Domain decomposition methods.

*(English)*Zbl 1132.65111
Kraus, Johannes (ed.) et al., Lectures on advanced computational methods in mechanics. Collection of lectures delivered during the special semester on computational mechanics, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, October 3 – December 16, 2005. Berlin: Walter de Gruyter (ISBN 978-3-11-019556-9/hbk). Radon Series on Computational and Applied Mathematics 1, 89-159 (2007).

From the introduction: These notes aim at the construction of preconditioners \(B\) for solving systems of grid equations approximating elliptic boundary value problems in domains with complex geometry. A preconditioner \(B\) can be used, for example, in iterative processes of the following form:

\[ B(u^{k+1}-u^k)=-\tau^k(Au^k-f),\tag{1} \]

where \(A\) is the stiffness matrix of the original system of grid equations. The convergence rate of the iterative process (1) depends on the constants \(c_1\) and \(c_2\) in the spectral equivalence inequalities

\[ c_1(Bu,u)\leq (Au, u)\leq c_2(B_u, u)\tag{2} \]

which should be valid for any vector \(u\). Here, we assume that \(A\) and \(B\) are Symmetric positive definite matrices. The constants \(c_1\) and \(c_2\) from (2) are independent of the mesh size, and, in order to perform the multiplication of \(B^{-1}\) with some vector, it is necessary to solve the system of grid equations corresponding to the five-point approximation of the Laplace operator on a uniform grid of a rectangle. The construction of a preconditioner \(B\) with similar characteristics in the case of boundary value problems in domains with complex geometry is of great interest.

For the entire collection see [Zbl 1123.65002].

\[ B(u^{k+1}-u^k)=-\tau^k(Au^k-f),\tag{1} \]

where \(A\) is the stiffness matrix of the original system of grid equations. The convergence rate of the iterative process (1) depends on the constants \(c_1\) and \(c_2\) in the spectral equivalence inequalities

\[ c_1(Bu,u)\leq (Au, u)\leq c_2(B_u, u)\tag{2} \]

which should be valid for any vector \(u\). Here, we assume that \(A\) and \(B\) are Symmetric positive definite matrices. The constants \(c_1\) and \(c_2\) from (2) are independent of the mesh size, and, in order to perform the multiplication of \(B^{-1}\) with some vector, it is necessary to solve the system of grid equations corresponding to the five-point approximation of the Laplace operator on a uniform grid of a rectangle. The construction of a preconditioner \(B\) with similar characteristics in the case of boundary value problems in domains with complex geometry is of great interest.

For the entire collection see [Zbl 1123.65002].

##### MSC:

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65F10 | Iterative numerical methods for linear systems |