Sine transform based preconditioners for elliptic problems

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

“…In this case this PCG shows a behaviour substantially equivalent to that one observed for the circulant preconditioning [9]. On the other hand, the technique described in [31] is evidently superior in this case (see the numerical experiments performed in [9,31,22]).…”

“…For this case the weak clustering property does not hold and in the case of a(x, y) and b(x, y) positive functions it is possible to prove only a result of linear convergence [21]. Actually the numerical experiments confirm this forecast: in the case of a = b we have a superlinear convergence even in the case of a having a zero, while in the case a = b the convergence is much slower and the behaviour of the proposed PCG is substantially equivalent to that one considered in [9] with circulant preconditioner. For this case the preconditioner proposed by [31] is superior and, for this reason, it would be interesting to analyze the convergence properties of this PCG method.…”

“…We do not make explicit comparison with circulant preconditioners [9,31,35] because, as shown in [22,23], τ and band Toeplitz based preconditioners perform much better than circulant ones in the case of Dirichlet boundary value problems. It is also true that circulant preconditioners are very effective in the case of y-periodic linear elliptic problems [35] with strictly positive smooth coefficient functions.…”

A Method for Solving Hermitian Pentadiagonal Block Circulant Systems of Linear Equations

“…The complexity of the proposed new algorithms is O(nm 3 ). For comparison algorithm based on the Fast Fourier Transform (FFT) for solving block circulant system with circulant blocks has complexity O(nm log(nm)) but it can be implemented only when the block size of the matrix W is a power of two and when all blocks of W are circulant [2]. Our method does not have these restrictions.…”

Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

“…[8]. This holds in particular for elliptic problems [3]. More generally, Manteuffel and Parter [12] have proved that if A and M are discrete elliptic operators, then the condition number of M −1 A is bounded as the grid is refined if and only if A and M has the same boundary conditions.…”