BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

An application of the BDDC method to the Navier-Stokes equations in 3-D cavity

summary:We deal with numerical simulation of incompressible flow governed by the Navier-Stokes equations. The problem is discretised using the finite element method, and the arising system of nonlinear equations is solved by Picard iteration. We explore the applicability of the Balancing Domain Decomposition by Constraints (BDDC) method to nonsymmetric problems arising from such linearisation. One step of BDDC is applied as the preconditioner for the stabilized variant of the biconjugate gradient (BiCGstab) method. We present results for a 3-D cavity problem computed on 32 cores of a parallel supercomputer

A study of two domain decomposition preconditioners.

Large-scale finite element analysis often requires the iterative solution of equations with many unknowns. Preconditioners based on domain decomposition concepts have proven effective at accelerating the convergence of iterative methods like conjugate gradients for such problems. A study of two new domain decomposition preconditioners is presented here. The first is based on a substructuring approach and can viewed as a primal counterpart of the dual-primal variant of the finite element tearing and interconnecting method called FETI-DP. The second uses an algebraic approach to construct a coarse problem for a classic overlapping Schwarz method. The numerical properties of both preconditioners are shown to scale well with problem size. Although developed primarily for structural mechanics applications, the preconditioners are also useful for other problems types. Detailed descriptions of the two preconditioners along with numerical results are included

Balancing domain decomposition by constraints associated with subobjects

A simple variant of the BDDC preconditioner in which constraints are imposed on a selected set of subobjects (subdomain subedges, subfaces and vertices between pairs of subedges) is presented. We are able to show that the condition number of the preconditioner is bounded by C(1+log(L/h))2, where C is a constant, and h and L are the characteristic sizes of the mesh and the subobjects, respectively. As L can be chosen almost freely, the condition number can theoretically be as small as O(1). We will discuss the pros and cons of the preconditioner and its application to heterogeneous problems. Numerical results on supercomputers are provided.Peer ReviewedPostprint (author's final draft

Balancing domain decomposition by constraints associated with subobjects

A simple variant of the BDDC preconditioner in which constraints are imposed on a selected set of subobjects (subdomain subedges, subfaces and vertices between pairs of subedges) is presented. We are able to show that the condition number of the preconditioner is bounded by C 1 + log(L/h)2, where C is a constant, and h and L are the characteristic sizes of the mesh and the subobjects, respectively. As L can be chosen almost freely, the condition number can theoretically be as small as O(1). We will discuss the pros and cons of the preconditioner and its application to heterogeneous problems. Numerical results on supercomputers are provided