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Iterative schemes for high order compact discretizations to the exterior Helmholtz equation∗

Yogi Erlangga, Eli Turkel (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider high order finite difference approximations to the Helmholtz equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear equations with a Krylov subspace iterative method. Since the method converges...

Mixed precision GMRES-based iterative refinement with recycling

Oktay, Eda, Carson, Erin (2023)

Programs and Algorithms of Numerical Mathematics

With the emergence of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes for solving linear systems A x = b have recently been developed. However, in certain settings, GMRES may require too many iterations per refinement step, making it potentially more expensive than the alternative of recomputing the LU factors in a higher precision. In this work, we incorporate the idea of Krylov subspace recycling, a well-known technique for reusing information across sequential invocations,...

Numerical study of two-level additive Schwarz preconditioner for discontinuous Galerkin method solving elliptic problems

Hammerbauer, Tomáš, Dolejší, Vít (2025)

Programs and Algorithms of Numerical Mathematics

The paper deals with the analysis and numerical study of the domain decomposition based preconditioner for algebraic systems arising from the discontinuous Galerkin (DG) discretization of the linear elliptic problems. We introduce the DG discretization of the model problem and present the spectral h p -bound of the corresponding linear algebraic systems. Moreover, we present the two-level additive Schwarz preconditioner together with the theoretical result related to the estimate of the condition number....

On adaptive BDDC for the flow in heterogeneous porous media

Bedřich Sousedík (2019)

Applications of Mathematics

We study a method based on Balancing Domain Decomposition by Constraints (BDDC) for numerical solution of a single-phase flow in heterogeneous porous media. The method solves for both flux and pressure variables. The fluxes are resolved in three steps: the coarse solve is followed by subdomain solves and last we look for a divergence-free flux correction and pressures using conjugate gradients with the BDDC preconditioner. Our main contribution is an application of the adaptive algorithm for selection...

On finite element approximation of fluid structure interaction by Taylor-Hood and Scott-Vogelius elements

Vacek, Karel, Sváček, Petr (2023)

Programs and Algorithms of Numerical Mathematics

This paper focuses on mathematical modeling and finite element simulation of fluid-structure interaction problems. A simplified problem of two-dimensional incompressible fluid flow interacting with a rigid structure, whose motion is described with one degree of freedom, is considered. The problem is mathematically described and numerically approximated using the finite element method. Two possibilities, namely Taylor-Hood and Scott-Vogelius elements are presented and implemented. Finally, numerical...

On fluid structure interaction problems of the heated cylinder approximated by the finite element method

Vacek, Karel, Sváček, Petr (2025)

Programs and Algorithms of Numerical Mathematics

This study addresses the problem of the flow around circular cylinders with mixed convection. The focus is on suppressing the vortex-induced vibration (VIV) of the cylinder through heating. The problem is mathematically described using the arbitrary Lagrangian-Eulerian (ALE) method and Boussinesq approximation for simulating fluid flow and heat transfer. The fluid flow is modeled via incompressible Navier-Stokes equations in the ALE formulation with source term, which represent the density variation...

On the parameter in augmented Lagrangian preconditioning for isogeometric discretizations of the Navier-Stokes equations

Jiří Egermaier, Hana Horníková (2022)

Applications of Mathematics

In this paper, we deal with the optimal choice of the parameter γ for augmented Lagrangian preconditioning of GMRES method for efficient solution of linear systems obtained from discretization of the incompressible Navier-Stokes equations. We consider discretization of the equations using the B-spline based isogeometric analysis approach. We are interested in the dependence of the convergence on the parameter γ for various problem parameters (Reynolds number, mesh refinement) and especially for...

Operator preconditioning with efficient applications for nonlinear elliptic problems

Janos Karátson (2012)

Open Mathematics

This paper is devoted to the numerical solution of nonlinear elliptic partial differential equations. Such problems describe various phenomena in science. An approach that exploits Hilbert space theory in the numerical study of elliptic PDEs is the idea of preconditioning operators. In this survey paper we briefly summarize the main lines of this theory with various applications.

Parallel solution of elasticity problems using overlapping aggregations

Roman Kohut (2018)

Applications of Mathematics

The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements...

Preconditioning of two-by-two block matrix systems with square matrix blocks, with applications

Owe Axelsson (2017)

Applications of Mathematics

Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution...

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev, Juan Galvis, Raytcho Lazarov, Joerg Willems (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations. The constant in the corresponding abstract...

Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms

Yalchin Efendiev, Juan Galvis, Raytcho Lazarov, Joerg Willems (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes’ and Brinkman’s equations....

Robust operator estimates and the application to substructuring methods for first-order systems

Christian Wieners, Barbara Wohlmuth (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz et al., SIAM J. Numer. Anal. 49 (2011) 1788–1809; Zitelli et al., J. Comput. Phys. 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space....

Robust preconditioners for the matrix free truncated Newton method

Lukšan, Ladislav, Matonoha, Ctirad, Vlček, Jan (2010)

Programs and Algorithms of Numerical Mathematics

New positive definite preconditioners for the matrix free truncated Newton method are given. Corresponding algorithms are described in detail. Results of numerical experiments that confirm the efficiency and robustness of the preconditioned truncated Newton method are reported.

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

Maria Lymbery, Svetozar Margenov (2012)

Open Mathematics

While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study...

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