with the same efficiency, but it turned out that this is a very difficult task. Textbooks mention that there are substantial difficulties, see [3, page 72], [11, page 212], [12, page 400], and also the review [7] for why in general iterative methods have difficulties when applied to the Helmholtz equation (1). Motivated by the early proposition in [2] to use the Laplacian to precondition the He...

We study a conservative 5-point cell-centered finite volume discretization of the high-contrast diffusion equation. We aim to construct preconditioners that are robust with respect to the magnitude of the coefficient contrast and the mesh size simultaneously. For that, we prove and numerically demonstrate the robustness of the preconditioner proposed by Aksoylu et al. (2008, Comput. Vis. Sci. 1...

A Newton-Krylov method is developed for the solution of the steady compressible NavierStokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element Line-Jacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (BlockILU(0)) is also presented as a precondit...

The Stokes problem and linear elasticity problems can be viewed as a mixed variational formulation. These formulations are discretized by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development an efficient preconditioner requires three ingredients, a preconditioner ...

The paper is devoted to designing an interface preconditioner for the mortar element method. After brief overview of the problem in Introduction, we discuss the mortar element method with different types of the Lagrange multiplier spaces. Next, we consider the domain decomposition technique for the solution of mortar element systems and outline the general framework of the solution of saddle-po...

A new multilevel algebraic preconditioner for symmetric positive definite matrices is proposed. The projection and smoothing steps typical for textbook multigrids are replaced by a special coarsening algorithm which is based on orthogonal projectors onto kernels of subdomain matrices. The preconditioner is compared numerically with a few other multigrid preconditioners.

An adapted deflation preconditioner is employed to accelerate the solution of linear systems resulting from discretization fracture mechanics problems with well-conditioned extended/generalized finite elements. The space typically used for elasticity enriched additional vectors, accounting enrichment functions used, thus effectively removing low frequency components error. To further improve pe...

In this paper, we propose a new finite element approach to simulate the time-dependent Ginzburg-Landau equations under temporal gauge, and design an efficient preconditioner for Newton iteration of resulting discrete system. The solves magnetic potential in H(curl) space by lowest order second kind Nédélec element. This offers simple way deal with boundary condition, leads stable reliable perfo...

In this paper, we describe a general theory of local preconditioning which assigns a unique preconditioner for any set of two-dimensional hyperbolic partial differential equations. The preconditioner is optimal in the sense that it gives the lowest possible condition number (the ratio of maximum to minimum wave speeds). Taking the two-dimensional magnetohydrodynamic (MHD) equations as an exampl...

We present a communication avoiding ILU0 preconditioner for solving large systems of linear equations by using iterative Krylov subspace methods. Recent research has focused on communication avoiding Krylov subspace methods based on so called s-step methods. However there is no communication avoiding preconditioner yet, and this represents a serious limitation of these methods. Our precondition...