3d,KL-02} and it is shown that the eigenvalues of the BDDC and FETI--DP methods are the same except possibly for an eigenvalue equal to 1.