In 2001 Randolph E. Bank and Peter K. Jimack [6] introduced a new domain de-composition method for the adaptive solution of elliptic partial di↵erential equations.
This new method is motivated by the work of [4], [37], and [36]. The novel feature of this algorithm is that each of the subproblems is defined over the entire domain, see Figure 4.1. In this figure we see on the left a mesh on the domain⌦= (0,1)⇥(0,1).
In the middle we see a a fine mesh on the subdomain⌦1 which is indicated in red, and a coarse mesh on⌦\⌦1. On the right we see a fine mesh on the subdomain⌦2 which is indicated in red, and a coarse mesh on ⌦\⌦2. The parallel technique that they introduced is designed to use standard sequential adaptive mesh algorithms based on local h-refinement, (see [3], [8], [12], [32], and [45]) with minimal modifications.
To describe the method, we consider a linear elliptic PDE on a domain ⌦, and two overlapping subdomains ⌦1 and ⌦2, ⌦ = ⌦1 [⌦2, see Figure 4.1. Discretizing the problem on a global fine mesh leads to a linear system Ku = f, where K is the sti↵ness matrix, u is the vector of unknown nodal values on the global fine mesh, andf is the load vector. We partition now the vectoru=⇥
u1,us,u2⇤T
, whereu1 is the vector of unknowns strictly inside subdomain ⌦1,us is the vector of unknowns shared by the two subdomains, and u2 is the vector of unknowns strictly inside subdomain⌦2. We can then write the linear system in block matrix form,
2 4
A1 B1 0 B1T As B2T
0 B2 A2 3 5
2 4
u1 us
u2 3 5=
2 4
f1 fs
f2 3
5. (4.2.1)
CHAPTER 4. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD37 The idea of the Bank-Jimack method is to consider two further meshes on ⌦, one identical to the original fine mesh in ⌦1, but coarse on ⌦\⌦1, see Figure 4.1 in the middle, and one identical to the original fine mesh in ⌦2, but coarse on ⌦\⌦2, see Figure 4.1 on the right. This leads to the two further linear systems
2
where we introduced the restriction matricesMj to restrictfjto corresponding coarse meshes. The Bank-Jimack method is then performing the following iteration
The Bank-Jimack Algorithm:
Bank and Jimack studied a generalization of this algorithm to the many subdomain case, see [6, pages 6-8]. In [4] they introduced a new parallel domain decomposition preconditioner for the solution of the sparse linear systems that arise from the parallel
CHAPTER 4. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD38 adaptive finite element discretization of a class of self-adjoint linear elliptic partial di↵erential equations. This technique requires each processor to work on the entire domain but with a coarse mesh which has been locally refined only in the subdomain for which that process is responsible. They also studied a generalization of this algorithm to a mesh-free parallel solver in [5] and showed that the main parallel solution technique developed in [4] may be generalized to allow the parallel solution of an arbitrary sparse matrix.
A convergence study of the Bank and Jimack domain decomposition method was done in 2008 by Bank and Vassilevski in [7]. They showed that for an idealized version of the algorithm, the rate of convergence is independent of both the global problem size and the number of subdomains used in the domain decomposition partition.
In 2004 Kraus presented in [28] a particular construction of neighborhood matrices to be used in the computation of the interpolation weights in an algebraic multigrid method. The method use the existence of simple interpolation matrices on a hierarchy of coarse meshes. This is one approach for finding the interpolation matrices from the fine mesh to the coarse mesh.
A further study of the class of parallel adaptive finite element method has been done by Loisel and Nguyen in [33]. They introduced a new adaptive Schwarz preconditioner to use in parallel finite elements where local meshes can be meshes of the whole domain, and showed that the advantage of this approach is that the local matrices obtained through assembling without communication and without forming the global matrix.
In this thesis we study the Bank-Jimack method and its convergence behavior. In this chapter we introduce the Bank-Jimack method in 1D for the Poisson equation and we prove that the Bank-Jimack method is equivalent to an optimized Schwarz method for a special Robin parameter and it converges after one iteration.
In Chapter 5 we introduce the Bank-Jimack method in 1D for the⌘ equation, and we show that the convergence behavior for the ⌘ equation changes depending on the choice of the coarse mesh. We show that for a uniform coarse mesh the convergence factor is 1 O(L12), where L is the size of the overlap between two subdomains, independent of how many mesh points we have on the coarse area of the mesh and for a stretched coarse mesh the convergence factor is 1 O(L2n1 ), where nis the number of the mesh points on the coarse area of the mesh. Finally we provide numerical results to support our theoretical findings.
In Chapter 6 we study the Bank-Jimack method in 2D for the Poisson equation and using Fourier analysis we show that its convergence behavior is similar to the⌘ equation in 1D. We finish this chapter by providing some numerical experiments.
In Chapter 7 we study the Bank-Jimack method for the ⌘ equation in 1D for an unbounded domain which greatly simplifies the analysis, and we show that its
CHAPTER 4. THE BANK-JIMACK DOMAIN DECOMPOSITION METHOD39
0 mh lh 1
n1 ns n2
h
0 mh lh 1
n1 ns m2
h1
h
0 mh lh 1
⌦1 ⌦2
m1 ns n2
h2 h
Figure 4.2: Global fine mesh, and two partially coarse meshes
convergence factor for a stretched coarse mesh is 1 O(L2n11), which is di↵erent from the bounded domain case. We present some numerical results to close this chapter.