Discretization Methods and Iterative Solvers Based on Domain Decomposition

Abstract

Singapore; Tokyo: Springer, 2001 (Lecture notes in computational science and engineering ; 17)

Challenges and Applications of Boundary Element Domain Decomposition Methods

“…While for matching grids the described formulation is a conform discretization scheme, it may be generalized to different local grids and different local trial spaces as well. This leads immediately to hybrid or mortar domain decomposition methods where the choice of local trial spaces is essential to ensure the local stability conditions, see, e.g., Wohlmuth [2001] and the references given therein. Since the approximation of the local Dirichlet to Neumann maps can be done by any available discreization scheme, the presented formulation allows the coupling of different discretization schemes such as finite and boundary element methods, and the coupling of locally different meshes and trial spaces.…”

Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

“…Thus our main objective here is to extend the stability analysis for the mortar method to the much more flexible class M K,ε introduced above where appropriate mesh-dependent norms now involve mesh functions. Specifically, we will focus on the dual basis mortar method [22,26] which has been shown in [22] to yield stable and accurate discretisations in the 3-dimensional case also provided that certain weak matching conditions along the boundary of interfaces between adjacent subdomains hold. Here we employ concepts from the previous sections to establish stability without any such matching conditions.…”

“…A common strategy is to choose the Lagrange multipliers also as continuous piecewise linear finite elements to keep them as close as possible to the traces on the nonmortar side. Here we consider an interesting alternative that has been recently proposed in [26] for the case d = 2 and in [22] for d = 3. In these papers the Lagrange multipliers are allowed to be discontinuous in favour of an additional practically very desirable feature, namely the fact that the Lagrange multiplier spaces are spanned by a basis which is dual to those of the corresponding trace spaces on the nonmortar sides.…”

Substructuring Preconditioners for the Bidomain Extracellular Potential Problem

“…Since biorthogonal basis functions are employed, the square matrix C S is diagonal and easily invertible (cf. [11]). The reduction in (16) may be written in matrix form as…”

“…To obtain the matrix form ofs(·, ·) we consider the space L of linear functions, used in the splitting of the trace space (11). Then, we introduce an interpolation map denoted by R T H (say piecewise interpolation) from the nodal value on V (vertices) onto all nodes of S. The matrix R H can be viewed as the weighted restriction map from S onto V .…”