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].