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RESEARCH STATEMENT 1. Overview. I Am Working on Developing RESEARCH STATEMENT XUEMIN TU 1. Overview. I am working on developing efficient numerical algorithms for solving the system obtained from the discretization of partial differential equations (PDEs). Much of my work has so far been supervised by Professor Olof Widlund. Usually the first step of solving an elliptic partial differential equation (PDE) nu- merically is its discretization. Finite difference, finite element, or other discretizations reduce the original PDE to an often huge and ill-conditioned linear or nonlinear sys- tem of algebraic equations. Limited by the memory and speed of the computers, the traditional direct solvers can often not handle such large linear systems. Moreover, iterative methods, such as Krylov space methods, may need thousands of iterations to obtain accurate solutions due to large condition numbers. Domain decomposi- tion methods provide efficient and scalable preconditioners that can be accelerated by Krylov space methods and have become popular in applications in computational fluid dynamics, structural engineering, electromagnetics, constrained optimization, etc. The basic idea of domain decomposition methods is to split the original huge problem into many small problems which can be handled by direct solvers, and then solve these smaller problems a number of times and accelerate the solution of the original problem with Krylov space methods. There are two main classes of domain decomposition methods: overlapping Schwarz methods and iterative substructuring methods. My research focuses on the second ap- proach. In iterative substructuring methods, the domain is decomposed into nonover- lapping subdomains. The unknowns in the interior of the subdomains are first elim- inated independently and we then work with the Schur complement with respect to the unknowns associated with the interface. Coarse problems are constructed using one or a few degrees of freedom for each subdomain. Among these algorithms, the Neumann-Neumann and finite element tearing and interconnecting methods (FETI) families are the best known and they have been tested in many applications. Recently, a new family of iterative substructuring methods, the balancing do- main decomposition by constraints (BDDC) algorithms, has been introduced by Clark Dohrmann in [6]. These methods have a Neumann-Neumann flavor. However, their coarse problems are given by sets of constraints enforced on the interface, which are similar to those of the dual-primal FETI (FETI-DP) methods [11]. It has been proved that the preconditioned operators for BDDC and FETI-DP have identical nontrivial eigenvalues except possibly for 0 and 1, see [21, 18, 3]. The condition number of the preconditioned operators have the bound: 2 H (1.1) κ ≤ C 1 + log ; h where H is the diameter and h is the typical mesh size of subdomains and C is constant independent of H and h. Combining this estimate and the convergence analysis of Krylov space method, we can conclude that the rate of convergence of FETI-DP and BDDC both are independent of the number of subdomains but depend slightly on the problem size of each subdomain. They are efficient and scalable algorithms. However, a shortcoming of both BDDC, FETI-DP, and all other domain decom- position methods is that the coarse problem needs to be assembled and the resulting matrix needs to be factored by a direct solver at the beginning of the computation. 1 Usually the size of the coarse problem is proportional to the number of subdomains. Nowadays some computer systems have more than 100,000 powerful processors, which allow very large and detailed simulations. The coarse component can therefore be a bottleneck if the number of subdomains is very large. Motivated by this fact, I have developed two three-level BDDC algorithms to remove this difficulty. I have also extended the BDDC algorithms to flow in porous media and established the same condition number estimate for the preconditioned BDDC operator as in (1.1). I also worked with Professor Maksymilian Dryja on the using domain decomposi- tion method directly for parabolic problems and have obtained the best error estimate, to the best of our knowledge, for this type of discretization. In the summer 2005, I worked as a Givens associate in Mathematics and Computer science division at Argonne National Laboratory with Dr. Barry Smith on nonlinear multigrid for solving nonlinear PDEs. During my master study at Worcester Polytechnic Institute, I worked with my advisor Dr. Marcus Sarkis on enhanced singular function mortar finite element for nonconvex domains. I also worked with Professor Haiyang Huang on modeling the dynamics of flour beetle population at Beijing Normal University during my graduate study there. In the following sections, I describe these projects in detail and related future work. 2. Three-level BDDC. 2.1. Current work: Three-level BDDC for scalar elliptic problems in two and three dimensions. The BDDC algorithms, previously developed for two levels [6, 20, 21], are similar to the balancing Neumann-Neumann algorithms. How- ever, the coarse problem, in BDDC, is given in terms of a set of primal constraints and is generated and factored by direct solvers at the beginning of the computation. The coarse components of the preconditioners can then ultimately become a bottleneck if the number of subdomains is very large. We try to remove this difficulty by using one or several additional levels. We proceed as follows: we group several subdomains together to form a sub- region. We could first reduce the original coarse problem to a subregion interface problem by eliminating independently the subregion interior variables, which are the primal variables on the subdomain interface and interior to the subregions. In one of the three-level BDDC algorithms, we do not solve the subregion interface prob- lem exactly, but replace it by one iteration of the BDDC preconditioner; Dohrmann has also suggested this approach in [6]. This means that we only need to solve sev- eral subregion local problems and one coarse problem on the subregion level in each iteration. We assume that all these problems are small enough to be solved by di- rect solvers. We have shown that the condition number estimate for the resulting three-level preconditioned BDDC operator is bounded by 2 2 H^ H (2.1) κ ≤ C 1 + log 1 + log ; H h ! where H^ , H, and h are the typical diameters of the subregions, subdomains, and mesh of subdomains, respectively. C is constant independent of H^ , H, h, and the coefficients of the original PDE, provided that the coefficients of the PDE vary moderately in each subregion. 2 ^ 2 H In order to remove the additional factor 1 + log H in (2.1), we can use a Chebyshev iteration method to accelerate the three-levelBDDC algorithms. With this device, the condition number bound is 2 H (2.2) κ ≤ CC(k) 1 + log ; h where C(k) depends on the eigenvalues of the preconditioned coarse problem, the two parameters chosen for the Chebyshev iteration, and k, the number of the Chebyshev iterations. C(k) goes to 1 as k goes to 1. H and h are the same as before. We first obtained these results for two dimensional problem with vertex con- straints, see [29]. We then extended these algorithms to the three dimensional cases, see [33]. In the three dimensional case, vertex constraints alone are not enough to obtain good polylogarithmic condition number bound (1.1) due to much weaker inter- polation estimate and constraints on the averages over edges or faces are needed. The new constraints lead to a considerably more complicated coarse problem and the need for new technical tools in the analysis. The same condition number bounds (2.1) and (2.2) with Chebyshev acceleration are obtained for three dimensional cases in [33]. 2.2. Future work. Our three-level BDDC algorithms are one of the inexact BDDC methods. Recently, there are several new results which have been reported, [32, 19, 16, 5]. BDDC has been extended to incompressible Stokes equations [17], flow in porous media [30, 31], and with mortar finite element discretization [15]. I will continue to work on extending our three-level BDDC algorithms to these problems, successfully extending the two-level BDDC methods. Among these, a three-level BDDC method with mortar finite element discretization is joint work with Dr. Hyea Hyun Kim. 3. BDDC algorithm for flow in porous media. We have extended BDDC algorithm for flow in porous media with two kinds of discretizations: mixed and hybrid finite element discretizations. 3.1. A BDDC algorithm for a mixed formulation of flow in porous me- dia. Using mixed formulations of flow in porous media, we obtain a saddle point problem which is closely related to that arising from the incompressible Stokes equa- tions. In a recent paper [17], the BDDC algorithms have been applied to the incom- pressible Stokes equation. Our situation is different. First of all, our problem is not originally formulated in the benign, divergence free subspace, and it will therefore be reduced to the benign subspace, as in [10, 22, 23, 24], at the beginning of the computation. In addition, only edge/face constraints are needed to force the iterates into the benign subspace and to ensure a good bound for the condition number, since Raviart-Thomas finite elements, see [4, Chapter III], are utilized. These elements have no degrees of freedom associated with vertices/edges in two/three dimensions. Also, the condition number estimate for the Stokes case can be simplified since the Stokes extension is equivalent to the harmonic extension, see [2]. However, this is not the case here, and different technical tools are required. An iterative substructuring method with Raviart-Thomas finite elements for vector field problems was proposed in [35, 27]. We have borrowed some technical tools from these papers and proven the condition number bound (1.1) for this BDDC algorithm, see [30].
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