Additive Schwarz preconditioner for the finite volume element discretization of symmetric elliptic problems

Abstract

A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner are proposed for the solution of a class of finite volume element discretization of the symmetric elliptic problem in two dimensions, with large jumps in the entries of the coefficient matrices across subdomains. It is shown that the converCommunicated by Michiel E. Hochstenbach.

Additive Average Schwarz Method for a Crouzeix-Raviart Finite Volume Element Discretization of Elliptic Problems

“…In this section we introduce the additive method for the discrete problem (5) and provide bounds on the convergence rate, both for the solution of the symmetric and nonsymmetric problem following the newly developed abstract framework of [10]. For each substructure Ω k define the restriction of V h to Ωk and the corresponding subspace with CR zero Dirichlet boundary conditions as…”

Robust BDDC algorithms for finite volume element methods

“…ROBUST BDDC FOR FVEM 67 are studied in [34] using an additive Schwarz framework, wherein the convergence is proved to be independent of the number of subdomains, depends poly-logarithmically on the subdomain problem sizes, and is robust with respect to jumps in the coefficient across the subdomain interface as well.…”

“…Therefore, in our analysis we only require that the mesh size is small enough to ensure convergence independent of the number of subdomains. Different from the additive Schwarz approach used in [34], we will combine the estimate of an average operator [43,47] and the connection between the linear systems from the FEMs and FVEMs for our analysis. In this paper, in addition to jumps of the coefficient across subdomain interfaces as in [34], we will also consider jumps of the coefficient inside subdomains.…”

“…Different from the additive Schwarz approach used in [34], we will combine the estimate of an average operator [43,47] and the connection between the linear systems from the FEMs and FVEMs for our analysis. In this paper, in addition to jumps of the coefficient across subdomain interfaces as in [34], we will also consider jumps of the coefficient inside subdomains. For the latter, we will use the deluxe scaling [20], and the primal variables in the BDDC algorithms are derived from the eigenvectors of some carefully chosen local generalized eigenvalue problems [24,35,36,54].…”

“…Our first test example considers G = ρ(2 + sin(xπ) sin(yπ)) 1 0 0 1 as in [34], where ρ has checkerboard patterns as displayed in Figure 6.2. In the second example it is used G = ρ 2 + x 0 0 2 + y .…”